ARTICLE | doi:10.20944/preprints201712.0188.v1
Subject: Keywords: sine-gordon expansion method; conformable time fractional EW equation; conformable time fractional modified EW Equation; exact solution; traveling solution
Online: 27 December 2017 (04:00:25 CET)
New exact solutions to conformable time fractional EW and modified EW equations are constructed by using Sine-Gordon expansion approach. The fractional traveling wave transform and homogeneous balance have significant roles in the solution procedure. The predicted solution is of the form of some finite series of multiplication of powers of cos and sin functions. The relation among trigonometric and hyperbolic functions in sense of Sine-Gordon expansion gives opportunity to construct the solutions in terms of hyperbolic functions.
ARTICLE | doi:10.20944/preprints201802.0141.v1
Subject: Mathematics & Computer Science, Numerical Analysis & Optimization Keywords: Ansatz method; fractional modified Korteweg–de Vries Equation; fractional modified Equal Width Equation; fractional Benney–Luke Equation; Conformable fractional derivative
Online: 22 February 2018 (10:45:02 CET)
The space time fractional Korteweg-de Vries equation, modified Equal Width equation and Benney-Luke Equations are solved by using simple hyperbolic tangent Ansatz method. A simple compatible wave transformation in one dimension is employed to reduce the governing equations to integer ordered ODEs. Then, the ansatz approximation is used to derive exact solutions. Some illustrative examples are presented for some particular choices of parameters and derivative orders.
ARTICLE | doi:10.20944/preprints201612.0004.v2
Subject: Physical Sciences, Mathematical Physics Keywords: conformable time fractional (3+1)-dimensional Kadomtsev-Petviashvili equation; conformable time fractional modified Kawahara equation; modified Kudryashov method; wave solution
Online: 5 December 2016 (10:31:30 CET)
The three dimensional conformable time fractional Kadomtsev-Petviashvili and the conformable time fractional modified Kawahara equations are solved by implementing the Kudryashov's procedure. The corresponding wave transformation reduces both equations to some ODEs. Balancing the nonlinear and the highest order derivative terms gives the structure of the solutions in the finite series form. The useful symbolic tools are used to solve the resultant algebraic systems. The solutions are expressed in explicit forms.
ARTICLE | doi:10.20944/preprints201712.0178.v1
Subject: Keywords: sine-gordon expansion method; conformable time fractional KdV equation; conformable time fractional modified KdV equation; exact solution, traveling wave solution
Online: 25 December 2017 (10:31:11 CET)
An expansion method based on time fractional Sine-Gordon equation is implemented to construct some real and complex valued exact solutions to the Korteweg-de Vries and modified Korteweg-de Vries equation in time fractional forms. Compatible fractional traveling wave transform plays a key role to be able to apply homogeneous balance technique to set the predicted solution. The relation between trigonometric and hyperbolic functions based on fractional Sine-Gordon equation allows to form the exact solutions with multiplication of powers of hyperbolic functions.
ARTICLE | doi:10.20944/preprints201901.0122.v1
Subject: Mathematics & Computer Science, Applied Mathematics Keywords: conformable fractional derivative; conformable partial fractional derivative; conformable double Laplace decomposition method; conformable Laplace transform; Singular one dimensional coupled Burgers equations
Online: 14 January 2019 (07:04:49 CET)
This article deals with the conformable double Laplace transforms and their some properties with examples and also the existence Condition for the conformable double Laplace transform is studied. Finally, in order to obtain the solution of nonlinear fractional problems, we present a modified conformable double Laplace that we call conformable double Laplace decomposition methods (CDLDM). Then, we apply it to solve, Regular and singular conformable fractional coupled burgers equation illustrate the effectiveness of our method some examples are given.
ARTICLE | doi:10.20944/preprints201812.0341.v1
Subject: Physical Sciences, Mathematical Physics Keywords: Time-fractional Phi-four equation; Conformable derivatives; exp a function approach; Hyperbolic function approach; Exact solutions
Online: 28 December 2018 (08:05:11 CET)
In this study, we actually want to explore the time-fractional Phi-four equation via two methods, the exp a function method and the hyperbolic function method. We transform a fractional order dierential equation into an ordinary differential equation using a wave transformation and the fractional derivative in conformable form. Then, the resulting equation has successfully been explored for new explicit exact solutions. The procured solutions are simply showed the effectiveness and plainness of the projected methods.
ARTICLE | doi:10.20944/preprints201809.0476.v1
Subject: Mathematics & Computer Science, Applied Mathematics Keywords: fractional power series; integro-differential equations; conformable derivative
Online: 25 September 2018 (05:06:12 CEST)
In this paper, we investigate an analytical solution of a class of nonlinear fractional integro-differential equation base on a generalized fractional power series expansion. The fractional derivatives are described in the conformable's type. The new approach is a modified form of the well-known Taylor series expansion. The illustrative examples are presented to demonstrate the accuracy and effectiveness of the proposed method
ARTICLE | doi:10.20944/preprints201612.0084.v1
Subject: Mathematics & Computer Science, Applied Mathematics Keywords: modified Kudryashov method; conformable time fractional RLW-Burgers Equation; conformable time fractional potential KdV Equation; conformable time fractional CRWP equation; conformable derivative
Online: 16 December 2016 (08:01:56 CET)
The exact solutions of some conformable time fractional PDEs are presented explicitly. The modified Kudryashov method is applied to construct the solutions to the conformable time fractional Regularized Long Wave-Burgers (RLW-Burgers, potential Korteweg-de Vries (KdV) and clannish random walker's parabolic (CRWP) equations. Initially, the predicted solution in the finite series of a rational form of an exponential function is substituted to the ODE generated from the conformable time fractional PDE by using wave transformation. The coefficients used in the finite series are determined by solving the algebraic system derived from the coefficients of the powers of the predicted solution.
ARTICLE | doi:10.20944/preprints201712.0183.v1
Subject: Keywords: Sine-Gordon Expansion Method; conformable time fractional RLW equation; conformable time fractional modified RLW equation; conformable time fractional symmetric-RLW equation
Online: 26 December 2017 (04:50:06 CET)
The Sine-Gordon expansion method is implemented to construct exact solutions some conformable time fractional equations in Regularized Long Wave(RLW)-class. Compatible wave transform reduces the governing equation to classical ordinary differential equation. The homogeneous balance procedure gives the order of the predicted polynomial-type solution that is inspired from well-known Sine-Gordon equation. The substitution of this solution follows the previous step. Equating the coefficients of the powers of predicted solution leads a system of algebraic equations. The solution of resultant system for coefficients gives the necessary relations among the parameters and the coefficients to be able construct the solutions. Some solutions are simulated for some particular choices of parameters.
ARTICLE | doi:10.20944/preprints201807.0025.v1
Subject: Mathematics & Computer Science, Applied Mathematics Keywords: fractional differential equations; conformable derivative; Bernoulli equation; exact solution
Online: 3 July 2018 (06:04:53 CEST)
In this paper we use the conformable fractional derivative to discuss some fractional linear differential equations with constant coefficients. By applying some similar arguments to the theory of ordinary differential equations, we establish a sufficient condition to guarantee the reliability of solving constant coefficient fractional differential equations by the conformable Laplace transform method. Finally, we analyze the analytical solution for a class of fractional models associated with Logistic model, Von Foerster model and Bertalanffy model is presented graphically for various fractional orders and solution of corresponding classical model is recovered as a particular case.
ARTICLE | doi:10.20944/preprints201609.0105.v1
Subject: Mathematics & Computer Science, Applied Mathematics Keywords: integral inequalities; special functions; fractional calculus; conformable fractional integral
Online: 27 September 2016 (11:05:29 CEST)
In this paper, we establish the generalized Qi-type inequality involving conformable fractional integrals. The results presented here would provide extensions of those given in earlier works.
ARTICLE | doi:10.20944/preprints201902.0040.v1
Subject: Mathematics & Computer Science, Applied Mathematics Keywords: Convex function; Integral inequalities; Hermite-Hadamard inequality; Conformable fractional integrals
Online: 4 February 2019 (16:38:15 CET)
In this work, we establish some new Hermite-Hadamard type inequalities for convex functions via conformable fractional integral. Moreover, we show that through the conformable fractional integral we can find some new Hermite-Hadamard type inequalities for convex functions via the classical integrals.
ARTICLE | doi:10.20944/preprints202108.0055.v1
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: integro-differential equation; fractional derivatives; fractional conformable derivatives; normalized systems method.
Online: 2 August 2021 (15:46:38 CEST)
The methods for constructing solutions to integro-differential equations of the Volterra type are considered. The equations are related to fractional conformable derivatives. Explicit solutions of homogeneous and inhomogeneous equations are constructed and a Cauchy-type problem is studied. It should be noted that the considered method is based on the construction of normalized systems of functions with respect to a differential operator of fractional order.
ARTICLE | doi:10.20944/preprints201902.0243.v1
Subject: Mathematics & Computer Science, Applied Mathematics Keywords: Convex function, Ostrowski inequality, Holder's inequality, Power mean inequality, Conformable integrals, Midpoint formula
Online: 26 February 2019 (13:10:40 CET)
In the article, by applied the concept of strongly convex function and one known identity, we establish several Ostrowski type inequalities involving conformable fractional integrals. As applications, some new error estimations for the midpoint formula are provided as well.
ARTICLE | doi:10.20944/preprints201812.0190.v1
Subject: Mathematics & Computer Science, Applied Mathematics Keywords: Space-time fractional Equal-Width equations; Conformable derivative; Hyperbolic function approach; Exact soliton solutions
Online: 17 December 2018 (10:47:03 CET)
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.
ARTICLE | doi:10.20944/preprints202111.0528.v2
Subject: Mathematics & Computer Science, Applied Mathematics Keywords: Conformable calculus; Fractional-order financial system; ESDDFD and NSFD methods; Hyperchaotic attractor; Market confidence; Ethics risk
Online: 6 December 2021 (12:48:05 CET)
Four discrete models using the exact spectral derivative discretization finite difference (ESDDFD) method are proposed for a chaotic five-dimensional, conformable fractional derivative financial system incorporating ethics and market confidence. Since the system considered was recently studied using the conformable Euler finite difference (CEFD) method and found to be hyperchaotic, and the CEFD method was recently shown to be valid only at fractional index , the source of the hyperchaos is in question. Through numerical experiments, illustration is presented that the hyperchaos previously detected is in part an artifact of the CEFD method as it is absent from the ESDDFD models.
ARTICLE | doi:10.20944/preprints202009.0440.v1
Subject: Mathematics & Computer Science, Applied Mathematics Keywords: Conformable fractional derivative; Conformable fractional integral; Conformable fractional differential equations; Sturm´s Theorems; Green´s Function
Online: 18 September 2020 (12:16:52 CEST)
Recently, the conformable derivative and its properties have been introduced. In this paper, we propose and prove some new results on conformable Boundary Value Problems. First, we introduce a conformable version of classical Sturm´s separation, and comparison theorems. For a conformable Sturm-Liouville problem, Green's function is constructed, and its properties are also studied. In addition, we propose the applicability of the Green´s Function in solving conformable inhomogeneous linear differential equations with homogeneous boundary conditions, whose associated homogeneous boundary value problem has only trivial solution. Finally, we prove the generalized Hyers-Ulam stability of the conformable inhomogeneous boundary value problem.
ARTICLE | doi:10.20944/preprints201611.0117.v1
Subject: Mathematics & Computer Science, Analysis Keywords: Gamma function; conformable Gamma function; inequality
Online: 23 November 2016 (09:59:05 CET)
In a recent paper, Sarikaya et al. introduced a new analogue of the classical Gamma function which they called, the conformable Gamma function. Motivated by their results, this paper seeks to establish some inequalities for the conformable Gamma and Polygamma functions. Among other analytical tools, the procedure relies on the generalized forms of some classical inequalities.
Subject: Mathematics & Computer Science, Applied Mathematics Keywords: Sturm-Liouville; conformable derivative; asymptotic formula; spectral data
Online: 25 December 2019 (03:13:13 CET)
In this study, we investigate spectral structure of conformable Sturm-Liouville problems and with this end, we obtain representation of solutions under different initial conditions and asymptotic formulas for eigenfunctions, eigenvalues, norming constants and normalized eigenfunctions. Consequently, we prove the existence of infinitely many eigenvalues. Also, we compare the solutions with graphics with different orders, different eigenvalues, different potentials and so, we observe the behaviors of eigenfunctions. We give an application to the α-orthogonality of eigenfunctions and reality of eigenvalues for conformable Sturm-Liouville problems defined by  in the last section.