The article introduces the reader to the new thermodynamic potential of classical thermodynamics, called the J-potential. For the case of a homogeneous system, it vanishes, but it perfectly serves multiphase (heterogeneous) systems. The J-potential is especially important for the thermodynamics of capillary systems in colloidal science. The J-potential simplifies the derivation of the Neumann and Young equations, and has found application in the thermodynamics of thin films and the theory of strength of materials. In this work, for example, it was used to study the dependence of the chemical potential of a substance of a small bubble on the bubble size.

This paper presents a numerical method for solving a two-dimensional subdiffusion equation with a Caputo fractional derivative. The problem considered assumes symmetry in both the equation’s solution domain and the boundary conditions, allowing for a reduction of the two-dimensional equation to a one-dimensional one. The proposed method is an extension of the fractional Crank-Nicolson method, based on the discretization of the equivalent integral-differential equation. To validate the method, the obtained results were compared with a solution obtained through Laplace transform. The analytical solution in the image of the Laplace transform was inverted using the Gaver-Wynn-Rho algorithm implemented in the specialized mathematical computing environment, Wolfram Mathematica. The results clearly show the mutual convergence of the solutions obtained by the two methods.

Non-standard Lagrangians do not display any discernible energy-like terms, yet they give the same equations of motion as standard Lagrangians, which have easily identifiable energy-like terms. A new method to derive non-standard Lagrangians for second-order nonlinear differential equations with damping is developed and the limitations of this method are explored. It is shown that the limitations do not exist only for those nonlinear dynamical systems that can be converted into linear ones. The obtained results are applied to selected population dynamics models for which non-standard Lagrangians and their corresponding null Lagrangians and gauge functions are derived, and their roles in the population dynamics are discussed.

When 2−1/Q<p≤2, we establish the C^{0,1} and C^{1,α}-regularities of weak solutions to quasi-linear p-Laplacian type non-homogeneous equations in the Heisenberg group.

This short paper demonstrates that the galactic spiral equations developed from ROTASE model can be used to morphologically simulate the phyllotactic spiral-ring patterns of plants. It is also noticed during this project that although the numbers in the Fibonacci sequence dominate the phyllotactic numbers, there are still significant phyllotactic numbers are not in the Fibonacci sequence, however, most of those numbers are well fit into a Fibonacci sibling sequence starting with numbers 0 and 2. Many plants show the phyllotactic number pairs of 10-16, 16-26 and 26-42 from the Fibonacci sibling sequence, which is the second dominated sequence in the phyllotaxis.

In this paper, the sequential prime numbers are used as variables for the galactic spiral equations which were developed from the ROTASE model. Special spiral patterns are produced when prime numbers are treated with the unit of radian. The special spiral patterns produced with the first 1000 prime numbers have 20 spirals arranged in two groups. The two groups have perfect central symmetry with each other and are separated with two spiral gaps. The special spiral pattern produced with natural numbers from 1 to 7919 shows 6 spirals in the central area and 44 spirals in the outer area. The whole 7919 spiral points can be plotted with either 6-spiral pattern or 44-spiral pattern. For the spirals only produced by the prime numbers in the 6-spiral pattern plotting, the spiral 2 and spiral 3 each has only one spiral point produced by prime number 2 and 3, respectively, all other spiral points produced by other prime numbers are located on the spiral 1 and spiral 5. The special spiral pattern is well explained with careful analysis, it is concluded that all prime numbers greater than 3 must meet one of the equations: P₁ = 1 + 6 * n (n > 0) P₅ = 5 + 6 * n (n ≥ 0) In other words, every prime number greater than 3 is either a P₁ prime number or a P₅ prime number, no exception. Matching one of the equations is a necessary condition for a number to be a prime number, not a sufficient condition. Hope such sufficient condition can be found in the future. The number of P₁ prime numbers roughly equal the number of P₅ prime number in the first 2 billion prime numbers. The galactic spiral equations with golden angle can duplicate Vogel’s result for the simulation of sunflower seed head pattern, and a pinwheel pattern can be produced also with galactic spiral equations and 1 degree more than golden angle.

A thermodynamic description of nano-porous media must handle the size- and shape-dependence of the media properties. Such dependencies are typically due to the presence of immiscible phases, contact areas and contact lines. We propose a way to obtain average densities suitable for integration on the course grained scale, applying Hill's thermodynamics for small systems to the subsystems. we argue that the average densities of the porous medium, when defined in a proper way, obey the Gibbs equation. All contributions are additive or weakly coupled. From the Gibbs equation and the balance equations, we derive the entropy production in the standard way, for transport of multi-phase fluids in a non-deformable, porous medium exposed to di¤erences in boundary pressures, temperatures, and chemical potentials. Linear relations between thermodynamic fluxes and forces follow for the control volume. Fluctuation- dissipation theorems are formulated for the first time, for the fluctuating contributions to fluxes in the porous medium. These give an added possibility for determination of porous media permeabilities. Practical possibilities are further discussed.

This paper presents a new analytical method for obtaining new deformations compatibility equations or, new Saint-Venant’s identities, into the relative coordinate system with projected deformations by applying the hypothesis of the lineal shell theory in general flexion state. The method proposed generalizes the compatibility conditions established by A.L. Goldenveizer for the shell theory. On the other hand, the new equations include the deformations compatibility equations by other authors: Flügge, Saint-Venant, Love-Kirchhoff, Timoshenko, Goldenveizer, and Reissner-Mindlin. The results showed an increase of knowledge in general shell theory, and provide inverse and semi-inverse solutions, whose systems solution correspond to the hyper-statics degrees of the physical model, and not to their degrees of freedom.

In this paper, we study the global asymptotically stability of following system of difference equations with quadratic terms: x_{n+1}=A+B((y_{n})/(y_{n-1}²)),y_{n+1}=A+B((x_{n})/(x_{n-1}²)) where A and B are positive numbers and the initial values are positive numbers. We also investigate the rate of convergence and oscillation behaviour of the solutions of related system.

For the solution existence condition of the Navier-Stokes equation, we propose a conjecture as follows: "\emph{The Navier-Stokes equation has a solution if and only if the determinant of flow velocity gradient is not zero, namely $\det (\bm \nabla \bm v)\neq 0$.}"

In this study we investigate the formation of dynamical energy cascades in higher order KdV-type equations. In the beginning we recall what is known about the dynamic cascades for the classical KdV (quadratic) and mKdV (cubic) equations. Then, we investigate further the mKdV case by considering a richer set of initial perturbations in order to check the validity and persistence of various facts previously established for the narrow-banded perturbations. Afterwards we focus on higher order nonlinearities (quartic and quintic) which are found to be quite different in many respects from the mKdV equation. Throughout this study we consider both the direct and double energy cascades. It was found that the dynamic cascade is always formed, but its formation is not necessarily accompanied by the nonlinear stage of the modulational instability. Direct cascade structure remains invariant regardless the size of the spectral domain. In contrast, the double cascade shape can depend on the size of the spectral domain, even if the total number of cascading modes remains invariant. Results obtained in this study can be potentially applied to plasmas, free surface and internal wave hydrodynamics.

The famous Toms effect (1948) consists of a substantial increase of the critical Reynolds number when a small amount of soluble polymer is introduced into water. The most noticeable influence of polymer additives is manifested in the boundary layer near solid surfaces. The goal of the present paper is a group analysis of the boundary layer equations in two mathematical models of the flow of aqueous polymer solutions: the second grade fluid (Rivlin and Ericksen, 1955) and the model derived by Pavlovskii (1971). The equations of the unsteady two-dimensional boundary layer in the Pavlovskii and Rivlin-Ericksen models are analyzed for the first time here. These equations have no definite type so that finding their exact solutions is very important in order to understand the mathematical nature of the above mentioned models. The problem of group classification with respect to the arbitrary function of the longitudinal coordinate and time present in the equations, which sets the pressure gradient of the external flow, arises. All functions for which an extension of the admitted Lie group occurs are found. The task includes the ratio of two characteristic length scales. One of them is the Prandtl scale, and another is defined as the square root of the normalized coefficient of relaxation viscosity (Frolovskaya and Pukhnachev, 2018) and does not depend on the characteristics of the motion. The paper contains a number of exact solutions in the Pavlovskii model including a solution describing the flow near a critical point. Among the solutions of the new model of the boundary layer, a special place is taken by the solution of the stationary problem of flow around a rectilinear plate. Within the framework of the Prandtl theory of the boundary layer, such a solution was constructed by Blasius (1908). As is well-known, this solution has a non-removable defect: the transverse velocity near the edge of the plate increases without bound. The introduction of a relaxation term into the model makes it possible to eliminate this singularity.

In this study, we investigate the formation of dynamical energy cascades in higher-order KdV-type equations. In the beginning, we recall what is known about the dynamic cascades for the classical KdV (quadratic) and mKdV (cubic) equations. Then, we investigate further the mKdV case by considering a richer set of initial perturbations in order to check the validity and persistence of various facts previously established for the narrow-banded perturbations. Afterwards, we focus on higher-order nonlinearities (quartic and quintic) which are found to be quite different in many respects from the mKdV equation. Throughout this study, we consider both the direct and double energy cascades. It was found that the dynamic cascade is always formed, but its formation is not necessarily accompanied by the nonlinear stage of the modulational instability. Direct cascade structure remains invariant regardless of the size of the spectral domain. In contrast, the double cascade shape can depend on the size of the spectral domain, even if the total number of cascading modes remains invariant. Results obtained in this study can be potentially applied to plasmas, free surface and internal wave hydrodynamics.

This article is devoted to Feller's diffusion equation which arises naturally in probabilities and physics (e.g. wave turbulence theory). If discretized naively, this equation may represent serious numerical difficulties since the diffusion coefficient is practically unbounded and most of its solutions are weakly divergent at the origin. In order to overcome these difficulties we reformulate this equation using some ideas from the Lagrangian fluid mechanics. This allows us to obtain a numerical scheme with a rather generous stability condition. Finally, the algorithm admits an elegant implementation and the corresponding Matlab code is provided with this article under an open source license.

Free vibration analysis of the porous functionally graded circular plates has been presented on the basis of classical plate theory. The three defined coupled equations of motion of the porous functionally graded circular/annular plate were decoupled to one differential equation of free transverse vibrations of plate. The one universal general solution was obtained as linear combination of the multiparametric special functions for the functionally graded circular and annular plates with even and uneven porosity distributions. The multiparametric frequency equations of functionally graded porous circular plate with diverse boundary conditions were obtained in the exact closed-form. The influences of the even and uneven distributions of porosity, power-law index, diverse boundary conditions and the negligibled effect of the coupling in-plane and transverse displacements on the dimensionless frequencies of the circular plate were comprehensively studied for the first time. The formulated boundary value problem, the exact method of solution and the numerical results for the perfect and imperfect functionally graded circular plates have not yet been reported.

Various sudden shifts in financial market conditions over the past decades have demonstrated the significant impact of market structural breaks on firms' credit behavior. To characterize such effect quantitatively, we develop a continuous-time modulated Markov model for firms' credit rating transitions with the possibility of market structural breaks. The model takes a semi-parametric multiplicative regression form, in which the effects of firms' observable covariates and macroeconomic variables are represented parametrically and nonparametrically, respectively, and the frailty effects of unobserved firm-specific and market-wide variables are incorporated via the integration form of the model assumption. We further develop a mixtured-estimating-equation approach to make inference on the effect of market variations, baseline intensities of all firms' credit rating transitions, and rating transition intensities for each individual firm. We then use the developed model and inference procedure to analyze the monthly credit rating of U.S. firms from January 1986 to December 2012, and study the effect of market structural breaks on firms' credit rating transitions.

In this paper we study the class of mixed-index time fractional differential equations in which different components of the problem have different time fractional derivatives on the left hand side. We prove a theorem on the solution of the linear system of equations, which collapses to the well-known Mittag-Leffler solution in the case the indices are the same, and also generalises the solution of the so-called linear sequential class of time fractional problems. We also investigate the asymptotic stability properties of this class of problems using Laplace transforms and show how Laplace transforms can be used to write solutions as linear combinations of generalisedMittag-Leffler functions in some cases. Finally we illustrate our results with some numerical simulations.

In this paper, we study the semilinear pseudo-parabolic equations $\displaystyle u_{t} - \triangle_{\mathbb{B}}u - \triangle_{\mathbb{B}}u_{t} = \left|u\right|^{p-1}u$ on a manifold with conical singularity, where $\triangle _{\mathbb{B}}$ is Fuchsian type Laplace operator investigated with totally characteristic degeneracy on the boundary $x_{1} = 0$. Firstly, we discuss the invariant sets and the vacuum isolating behavior of solutions with the help of a family of potential wells. Then, we derive a threshold result of existence and nonexistence of global weak solution: for the low initial energy $J(u_{0})<d$, the solution is global in time with $I(u_{0}) >0$ or $\displaystyle\Vert \nabla_{\mathbb{B}}u_{0}\Vert_{L_{2}^{\frac{n}{2}}(\mathbb{B})} = 0$ and blows up in finite time with $I(u_{0}) < 0$; for the critical initial energy $J(u_{0}) = d$, the solution is global in time with $I(u_{0}) \geq0$ and blows up in finite time with $I(u_{0}) < 0$. The decay estimate of the energy functional for the global solution and the estimates of the lifespan of local solution are also given.

We primarily investigate the existence of solutions for fractional neutral integro-differential equations subjected to Neumann-type boundary conditions, which is crucial for understanding natural phenomena. Taking into account factors such as neutral type, fractional-order integrals, and fractional-order derivatives, we employ probability density functions, Laplace transforms, and resolvent operators to formulate a well-defined concept of a mild solution for the specified equation. Following this, by integrating fixed point theorems, we establish the existence of mild solutions under more relaxed conditions.

Our starting point is the covariant coordinate transformation equation of a relativistic positioning system in Minkowski space-time, which maps the receiver’s emission coordinates (proper times broadcast by the emitters) to its coordinates in some inertial reference frame. Bancroft’s analytical (closed-form) solution to the basic pseudorange navigation equations with four emitters is recovered and the subjacent elements are geometrically interpreted. The case of four static beacons is analized as a clarifying situation.

We consider the second order differential equation y'' + ω^(2)ρ(x)y = 0 where ω is a positive parameter. The principal concern here is to find conditions on the function ρ^(−1/2)(x), which ensure that the consecutive differences of sequences constructed from the zeros of a nontrivial solution of the equation are regular in sign for ω sufficiently large. In particular, if c_(νk)(α) denotes the k-th positive zero of the general Bessel (cylinder) function C_(ν)(x; α) = J_(ν)(x) cos α−Y_(ν)(x) sin α of order ν, and if |ν| < 1/2, we prove that (−1)^(m)∆^(m+2)c_(νk)(α) > 0 (m = 0, 1, 2, ...; k = 1, 2, ...), where ∆a_(k) = a_(k+1) − a_(k). This type of inequalities was conjectured by Lorch and Szego in 1963. We also show that the differences of the zeros of various orthogonal polynomials with higher degrees possess the sign-regularity.

Given any sequence a = (an)n1 of positive real numbers and any set E of complex sequences, we write Ea for the set of all sequences y = (yn)n1 such that y=a = (yn=an)n1 2 E. In this paper, we use the spaces w1, w0 and w of strongly bounded, summable to zero and summable sequences, that are the sets of all sequences y such that ? n?1Pn k=1 jykj
n is bounded, tends to zero and such that y ? le 2 w0, for some scalar l, respectively, (cf. [24, 22]). These sets where used in the statistical convergence, (cf. [17, Chapter 4]). Then we deal with the solvability of each of the (SSIE) F
 E + F0x where E is a linear space of sequences, F = c0, c, `1, w0, w, or w1 and F0 = c0, c, or `1. For instance, the solvability of the (SSIE) w
 w0+s(c) x consists in determining the set of all sequences x = (xn)n1 2 U+ that satisfy the following statement. For every sequence y that satisfy the condition limn!1 n?1Pn k=1 jyk ? yk?1 ? lj = 0, there are two sequences u and v, with y = u+v such that limn!1 n?1Pn k=1 jukj = 0 and limn!1 (vn=xn) = L for some scalars l and L. These results extend those stated in [11, 12, 10].

This paper is related to the fractional view analysis of Helmholtz equations, using innovative analytical techniques. The fractional analysis of the proposed problems has been done in terms of Caputo-operator sense. In the current methodology, first, we applied the r-Laplace transform to the targeted problem. The iterative method is then implemented to obtain the series form solution. After using the inverse transform of the r-Laplace, the desire analytical solution is achieved. The suggested procedure is verified through specific examples of the fractional Helmholtz equations. The present method is found to be an effective technique having a closed resemblance with the actual solutions. The proposed technique has less computational cost and a higher rate of convergence. The suggested methods are therefore very useful to solve other systems of fractional order problems.

When solving the Einstein's equations for an isolated system of masses, V. Fock introduces harmonic reference frame and obtains an unambiguous solution. Further, he concludes that there exists a harmonic reference frame which is determined uniquely apart from a Lorentz transformation if suitable supplementary conditions are imposed. It is known that wave equations keep the same form under Lorentz transformations. Thus, we speculate that Fock's special harmonic reference frames may have provided us a clue to derive the Einstein's equations in some special class of non-inertial reference frames. Following this clue, generalized Einstein's equations in some special non-inertial reference frames are derived based on the theory of vacuum mechanics. If the field is weak and the reference frame is quasi-inertial, these generalized Einstein's equations reduce to Einstein's equations. Thus, this theory may also explain all the experiments which support the theory of general relativity. There exist some differences between this theory and the theory of general relativity.

Thermodynamics has been the foundation of many models of biological and technological systems. But thermodynamics is static and is misnamed. A more suitable name is thermostatics. Thermostatics does not include time as a variable and so has no velocity, flow or friction. Indeed, as usually formulated, thermostatics does not include boundary conditions. Devices require boundary conditions to define their input and output. They usually involve flow and friction. Thermostatics is an unsuitable foundation for understanding technological and biological devices. A time dependent generalization of thermostatics that might be called thermal dynamics is being developed by Chun Liu and collaborators to avoid these limitations. Electrodynamics is not restricted like thermostatics, but in its classical formulation involves drastic assumptions about polarization and an over-approximated dielectric constant. Once the Maxwell equations are rewritten without a dielectric constant, they are universal and exact. Conservation of total current, including displacement current, is a restatement of the Maxwell equations that leads to dramatic simplifications in the understanding of one dimensional systems, particularly those without branches, like the ion channel proteins of biological membranes and the two terminal devices of electronic systems. The Brownian fluctuations of concentrations and fluxes of ions become the spatially independent total current, because the displacement current acts as an unavoidable low pass filter, a consequence of the Maxwell equations for any material polarization. Electrodynamics and thermal dynamics together form a suitable foundation for models of technological and biological systems.

In this study, the hot deformation of Cu-0.55Sn-0.08La (wt%) alloy was studied by gleeble-3180 testing machine at deformation temperatures of 400-700oC and various strain rates. The stress-strain curve showed that the hot deformation behavior of Cu-0.55Sn-0.08La (wt%) alloy was greatly affected by work hardening, dynamic recovery, and dynamic recrystallization. The activation energy Q was calculated as 287.141 kJ/mol , and hot compression constitutive equation was determined as ε ̇=[sinh(0.006388σ)]^12.123∙exp⁡(37.7505-287.141/RT). The microstructural evolution of the alloy during deformation at 400oC revealed the presence of both slip and shear bands in the grains. At 700oC, dynamic recrystallization grains were observed but recrystallization was incomplete. In sum, these results provided the theoretical basis for continuous extrusion process of alloys with promising future use.

In this paper, we compare the performance between systems of ordinary and (Caputo) fractional differential equations depicting the susceptible-exposed-infectious-recovered (SEIR) models of diseases. In order to understand the origins of both approaches as mean-field approximations of integer and fractional stochastic processes, we introduce the fractional differential equations as approximations of some type of fractional nonlinear birth--death processes. Then, we examine validity of the two approaches against empirical courses of epidemics; we fit both of them to case counts of three measles epidemics that occurred during the pre-vaccination era in three different locations. While FDEs appear more flexible in fitting empirical data, our ODEs offered better fits to two out of three data sets. Important differences in transient dynamics between these modeling approaches are discussed.

The present paper discusses a study of a class of Charlier matrix polynomials and its generalized analogue. Certain generating matrix functions, recurrence matrix relations, matrix differential equation, summation formulas and many new results have been discussed for these matrix polynomials. Weisner's group theoretic method is used to obtain matrix generating relations for Charlier matrix polynomials and the details of this method were given in this paper. Finally, we will discuss only briefly the procedure followed.

This paper deals with the form, the stability character, the periodicity and the global behavior of solutions of the following four rational difference equations x_{n+1} = ((±1)/(x_{n}(x_{n-1}±1)-1)) x_{n+1} = ((±1)/(x_{n}(x_{n-1}∓1)+1)).

Dietary patterns established during adolescence might play a role in adulthood disease. We examined the stability of dietary patterns (DPs) from childhood through adolescence and into young adulthood (from age 8 to 34 years). Data from 130 participants (53 females) of Saskatchewan Pediatric Bone Mineral Accrual Study (aged 8-15 years, at baseline) were included. Multiple 24-hour recalls were collected annually from 1991 to 1997, 2002 to 2005 and 2010 and 2011. Using principal component analysis, “Vegetarian-style”, “Western-like”, “High-fat, high-protein”, “Mixed” and “Snack” DPs were derived at baseline. Applied DP scores for all annual measurements were calculated using factor loading of baseline DPs and energy-adjusted food group intakes. We analyzed data using generalized estimating equations. The tracking coefficient represents correlation between baseline dietary pattern scores and all other follow-up dietary pattern scores. We found a moderate tracking for the “Vegetarian-style” (ß=0.44, P<0.001) and “High-fat, high-protein” (ß=0.39, P<0.001) DPs in females and “Vegetarian-style” DP (ß=0.30, P<0.001) in males. Remaining DPs showed a poor-to-fair tracking in both sexes. No tracking for “Western-like” DP in females was observed. Assessing overall change in DP scores from childhood to young adulthood showed an increasing trend in adherence to “Vegetarian-style” DP and decreasing trend in adherence to “High-fat, high-protein” DP by age in both sexes (P<0.001). While “Western-like” and “Mixed” DP scores increased only in males (P<0.001). These findings suggest that healthy dietary habits established during childhood and adolescence moderately continue into adulthood.

The study of different forms of preconditioners for solving a system of nonlinear equations, by using Newton’s method, is presented. The preconditioners provide numerical stability and rapid convergence with reasonable computation cost, whenever chosen accurately. Different families of iterative methods can be constructed by using a different kind of preconditioners. The multi-step iterative method consists of a base method and multi-step part. The convergence order of base method is quadratic and each multi-step add an additive factor of one in the previously achieved convergence order. Hence the convergence of order of an m-step iterative method is m + 1. Numerical simulations confirm the claimed convergence order by calculating the computational order of convergence. Finally, the numerical results clearly show the benefit of preconditioning for solving system of nonlinear equations.

A cosmological model described by the perturbation metric is proposed, in which space is entrained by matter expanding as a result of the Big Bang. In the absence of an input of external energy and momentum, the solution of the Einstein equations assumes a pressure equal to zero and an accelerated expansion. The cosmological perturbation model explains the difference in measurements of the Hubble constant using gravitational lensing and a distance ladder.

Based on recent progress in cosmological thermodynamics, we present a new thermody- namic formulation of the Friedmann equations in the new Haug-Tatum cosmology model, as well as for other similar models put forward in the literature from different solutions to Einstein’s field equation. Since the CMB temperature has been measured much more accu- rately than the Hubble constant, we demonstrate that our thermodynamic formulation of the Friedmann equation dramatically narrows the acceptable range for the critical density of the universe. Furthermore, we show how our thermodynamic formulation leads to a more precise cosmological constant estimate.

Polymer electrolyte was used as a medium for testing the performance of microband electrodes under conditions of linear diffusion. Cyclic voltammetry and chronoamperometry experiments were performed in such viscous medium, where diffusion rates are much slower than in fluid solutions. Confirmation of the existence of such conditions from the log i vs. log v or log i vs. log t relationships in the current equation always yielded values lower than the expected value of 0.5. This could indicate impure linear diffusion profile, i. e. some contribution from radial diffusion (edge effect). However, performing these tests in monomeric solvents of similar viscosities, such as glycerol or propylene glycol, gave the desired value of 0.5. These results lead to the conclusion that current equations which are based on Fick’s laws may not be applicable for polymer electrolytes, where various obstruction to diffusion make it a more complicated process than in monomeric solvents.

This study aims to reveal the revolutionary Gamma Distribution Manifold (GDM) info-geometric analysis combined with potential Gamma distribution (GD) applications to advance Satellite Earth Observations (SEOs). An exposition of the GDM Fisher Information Matrix (FIM) was undertaken, together with the analytic investigation of the existence of FIM’s inverse. The derivation of GDM’s Geodesic Equations (GEs) is undertaken. The solution of the obtained Geodesic Equations is optimally complicated, and it provides more extensions to numerous open problems. Notably, the current paper provides a first-time unification between three multi-interdisciplinary fields, namely, information geometry theory, probability theory, and Satellite earth observations. Looking at the other side of the coin, the provided revolutionary approach in this current work reports the significant impact of info-geometric analysis on the probabilistic dynamics as well as the influential role of information geometry on satellite and space industry. This poses numerous open issues that initiates a new form of contemporary unification between these inter-disciplinaries to discover another visionary thinking out of traditional statistical frames. In other words, refraining ourselves from using classical statistical frames of thinking and delving into a new info-geometric vision of statistical distributions as well as examining real life applications of information geometry to advance satellite earth observations. The curtain is drawn on this paper by providing several challenging open problems in combination with a concluding remark and the next phase of research.

To accurately predict the matching relationships between the various components and the engine performance in the whole aero-engine environment, this study introduces a two-dimensional throughflow simulation method for the whole aero-engine. This method is based on individual throughflow solvers for the turbo-machinery and the combustor. It establishes a throughflow simulation model for the whole engine by integrating with the compressor-turbine co-operating equations and boundary conditions. The turbo-machinery throughflow solver employs a circumferentially averaged form of the time-dependent Navier-Stokes equations (N-S) as the governing equation. The combustor solver uses the Reynolds Average Navier-Stokes (RANS) method to solve flow and chemical reaction processes by constructing turbulence, combustion, and radiation models. The accuracy of the component solver is validated using Pratt &amp; Whitney’s three-stage axial compressor (P&amp;W3S1) and General Electric’s high-pressure turbine (GE-EEE HPT), and the predicted results are consistent with the experimental data. Finally, the developed throughflow method is applied to simulate the throttling characteristics of the WZ-X turboshaft engine. The results predicted by the throughflow program are consistent with the GasTurb calculations, including the trends of shaft power delivered, specific fuel consumption (SFC), inlet airflow, and total pressure ratio of the compressor. The developed method to perform throughflow simulation of the whole aero-engine eliminates the dependence on a general component map. It can quickly obtain the meridian flow field parameters and overall engine characteristics, which is expected to guide the design and modification of the engine in the future.

This paper provides the numerical investigation of two-dimensional stochastic Navier-Stokes equation driven by additive white noises. The vorticity-stream function method is presented to calculate the incompressible fluid flow. In order to solve the formulation of steam function equation, we propose a Crank-Nicolson Fourier pseudo-spectral method. Numerical results obviously show that the presence of stochastic forcing can have an impact upon the shapes of vortex in fluid flow. To a certain extent, the phenomenon of eddy shedding and the formation of some new vortex in a random medium can be observed, which make it possible for us to conduct optimal control experiments in the development of vortex structures.

This is the second paper in a cycle investigating the exact solution of loop equations in decaying turbulence. We perform numerical simulations of the Euler ensemble, suggested in the previous work, as a solution to the loop equations. We designed novel algorithms for simulation, which take a small amount of computer RAM so that only the CPU time grows linearly with the size of the system and its statistics. This algorithm allows us to simulate systems with billions of discontinuity points on our fractal curve, dual to the decaying turbulence. For the vorticity correlation function, we obtain \textbf{quantum fractal laws} with regime changes between universal discrete values of indexes. The traditional description of turbulence with fractal or multifractal scaling laws does not apply here. The measured conditional probabilities of fluctuating variables are smooth functions of the logarithm of scale, with statistical errors being negligible at our large number of random samples $T = 167,772,160$ with $N = 200,000,000$ points on a fractal curve. The quantum jumps arise from the analytical distribution of scaling variable $X = \frac{\cot^2(\pi p/q)}{q^2}$ where $\frac{p}{q}$ is a random fraction. This distribution is related to the totient summatory function and is a discontinuous function of $X$. In particular, the energy spectrum is a devil's staircase with tilted uneven steps with slopes between $-1$ and $-2.03$. The logarithmic derivative of energy decay $n(t)$ as a function of time is jumping down the stairs of universal levels between $1.50$ and $1.53$. The quantitative verification of this quantization would require more precise experimental data.

Exercising surds to represent figures is a common practice in scientific and Engineering fields, especially in scripts where calculators are banned or unapproachable. Peter Chew Theorem make result becomes simple when dealing with converting Quadratic Surds. The substance of the Peter Chew Theorem lies in enabling the forthcoming generation to simple break problems related to Quadratic Surds more effectively, easing a direct comparison with contemporary results. By employing the Peter Chew Theorem, one can streamline the tutoring and literacy of math, particularly concerning second- order direct equations with constant portions. This theorem's objective aligns with Albert Einstein's famed quotation Everything should be made as simple as possible, but not simpler.

Partial differential equations (PDEs) are used to describe a wide range of phenomena, such as heat transfer, fluid dynamics, and quantum mechanics. By solving PDEs, we can ob- tain insights into the behavior of the system and make predictions about its future evolution. Conventional numerical methods for obtaining the approximate solutions of PDEs may re- quire extensive computational resources and time, especially for complex PDEs and large- scale problems. The recently developed physics-informed neural network (PINN) has shown promise in a variety of scientific and engineering fields by incorporating physical laws into the loss functions of the neural network (NN). In addition, the training of PINN does not require ground truth data, but it has poor generalization ability to unseen domains. On the other hand, a convolutional neural network (CNN) has fast inference and better generalization ability, but it requires a large amount of training data. Taking the advantages of PINN and CNN by using Legendre multiwavelets (LMWs) as basis functions, we introduce a new method to approach the PDEs in this paper, namely Physics-Informed Legendre Multiwavelets CNN (PiLMWs-CNN), in order to continuously approximate a grid-based state representation that can be handled by a CNN. PiLMWs-CNN enable us to train our models using only physics-informed loss functions without any pre- computed training data, simultaneously providing fast and continuous solutions that gener- alize to previously unknown domains. In particular, the LMWs can simultaneously possess compact support, orthogonality, symmetry, high smoothness, and high approximation order. Compared to orthonormal polynomial (OP) bases, the approximation accuracy can be greatly increased and computation costs can be significantly reduced by using LMWs. We applied PiLMWs-CNN to approximate the damped wave equation, incompressible Navier-Stokes (N- S) equation, and two-dimensional heat conduction equation. The experimental results show that this method provides more accurate, efficient, and fast convergence with better stability when approximating the solution of PDEs.

The fractional diffusion equations with integrals play a significant role in describing anomalous diffusion phenomena. We first construct an appropriate resolvent family, and use the convolution theorem of Laplace transform, the probability density function, Cauchy integral formula and Fubini theorem to study its related equicontinuity, strongly continuity, compactness, etc. Then, a reasonable mild solution is defined. Finally, by combining some fixed point theorems, we obtain some conclusions on the existence and uniqueness of mild solutions.

In this paper, we introduce as a pedagogical strategy an internal division-free, straightforward, and symmetrically progressing algorithm in manually computing matrix inverse and solving systems of linear equations by revisiting the application of elementary row operations in the Gauss-Jordan reduction method and connecting it to the determinant method. The proposed cross-multiplication-flip (CMF) algorithm employs cross-multiplication similar to the butterfly movement in computing determinants as a strategic application of elementary row operations to efficiently reduce the rows and then applies flipping of rows and entries to put an upper triangular matrix into lower triangular form to continue the reduction process.

The correspondence between classical and quaternion quantum equations, permits considering the universe (vacuum) as an ideal elastic solid. Elementary particles would have to be standing or soliton-like waves. Tension induced by the compression and twisting of the elastic medium would increase the energy density, consequently generate a gravity forcing and affect the wave speed. Consequently the gravity could be described by an index of refraction. Theory was created by combining the Cauchy model of the elastic continuum with the Planck-Kleinert crystal hypothesis. The quaternion-imaginary Lagrange’an, the quaternion motion equation and the quaternionic oscillator allowed deriving: - The Schrödinger equation from the functional integral, which identifies the quaternion-imaginary quantum Hamiltonian. - The 2nd order wave equation system describing both the bosons and the gravity in terms of quaternionic Poisson equation. - The first order quaternionic wave equation system. - The family of the second order wave equation systems describing both the particles and the generated quaternionic force-fields (four-potentials). - The fundamental Planck and gravity constants. - The quaternionic continuity equation in an ideal elastic solid.

In the last years, a recent growing line of research has been developed with fruitful results by using fractional and conformable derivatives in the iterative procedures of classical methods for solving nonlinear equations. In that sense, the use of conformable derivatives has shown better behavior than fractional ones, not only in the theory, but also in the practice. In this work, we adapt the approximation of conformable derivatives in order to design the first conformable derivative-free iterative schemes to solve nonlinear equations: a Steffensen’s type method and a Secant type method; the latter with memory. Convergence analysis is made, preserving the order of classical cases, and the numerical performance is studied in order to confirm the results in the theory. It is shown that these methods can present some numerical advantages versus their classical partners, with wide sets of converging initial estimations.

In this paper, we report orthogonal fuzzy versions of some celebrated iterative mappings. We provide various concrete conditions on the real valued functions J,S:(0,1]→(−∞,∞) for the existence of fixed-points of (J,S)-fuzzy iterative mappings. We obtain many fixed point theorems in orthogonal fuzzy metric spaces. We apply (J,S)-fuzzy version of Banach fixed point theorem to show the existence and uniqueness of the solution. We support these results with several non-trivial examples and applications to Volterra-type integral equations and fractional differential equations.

The splitting method developed in a previous study for linear, multi – component, convection-diffusion mass transfer equations is extended to non-linear, multi-component, convection-diffusion mass transfer equations. The diffusion term is non-linear (the diffusion coefficients depend on chemical species concentrations). Theoretical results about the splitting error are presented, as well as numerical simulations for mixtures with n ≤ 5. The numerical experiments show second-order accuracy and stability for the present splitting algorithm.

A cloud of relativistic material particles is considered, for which the distortions of length and time caused by the presence of the Lorentz factor an order of magnitude greater than curvature of space-time by gravity. The gravitational mass of the cloud is found in the region where its size is insignificant. It is established that the coefficient of dependence of the cloud mass on the total rest mass of its particles includes a complete elliptic integral of the 2nd kind.

This work emphasizes some geometrical properties of the Maxwell--Bloch equations. Based on these properties the closed-form solutions of their equations are established. Thus, the Maxwell-Bloch equations are reduced to a nonlinear differential equation depending on an auxiliary unknown function. The approximate analytical solutions were built using the Optimal Homotopy Asymptotic Method (OHAM). A good agreement between the analytical and corresponding numerical results was found. The accuracy of the obtained results is validated through the representative figures. This procedure could be successfully applied for more dynamical systems with geometrical properties.

Cell migration is essential for physiological, pathological and biomedical processes such as, in embryogenesis, wound healing, immune response, cancer metastasis, tumour invasion and inflammation. In light of this, quantifying mechanical properties during the process of cell migration is of great interest in experimental sciences, yet few theoretical approaches in this direction have been studied. In this work, we propose a theoretical and computational approach based on the optimal control of geometric partial differential equations to estimate cell membrane forces associated with cell polarisation during migration. Specifically, cell membrane forces are inferred or estimated by fitting a mathematical model to a sequence of images, allowing us to capture dynamics of the cell migration. Our approach offers a robust and potentially accurate framework to compute geometric mechanical membrane forces associated with cell polarisation during migration and also yields geometric information of independent interest, we illustrate one such example that involves quantifying cell proliferation levels which are associated with cell division, cell fusion or cell death.

We suggest progress regarding the following six physics opportunities. List all elementary particles. Describe dark matter. Explain ratios of dark matter to ordinary matter. Explain eras in the history of the universe. Link properties of objects. Interrelate physics models. We use models based on Diophantine equations.

Formulating a relativistic equation for particles with arbitrary spin remains an open challenge in theoretical physics. In this study, the main algebraic approaches used to generalize the Dirac and Kemmer–Duffin equations for particles of arbitrary spin are investigated. It is proved that an irreducible relativistic equation formulated using spin matrices satisfying the commutation relations of the de Sitter group leads to inconsistent results, mainly as a consequence of violation of unitarity and the appearance of a mass spectrum that does not reflect the physical reality of elementary particles. However, the introduction of subsidiary conditions resolves the problem of unitarity and restores the physical meaning of the mass spectrum. The equations obtained by these approaches are solved and the physical nature of the solutions is discussed.

This study suggests a novel cooperative multi-simplex algorithm that generalizes a local search optimizer to design a novel global search heuristic algorithm. The proposed algorithm exploits the vertex sharing strategy to enhance the search abilities of the working simplexes. The vertex sharing among the simplexes is carried out through cooperative step that is based on fitness of the underlying simplex. The proposed algorithm is applied to solve some systems of nonlinear equations by transforming them to optimization problems. Comparative analysis of results shows that the proposed method is practical and effective.

The asymptotic dynamics of the classes of rational difference equations (RDEs) of third order defined over the positive real-line as $$\displaystyle{x_{n+1}=\frac{x_{n}}{ax_n+bx_{n-1}+cx_{n-2}}}, \displaystyle{x_{n+1}=\frac{x_{n-1}}{ax_n+bx_{n-1}+cx_{n-2}}}, \displaystyle{x_{n+1}=\frac{x_{n-2}}{ax_n+bx_{n-1}+cx_{n-2}}}$$ and $$\displaystyle{x_{n+1}=\frac{ax_n+bx_{n-1}+cx_{n-2}}{x_{n}}}, \displaystyle{x_{n+1}=\frac{ax_n+bx_{n-1}+cx_{n-2}}{x_{n-1}}}, \displaystyle{x_{n+1}=\frac{ax_n+bx_{n-1}+cx_{n-2}}{x_{n-2}}}$$ is investigated computationally with theoretical discussions and examples. It is noted that all the parameters $a, b, c$ and the initial values $x_{-2}, x_{-1}$ and $x_0$ are all positive real numbers such that the denominator is always positive. Several periodic solutions with high periods of the RDEs as well as their inter-intra dynamical behaviours are studied.

In this study, we first present a time-fractional L$\hat{e}$vy diffusion equation of the exponential option pricing models of European option pricing and the risk-neutral parameter. Then, we modify a particular L$\hat{e}$vy-time fractional diffusion equation of European-style options. Introduce a more general model from the models based on the L$\hat{e}$vy-time fractional diffusion equation and review some recent findings regarding of the Europe option pricing of risk-neutral free.

Nowadays, all the stakeholders, policy makers, regulators, aircraft designers, producers, operators, etc.) are intensively working on development of the aircraft with full electric and hybrid propulsion systems. However, the technical, technological constrains (like limit on accumulator energy density) require introducing a new approach to conceptual design of such aircraft. The new methods is based on energy and mass balance evaluation. This paper analyses the identified constrains; integrates the energy and mass balance equations into the preliminary definition and calculations of the aircraft performance. By this way, the technological constrains might be transferred into the limitation on the aircraft energy and mass breakdown, that initiates a new approach to aircraft conceptual design uses the knowledge based multidisciplinary optimization. The paper describes the developed methodology for conceptual design of aircraft. It show results of implementing this new development philosophy to conceptual design of a four-seat small electric/hybrid aircraft and a special hybrid cargo UAV. The discussion of the results including got by using the emerging and enabling new technologies and new methods and solutions (including for example distributed propulsion system, unconventional forms, morphing, biomimics, etc.), demonstrates the possible implementation of the new development philosophy, new approach to aircraft conceptual design.

In this study, we investigate Quantum Entanglement Asymmetry (QEA) as a potential explanation for the matter-antimatter imbalance observed in the universe. By integrating quantum mechanics with cosmological phenomena, the study proposes a model in which quantum entanglement among particle-antiparticle pairs in the early universe leads to an asymmetry favoring matter. Through theoretical development and numerical simulations, we explore how this model predicts unique patterns in Cosmic Microwave Background (CMB) anisotropies and the distribution of matter, setting a foundation for empirical verification. The study aims to enhance the understanding of cosmological asymmetries and highlight the significance of quantum entanglement in the universe's evolution.

This paper brings to fruition recently found \cite{migdal2023exact} exact reduction of decaying turbulence in the Navier-Stokes equation in $3 + 1$ dimensions to a Number Theory problem of finding the statistical limit of the Euler ensemble. We reformulate the Euler ensemble as a Markov chain and show the equivalence of this formulation to the \textbf{quantum} statistical theory of $N$ fermions on a ring, with an external field related to the random fractions of $\pi$. We find the solution of this system in the turbulent limit $N\to \infty, \nu \to 0$ in terms of a complex trajectory (instanton) providing a saddle point to the path integral over the density of these fermions. This results in an analytic formula for the observable correlation function of vorticity in wavevector space. This is a full solution of decaying turbulence from the first principle without approximations or fitted parameters. The energy spectrum decays as $k^{-\frac{7}{2}}$. We compute resulting integrals in \Mathematica{} and present effective index $n(t) = -t\partial_t \log E $ for the energy decay as a function of time Fig.\ref{fig::NPlot}. The asymptotic value of the effective index in energy decay $n(\infty) = \frac{5}{4}$, but the universal function $n(t)$ is neither constant nor linear.

New possibilities of Gramian computation by using canonical transformations into diagonal, controllable and observable canonical forms are shown. With the help of such a technique the Gramian matrices can be represented in the form of products of Hadamard matrices of multipliers and matrices of the transformed right-hand side of Lyapunov equations. It is shown that the multiplier matrices are invariant under various canonical transformations of linear continuous systems. The modal Lyapunov equations for continuous SISO LTI systems in diagonal form are obtained and their new solutions based on Hadamard decomposition are proposed. New algorithms for element-by-element computation of Gramian matrices for stable continuous MIMO LTI systems are developed. For continuous SISO LTI systems given by equations of state in controllable and observable canonical forms, new algorithms for the computation of controllability Gramians and their traces in the form of Hadamard products in the form of Xiao matrices are developed. The application of transformations to the canonical forms of controllability and observability allowed to simplify the formulas of spectral decompositions in the form of Xiao matrices. In the paper new spectral decompositions in the form of Hadamard’s products for solutions of Sylvester algebraic and differential equations of MIMO LTI systems, including spectral decompositions of finite and infinite cross-Gramians of continuous MIMO LTI systems. Recommendations on the use of the obtained results are given.

We present a modified characteristic finite element method that exhibits second-order spatial accuracy for solving convection-reaction-diffusion equations on surfaces. The temporal direction adopted backward-Euler method, while the spatial direction employed surface finite element method. In contrast to regular domains, it is observed that the point in the characteristic direction traverses the surface only once within a brief time. Thus, the good approximation of the solution in the characteristic direction holds significant importance for the numerical scheme. In this regard, Taylor expansion is employed to reconstruct the solution beyond the surface in the characteristic direction. The stability of our scheme is then proved. A comparison is made with an existing characteristic finite element method based on face mesh. Numerical examples are provided to validate the effectiveness of our proposed method.

The coupled Dirac equations for solar e- and μ-neutrinos are solved in the presence of a small uniform interaction of the e-neutrinos with matter (the Sun). It is shown that the interaction leads to an increase of the amplitude of the neutrinos oscillations and a decrease of the lo- calization probability of the e-neutrinos. However, the strength of this interaction is insufficient to explain the magnitude of the discrepancy between the solar neutrino flux and the neutrino flux measured on the Earth. An alternative interpretation related to a possible superfluid neutrino state is advanced, which may throw some light on the solar neutrino problem.

Recently, a paper was written to establish semi-structured complex numbers ℍ (a new number set invented to enable division by zero) as a useful mathematical tool providing novel results from its application in engineering and science. Whilst the paper was a milestone, the application of this number set to quantum and classical physics had not been fully explored. Consequently, the aim of this research was to use semi-structured complex numbers to develop a new mathematical model for the mass of an object and in course unite classical and quantum physics in a simple yet effective manner. As its major contributions this paper: (1) develops a new mathematical model for the mass of an object called the wave-mass function using De Broglie’s mass and frequency relation and semi-structured complex numbers; (2) uses the new wave-mass function to determine what happens to an object traveling at the speed of light; (3) derives a relation between Schrödinger’s wave function and the new wave-mass function; (4) derives a relation between Einstein’s Field Equation and the new wave-mass function; (5) uses the wave-mass function to combine Einstein’s Field Equation with Schrödinger’s wave function to form a new quantum gravity equation. These results and their applications provide a firm foundation to advance the number set ℍ as a useful mathematical tool.

We have found an infinite dimensional manifold of exact solutions of the Navier-Stokes loop equation for the Wilson loop in decaying Turbulence in arbitrary dimension $d >2$. This solution family is equivalent to a fractal curve in complex space $\mathbb C^d$ with random steps parametrized by $N$ Ising variables $\sigma_i=\pm 1$, in addition to a rational number $\frac{p}{q}$ and an integer winding number $r$, related by $\sum \sigma_i = q r$. This equivalence provides a \textbf{dual} theory describing a strong turbulent phase of the Navier-Stokes flow in $\mathbb R_d$ space as a random geometry in a different space, like ADS/CFT correspondence in gauge theory. From a mathematical point of view, this theory implements a stochastic solution of the unforced Navier-Stokes equations. For a theoretical physicist, this is a \textbf{quantum} statistical system with integer-valued parameters, satisfying some number theory constraints. Its long-range interaction leads to critical phenomena when its size $N \rightarrow \infty$ or its chemical potential $\mu \rightarrow 0$. The system with fixed $N$ has different asymptotics at odd and even $N\rightarrow \infty$, but the limit $\mu \rightarrow 0$ is well defined. The energy dissipation rate is analytically calculated as a function of $\mu$ using methods of number theory. It grows as $\nu/\mu^2$ in the continuum limit $\mu \rightarrow 0$, leading to anomalous dissipation at $\mu \propto \sqrt{\nu} \to 0$. The same method is used to compute all the local vorticity distribution, which has no continuum limit but is renormalizable in the sense that infinities can be absorbed into the redefinition of the parameters. The small perturbation of the fixed manifold satisfies the linear equation we solved in a general form. This perturbation decays as $t^{-\lambda}$, where the anomalous dimensions $\lambda$ satisfy the spectral equation \eqref{spectralEq}. The spectrum becomes a continuum in the statistical limit $N \to \infty$, presumably leading to multifractal phenomena.

Physics-informed neural networks are a promising method to yield surrogate models of flow fields. We present a metamodeling technique for variable geometries based on physics-informed neural networks. The method was applied to the DU99W350 airfoil at a Reynolds number of 1×10⁵. The model predicted the Reynolds-averaged velocity and pressure field around the airfoil for arbitrary angles of attack between 10.0° and 17.5°. The model was trained with data from CFD simulations for a limited set of angles of attack. Additionally, satisfaction of the a priori known boundary conditions as well as the Reynolds-averaged Navier-Stokes equations were trained. A sensitivity analysis concerning the Reynolds number, the amount and distribution of training data, and the turbulence model was conducted showing the superiority of the pseudo-Reynolds stress method and the demand of labeled training data in the domain. The trained network was capable of predicting the developing flow separation on the suction surface and exhibited excellent agreement with CFD results even in the proximity to the wall for interpolations as well as extrapolations from the labeled data set.

The deep energy method (DEM), a type of physics-informed neural network, is evolving as an alternative to finite element analysis. This method employs the principle of minimum potential energy to predict deformations under static loading conditions. However, the model’s accuracy is contingent upon choosing the appropriate architecture for the model, which can be challenging due to the high interactions between hyperparameters, large search space, difficulty in identifying objective functions, and non-convex relationships with the objective functions. To improve DEM’s accuracy, we first introduce random Fourier feature (RFF) mapping. RFF mapping helps with the training of the model by reducing bias towards high frequencies. The effects of six hyperparameters are then studied under compression, tension, and bending loads in planar linear elasticity. Based on this study, a systematic automated hyperparameter optimization approach is proposed. Due to the high interaction between hyperparameters and the non-convex nature of the optimization problem, Bayesian optimization algorithms are used. The models trained using optimized hyperparameters and having Fourier feature mapping can accurately predict deflections compared to finite element analysis. Additionally, the deflections obtained for tension and compression load cases are more sensitive to variations in hyperparameters than bending.

In this paper, we introduced a generalized ∆-implicit locally contractive condition and give some examples to support it and to show its significance in fixed point theory. We prove that the mappings satisfying generalized ∆-implicit locally contractive condition admits a common fixed point, where, the ordered multiplicative GM−metric space is chosen as underlying space. The obtained fixed point theorems generalize many earlier fixed point theorems on implicit locally contractive mappings. In addition, some nontrivial and interesting examples are provided to support our findings. To demonstrate the originality of our new main result, we apply it to show existence of solutions to a system of nonlinear -Volterra type- integral equations.

Many authors proposed analytical methods for solving fully fuzzy Sylvester matrix equation (FFSME) based on Vec-operator and Kronecker product. However, these methods are restricted to nonnegative fuzzy numbers and cannot be extended to FFSME with near-zero fuzzy numbers. The main intention of this paper is to develop a new numerical method for solving FFSME with near-zero trapezoidal fuzzy numbers that provides a wider scope of trapezoidal fully fuzzy Sylvester matrix equation (TrFFSME) in scientific applications. This numerical method can solve the trapezoidal fully fuzzy Sylvester matrix equation with arbitrary coefficients and find all possible finite arbitrary solutions for the system. In order to obtain all possible fuzzy solutions, the TrFFSME is transferred to a system of non-linear equations based on newly developed arithmetic fuzzy multiplication between trapezoidal fuzzy numbers. The fuzzy solutions to the TrFFSME are obtained by developing a new two-stage algorithm. To illustrate the proposed method numerical example is solved.

We provide methodology and numerical results for the Hattendorff differential equa- tion for the continuous time evolution of the variance of L(j)t , the loss at time t random variable for a multi-state process, given that the state at time t is j.

In this paper we study solutions and drift homogenization for a class of viscous lake equations by using the method of semigroups of bounded operators. Suppose that the initial value i.e.,for some Hölder continuous function onwith smooth function value satisfying and Then the initial value problem (2) for viscous lake equations has a unique smooth local strong solution. Using this result we study the drift homogenization for three-dimensional stationary Stokes equation in the usual sense

Fractional differential equations have become a fundamental modelling approach for understanding and simulating the many aspects of nonlocality and spatial heterogeneity of complex materials and systems. Yet, while real-order fractional operators are nowadays widely adopted, little progress has been made in extending such operators to complex-order counterparts. In this work, we introduce new definitions for the complex-order fractional Laplacian, fully consistent with the theory of averaging of smooth functions over fractal sets, and present tailored spectral methods for their numerical treatment. The proposed complex-order operators exhibit a dual particle-wave behaviour, with solutions manifesting wave-like features depending on the magnitude of the imaginary part of the fractional order. Reaction-diffusion systems driven by the complex-order fractional Laplacian exhibit unique spatio-temporal dynamics, such as equilibrium of diffusion in random materials by interference of scattered waves, conduction block and highly fractionated propagation, or the generation of completely novel Turing patterns. Taken together, our results support that the proposed complex-order operators hold unparalleled capabilities to advance the description of multiscale transport phenomena in physical and biological processes highly influenced by the heterogeneity of complex media.

In this paper, a relativistic scalar particle under Lorentz symmetry breaking effects in the presence of a scalar potential is investigated. We introduce the scalar potential by modifying the mass via transformation M → M+S(r) in the wave equation and analyze the behaviour of a scalar particle. We see that the analytical solution to the KleinGordon equation can be achieved, and the energy eigenvalues and the wave function depends on the Lorentz symmetry breaking parameters as well as potential

The relativistic quantum dynamics of a spin-0 scalar particle under the effects of the violation of Lorentz symmetry in the presence of a non-electromagnetic potential is analyzed. The central potential induced by the Lorentz symmetry violation is a linear electric and constant magnetic field and, analyze the effects on the eigenvalues and the wave function. We see there is a dependence of the linear charge density on the quantum numbers of the system

In this work, we investigate the behaviour of relativistic quantum oscillator under the effects of Lorentz symmetry violation determined by a tensor $(K_F)_{\mu\nu\alpha\beta}$ out of the Standard Model Extension. We analyze this relativistic system under an inverse radial electric field and a constant magnetic field induced by Lorentz symmetry violation. We see that the presence of Lorentz symmetry breaking terms modified the energy spectrum of the system, and a quantum effect arise due to the dependence of the linear charge density on the quantum numbers of the system

In this work, we investigate the behaviour of a relativistic scalar particle in the background of the Lorentz symmetry violation determined by a tensor (KF)µναβ out of the Standard Model Extension. A linear electric field and a uniform magnetic can be induced by the violation of the Lorentz symmetry breaking effects, and analyze the behaviour of the scalar particle. We see that the analytical solution to the KG-equation can be achieved, and a quantum effect characterized by the dependence of the magnetic field on the quantum numbers is observed

We investigate a scalar particle under Lorentz symmetry breaking effects determined by a tensor out of the Standard Model Extension (SME) in the presence of a Cornell-type potential by modifying the mass term M → M +S in the KG-equation. The field configuration is such that a Coulomb-type radial electric field and a constant magnetic field can be induced by Lorentz symmetry violation, and analyze the behaviour of a scalar particle. One can see that the bound states solution to the KG-equation under the consider effects can be achieved, and a quantum effect characterized by the dependence of charge density distribution parameter on the quantum numbers of the system is observed.

In this article, author performs computational analysis for the generalized zeta functions by using computational software Mathematica. To achieve the purpose recently obtained difference equations are used. These difference equations have a computational power to compute these functions accurately while they can not be computed by using their known integral represenations. Several authors investigated such functions and their analytic properties, but no work has been reported to study the graphical representations and zeors of these functions. Author performs numerical computations to evaluate these functions for different values of the involved parameters. Taylor series expansions are also presented in this research.

In this paper, we proposed the numerical method called the variational iteration orthogonal collocation method (VIOCM), for the approximate solution of the deadly Corona virus model using Mamadu-Njoseh polynomials as basis functions. The proposed method is an elegant mixture of the variational iteration method (VIM) and the orthogonal collocation method (OCM). It was observed that the proposed method converges rapidly to the exact solution even as N increases. This suggests that the use of orthogonal polynomials as trial functions for the SEIR model is indeed an effective approximant as it produces the analytic solution at just few iterations. Resulting numerical evidences were compared with the standard variational iteration method as available in literature. All computational frameworks were executed with MAPLE software.

The so-called Born-Huang ansatz is a fundamental tool in the context of ab-initio molecular dynamics, viz., it allows to effectively separate fast and slow degrees of freedom and thus treating electrons and nuclei at different mathematical footings. Here we consider the use of a Born-Huang-like expansion of the three-dimensional time-dependent Schr\"odinger equation to separate transport and confinement degrees of freedom in electron transport problems that involve geometrical constrictions. The resulting scheme consists of an eigenstate problem for the confinement degrees of freedom (in the transverse direction) whose solution constitutes the input for the propagation of a set of coupled one-dimensional equations of motion for the transport degree of freedom (in the longitudinal direction). This technique achieves quantitative accuracy using an order less computational resources than the full dimensional simulation for a prototypical two-dimensional constriction.

This paper deals with the study of the existence and non existence of solutions of a three parameter's family of nonlinear fractional differential equation with mixed-integral boundary value conditions. We consider the $\alpha$-Riemann-Liouville fractional derivative, with $\alpha \in (1,2]$. In order to deduce the existence and non existence results, we first study the linear equation, by deducing the main properties of the related Green's functions. We obtain the optimal set of parameters where the Green's function has constant sign. After that, by means of the index theory, the nonlinear boundary value problem is studied. Some examples, at the end of the paper, are showed to illustrate the applicability of the obtained results.

By defining a constant probabilistic orbit of and iterated functions, the stability dynamics of these functions in possible interactions through connectivity provides the formation of a dynamic fixed point as a metric space between both iterated functions. The presence of a dynamic fixed point identifies qualitatively phases of iteration time lengths and interaction orbits of the event. Qualitative results show that the greater the average distance from one of the functions to the fixed point of the other (all possible solutions), the higher the iteration expression on time (false asymptotic effect) of one of the functions and in the opposite hand, the lower the average distance, the higher orbit’s interactions proximity between iterated functions (stability). This feature reveals asymptotic (well-defined) behavior between functions f and g within a well-defined Lyapunov stability.

Exact and soliton type solutions have great importance in propagation of surface waves, fluid dynamics, optics, and many other elds of nonlinear sciences. In this study, the explicit and exact soliton type solutions for two space-time fractional Equal- Width (FEW) equations with conformable derivative are procured via the hyperbolic function approach. The wave type solutions are represented in some hyperbolic and trigonometric functions.

Complex and real valued exact solutions to some reaction-diffusion equations are suggested by using homogeneous balance and Sine-Gordon equation expansion method. The predicted solution of finite series of some hyperbolic functions are determined by using some relations between the hyperbolic functions and trigonometric functions based on Sine-Gordon equation and traveling wave transform. The Newel-Whitehead-Segel and Zeldovich equations are solved explicitly. Some real valued solutions are depicted for some particular choice of parameters.

Splines and quasi-interpolation operators are important both in approximation theory and applications. In this paper, we construct a family of quasi-interpolation operators for the bivariate quintic spline spaces S₅³ (∆_mn⁽²⁾). Moreover, the properties of the proposed quasi-interpolation operators are studied, as well as its applications for solving two-dimensional Burgers’ equation and image reconstruction. Some numerical examples show that these methods, which are easy to implement, provide accurate results.

This paper describes developing and analyzing an analytical model of Microelectromechanical Systems (MEMS) internal vibrating ring gyroscope. The internal ring structure consists of eight semicircular beams attached to the externally placed anchors. The paper analyses the vibrating ring gyroscope inplane displacement behavior and the resulting elliptical vibrational modes. The elliptical vibrational modes appear as pairs with the same resonance frequency due to the symmetric nature of the design. The analysis commences by conceptualizing the ring as a geometric structure with a circular shape, possessing specific dimensions such as thickness, height, and radius. We construct a linear model that characterizes the vibrational dynamics of the internal vibrating ring. The analysis encompasses comprehensive mathematical formulations for the radial and tangential displacements in local polar coordinates, considering the inextensional displacement of the ring structure. By utilizing the derived equations, we highlight the underlying relationships driving the vibrational characteristics of the MEMS vibrating ring gyroscope. These dynamic vibrational relationships are essential in enabling the vibrating ring gyroscope's future utilization in accurate navigation and motion sensing technologies.

The relativistic quantum motion of a scalar particle under the effects of violation of the Lorentz symmetry in the presence of a linear confining potential is investigated. We see that the solution of the bound state to the modified Klein-Gordon equation can be obtained and a quantum effect characterized by the dependence of the magnetic field on the quantum numbers of the system is observed

In this work, quantum dynamics of a spin-0 particle under the effects of Lorentz symmetry violation in the presence of Coulombtype non-electromagnetic potential $(S(r) ∝ \frac{1}{r})$ is investigated. The non-electromagnetic (or scalar) potential is introduced by modifying the mass term via transformation $M → M + \frac{η_c}{r}$ in the relativistic wave equation. The linear central potential induced by the Lorentz symmetry violation is a linear radial electric and constant magnetic field and, analyze the effects on the spectrum of energy and the wave function

In this paper, we found a new model for compact star with charged anisotropic matter distribution considering an extended version of the Chaplygin equation of state. We specify a particular form of the metric potential Z(x) that allows us to solve the Einstein-Maxwell field equations. The obtained model satisfies all physical properties expected in a realistic star such that the expressions for the radial pressure, energy density, metric coefficients, measure of anisotropy and the mass are fully well defined and are regular in the interior of star. The solution obtained in this work can have multiple applications in astrophysics and cosmology.

We show that the quaternion quantum mechanics has well-founded mathematical roots and can be derived from the model of elastic continuum by French mathematician Augustin Cauchy, i.e., it can be regarded as representing physical reality of elastic continuum. Starting from the Cauchy theory (classical balance equations for isotropic Cauchy-elastic material) and using the Hamilton quaternion algebra we present a rigorous derivation of the quaternion form of the non- and relativistic wave equations. The family of the wave equations and the Poisson equation are a straightforward consequence of the quaternion representation of the Cauchy model of the elastic continuum. This is the most general kind of quantum mechanics possessing the same kind of calculus of assertions as conventional quantum mechanics. The problem of the Schrödinger equation, where imaginary ‘i’ should emerge, is solved. This interpretation is a serious attempt to describe the ontology of quantum mechanics, and demonstrates that, besides Bohmian mechanics, the complete ontological interpretations of quantum theory exists. The model can be generalized and falsified. To ensure this theory to be true, we specified problems allowing exposing its falsity.

Interaction of an electromagnetic field with matter in a laser cavity without the assumption of a fixed direction of the transverse electric field, described by the two-level Maxwell-Bloch equations, is studied. By using a perturbative nonlinear analysis, performed near the laser threshold, we report on the derivation of the laser (3+1)D vectorial complex cubic-quintic complex Ginzburg-Landau equation. Furthermore, we study the modulational instability of the plane waves both theoretically using the linear stability analysis, and numerically, using direct simulations via the split-step Fourier method. The linear theory predicts instability for any amplitude of the primary waves. Our numerical simulations confirm the theoretical predictions of the linear theory as well as the threshold of the amplitude of perturbations. The system understudy shows a deep dependence on the laser cavity parameters, for which there appear wave patterns in accordance with the predictions from the gain spectrum.

In this paper, a local meshless method (LMM) based on radial basis functions (RBFs) is utilized for the numerical solution of various types of PDEs, due to the flexibility with respect to geometry and high order of convergence rate. In case of global meshless methods, the two major deficiencies are the computational cost and the optimum value of shape parameter. Therefore, research is currently focus towards localized radial basis function approximations, as the local meshless method is proposed here. The local meshless procedures is used for spatial discretization whereas for temporal discretization different time integrators are employed. The proposed local meshless method is testified in terms of efficiency, accuracy and ease of implementation using regular and irregular domains.

By using the new recently proposed extended rational methods, we set some solitary and multi wave solutions of the coupled Higgs system in the present study. Some new families of hyperbolic and trigonometric solutions of the coupled Higgs equations are successfully obtained. Some solutions are real as some are complex valued functions. Particular forms of some solutions derived by choice of some parameters are demonstrated in three dimensional spaces. The complex valued solutions are also depicted by plotting both real and complex component.

A formulation of nonequilibrium thermo field dynamics has been performed using the nonequilibrium statistical operator method by D.N. Zubarev. Generalized transfer equations for a consistent description of kinetics and hydrodynamics of the dense quantum-field system with strongly coupled states are derived.

In this paper, we study the third order semilocal convergence of the Newton-like method for finding the approximate solution of nonlinear operator equations in the setting of Banach spaces. First, we discuss the convergence analysis under ω-continuity condition, which is weaker than the Lipschitz and Hölder continuity conditions. Second, we apply our approach to solve Fredholm integral equations, where the first derivative of involved operator not necessarily satisfy the Hölder and Lipschitz continuity conditions. Finally, we also prove that the R-order of the method is 2p + 1 for any p $\in$ (0,1].

An eigenvalue equation representing symmetric (dual) quantum equation is introduced. The particle is described by two scalar wavefunctions, and two vector wavefunctions. The eigenfunction is found to satisfy the quantum Telegraph equation keeping the form of the particle fixed but decaying its amplitude. An analogy with Maxwellian equations is presented. Massive electromagnetic field will satisfy a quantum Telegraph equation instead of a pure wave equation. This equation resembles the motion of the electromagnetic field in a conducting medium. With a particular setting of the scalar and vector wavefunctions, the dual quantum equations are found to yield the quantized Maxwell's equations. The total energy of the particle is related to the phase velocity ($v_p$) of the wave representing it by $E=p\,|v_p|$, where $p$ is the matter wave momentum. A particular solution which describes the process under which the particle undergoes a creation and annihilation is derived. The force acting on the moving particle is expressed in a dual Lorentz-like form. If the particle moves like a fluid, a dissipative (drag) quantum force will arise.

In this paper, by means of the Banach fixed point theorem and the Krasnoselskii's fixed point theorem, we investigate the existence of solutions for some fractional neutral functional integro-differential equations involving infinite delay. This paper deals with the fractional equations in the sense of Caputo fractional derivative and in the Banach spaces. Our results generalize the previous works on this issue. Also, an analytical example is presented to illustrate our results.

This article focuses on obtaining the analytical solutions for parabolic Volterra integro- differential equations in d-dimensional with different types frictional memory kernel. Based on theories of Laplace transform, Fourier transform, the properties of Fox-H function and convolution theorem, analytical solutions of the equations in the infinite domain are derived under three frictional memory kernel functions respectively. The analytical solutions are expressed by infinite series, the generalized multi-parameter Mittag-Leffler function, Fox-H function and convolution form of Fourier transform. In addition, the graphical representations of the analytical solution under different parameters are given for one-dimensional parabolic Volterra integro-differential equation with power-law memory kernel. It can be seen that the solution curves subject to Gaussian decay at any given moment.

This paper presents a thorough investigation into the shear strength capacity of reinforced concrete deep beams, with a focus on improving predictive accuracy beyond existing code provisions. Analyzing 198 deep beams from 15 investigations, the study considers parameters such as concrete compressive strength (fc’), shear span to effective depth ratio (av/d), and reinforcement ratios (ρs, ρv, ρh ). Introducing a novel predictive model (Equation 7), the study rigorously evaluates it using nonlinear regression analysis and statistical metrics (MAE, RMSE, R2). The proposed model demonstrates a significant reduction in the coefficient of variation (COV) to 27.08%, surpassing existing codes' limitations. Comparative analyses highlight the model's robustness, revealing improved convergence of data points and minimal sensitivity to variations in key parameters. The findings suggest that the proposed model offers enhanced predictive accuracy across diverse scenarios, making it a valuable tool for structural engineers. This research contributes to advancing the understanding of shear strength in reinforced concrete deep beams, offering a reliable and versatile predictive model with implications for refining design methodologies and enhancing the safety and efficiency of structural systems.

This paper discusses how mathematical models are used to predict environmental conditions. While these models are successful for larger scale phenomena, it becomes more complex for local weather or wave processes. Several approaches have been proposed to address this problem, including neural networks and statistical and mathematical models. The novelty in this approach is the use of statistical manifolds, which provide a more accurate way to estimate distances and cost functions. This work focuses on advancements in differential and information geometry for creating different types of models that can be used for environmental simulation and forecasting. These fields are vital for many institutes and centres worldwide, as they help solve challenges in areas such as meteorology, renewable energy, and aviation safety. Some potential IG applications to advance energy were given. Closing remarks with new open problems combined with the next research phase are provided.

The article proposes asymptotic methods for calculating Sommerfeld integrals, which enable us to calculate the integral using the expansion of a function into an infinite power series at the saddle point, where the role of a rapidly oscillating function under the integral can be fulfilled either by an exponential or by its product by the Hankel function. The proposed types of Sommerfeld integrals are generalized on the basis of integral representations of the Hertz radiator fields in the form of the inverse Hankel transform with the subsequent replacement of the Bessel function by the Hankel function. It is shown that the numerical values of the saddle point are complex. During integration, auxiliary or so-called standard integrals, which contain the main features of the integrand function, were used. As a demonstration of the accuracy of the technique, a previously known asymptotic formula for the Hankel functions was obtained in the form of an infinite series. The proposed method for calculating Sommerfeld integrals can be useful in solving the half-space Sommerfeld problem. An example is the expression in the form of an infinite series for the magnetic field of reflected waves, obtained directly through the Sommerfeld integral (SI).

Empirical equations representing, interpolating and smoothing groups of measurement results by regression methods are in widespread use in metrology and other fields of science and engineering. Standard equations for the propagation of measurement uncertainties to values computed from such equations are available but suffer from a lack of general acceptance and are only infrequently applied in practice. One reason for the slow uptake is the lack of clear methods to account for systematic error. In this paper, uncertainty propagation equations in terms of covariance matrices are generalized to allow for systematic errors in least-squares regression, and effects of the resulting uncertainties are investigated analytically. A stochastic ensemble model of systematic error is proposed for the computation of the non-diagonal elements of the weight matrix of the Generalized Least-Squares method (GLS) from the measurement uncertainty. A GLS projector formalism is described which separates the effects of measurement scatter on values calculated by the equation from those on the related “residuals”, i.e., the residual errors in the fitted data. The same projectors act similarly on the associated uncertainties and covariance matrices and permit the effects of systematic errors on the simulation covariance to be quantified. It is demonstrated that, in order to include systematic errors in the uncertainty estimates of GLS, covariance matrices may be substituted by novel, specifically defined dispersion matrices that are not specified yet by the GUM[1]. Systematic measurement errors may be estimated from various sources; a particularly easy way suggested here involves analyzing the structure in the regression residuals. It is demonstrated that GLS projectors exhibit inherent features of “error blindness” also with respect to systematic errors. [1] GUM: Guide to the Expression of Uncertainty in Measurement, http://www.bipm.org/en/publications/guides

In this paper, existence and stability results of non-local integro-differential equation with Hilfer derivative of order ω∈(1,2) have been investigated. In this derivative is applied in RLC circuit equation of various fractional order. To proposed problem the existence solutions are derived using Schaefer’s fixed point theorem and the Banach contraction principle is using for uniqueness results. To demonstrate the effectiveness and applicability of our theoretical conclusions and two-step Lagrange polynomial interpolation were used to solve four numerical results.