ARTICLE | doi:10.20944/preprints202103.0148.v1
Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Adomian decomposition method; variational iteration method; Volterra-Fredholm integro-differential equation; approximate solution.
Online: 4 March 2021 (09:20:22 CET)
In this article, we present a comparative study between the Adomian Decomposition Method (ADM) and Variational Iteration Method (VIM). The study outlines the significant features of the two methods, for solving nonlinear Volterra-Fredholm integro-differential equations. From the computational viewpoint, the VIM is more efficient, convenient and easy to use. Moreover, we proved the existence and uniqueness results and convergence of the solution. Finally, an example is included to demonstrate the validity and applicability of the proposed techniques.
ARTICLE | doi:10.20944/preprints201608.0129.v1
Subject: Computer Science And Mathematics, Mathematics Keywords: parabolic Volterra integro-differential equations; memory kernel; Laplace transform; Fourier transform; convolution theorem; analytical solution
Online: 11 August 2016 (11:47:10 CEST)
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.
ARTICLE | doi:10.20944/preprints202305.1762.v1
Subject: Computer Science And Mathematics, Applied Mathematics Keywords: Fractional Volterra integro-differential equation; Laplace fractional power series; Caputo fractional derivative, Laplace transform.
Online: 25 May 2023 (08:06:04 CEST)
In this paper, we compile the fractional power series method and the Laplace transform to design a new algorithm for solving the fractional Volterra integro-differential equation. For that, we assume the Laplace power series (LPS) solution in terms of power q=1m,m∈Z+, where the fractional derivative of order α=qγ for which γ∈Z+. This assumption will help us to write the integral, the kernel, and the nonhomogeneous terms as a LPS with the same power. The recurrence relations for finding the series coefficients can be constructed using this form. To demonstrate the algorithm's accuracy, the residual error is defined and calculated for several values of the fractional derivative. Two strongly nonlinear examples are discussed to provide the efficiency of the algorithm. The algorithm gains powerful results for this kind of problem. Under Caputo meaning the obtained results are illustrated numerically, and graphically. Geometrically, the behavior of the solution declares that the changing of the fractional derivative parameter values' in their domain alters the style of the attained solution in a symmetrical meaning and fully coinciding to the ordinary derivative value'. From these simulations, the results report that the recommended novel algorithm is a straightforward, accurate, and superb tool to generate analytic-approximate solutions for Integral, and integro-differental equations of fractional order.
ARTICLE | doi:10.20944/preprints201811.0151.v1
Subject: Computer Science And Mathematics, Applied Mathematics Keywords: time-dependent order of differentiation; fractional calculus; fractional derivative; differential equations; complex systems; Volterra integral equation; VODE; DODE; dynamical evolution
Online: 7 November 2018 (09:15:19 CET)
A new type of ordinary differential equation is introduced and discussed, namely, the time-dependent order ordinary differential equations. These equations can be solved via fractional calculus and are mapped into Volterra integral equations of second kind with singular integrable kernel. The solutions of the time-dependent order differential equations smoothly deforms solutions of the classical integer order ordinary differential equations into one-another, and can generate or remove singularities. An interesting symmetry of the solution in relation to the Riemann zeta function and Harmonic numbers was also proved.
ARTICLE | doi:10.20944/preprints202004.0288.v1
Subject: Computer Science And Mathematics, Analysis Keywords: branch points; bifurcation points; Fredholm operator; uniformization; asymptotics; iterations; regularization
Online: 17 April 2020 (01:34:37 CEST)
The necessary and sufficient conditions of existence of the nonlinear operator equations' branches of solutions in the neighbourhood of branching points are derived. The approach is based on reduction of the nonlinear operator equations to finite-dimensional problems. Methods of nonlinear functional analysis, integral equations, spectral theory based on index of Kronecker-Poincare, Morse-Conley index, power geometry and other methods are employed. Proposed methodology enables justification of the theorems on existence of bifurcation points and bifurcation sets in the nonstandard models. Formulated theorems are constructive. For a certain smoothness of the nonlinear operator, the asymptotic behaviour of the solutions is analysed in the neighbourhood of the branch points and uniformly converging iterative schemes with a choice of the uniformization parameter enables the comprehensive analysis of the problems details. General theorems are illustrated on the nonlinear integral equations.
ARTICLE | doi:10.20944/preprints202309.1296.v1
Subject: Biology And Life Sciences, Aquatic Science Keywords: Sustainable; Fisheries; Predator; Prey; Lotka-Volterra Equations
Online: 20 September 2023 (02:44:49 CEST)
This study delves into the intricate dynamics of sustainable fisheries through the lens of mathematical modeling, specifically employing the Lotka-Volterra equations. The Lotka-Volterra equations, originally conceived to understand predator-prey interactions, offer a potent framework to model and predict fish population dynamics within aquatic ecosystems. The aim of this research is to demonstrate the applicability and efficacy of the Lotka-Volterra equations in understanding the delicate balance of fish populations and aiding sustainable fisheries management. Through this mathematical approach, we seek to provide a structured methodology and present results that showcase the potential of this model in predicting and managing fish populations. In our analysis, we adjust the parameters of the Lotka-Volterra equations to simulate various scenarios, enabling a comprehensive understanding of sustainable fishing practices. Our findings underscore the significance of mathematical models, particularly the Lotka-Volterra equations, in informing strategies to maintain fish populations in a sustainable and ecologically balanced manner, thereby ensuring the longevity of marine ecosystems and the fisheries thatrely upon them.
ARTICLE | doi:10.20944/preprints202110.0279.v1
Subject: Engineering, Energy And Fuel Technology Keywords: EROEI; Lotka Volterra; resource exploitation; dissipative systems
Online: 19 October 2021 (13:57:02 CEST)
The Energy Return on Energy Invested, EROEI, is known as an important parameter for evaluating the efficiency of energy-producing technologies. In this paper we examine the concept of EROEI from a general viewpoint, giving insights on a wider range of applications. In general, natural resources can be seen as energy stocks characterized by a “potential” that can be exploited by creating intermediate stocks. This transformation is typical of dissipative systems and for the first time we found that Lotka-Volterra model, usually confined to the study of biology of populations, can represent a powerful tool to estimate EROEI for some such systems, providing an understanding of the reason for the overexploitation phenomenon and, in some cases, the collapse of the exploiting system.
ARTICLE | doi:10.20944/preprints202306.0570.v1
Subject: Engineering, Other Keywords: optical fiber data communication system; EML; PAM4; Volterra; DFE
Online: 8 June 2023 (03:00:45 CEST)
A novel simplifying Volterra structure algorithm is proposed for intensity modulation direct detection (IM-DD) optical fiber short distance communication system by using the decision feedback equalization algorithm (DFE). Based on this algorithm, the signal damage for four-level pulse amplitude modulation signal (PAM-4) is compensated, which is caused by device bandwidth limitation and dispersion during transmission. Experiments have been carried out using a 25GHz Electro-absorption Modulated Laser (EML), showing that PAM-4 signals can transmit over 10km in standard single-mode fiber (SSMF). The 112Gbps and 128Gbps signals can reach the error rate threshold of KP4-FEC (BER=2*10-4) and HD-FEC (BER=3.8*10-3), respectively. The simplified principle and process of proposed Volterra-based equalization algorithm are presented. Experimental results show that the algorithm complexity is greatly reduced by 75%, which provides an effective theoretical support for the commercial application of this algorithm.
ARTICLE | doi:10.20944/preprints202211.0216.v1
Subject: Business, Economics And Management, Econometrics And Statistics Keywords: Ten ASEAN countries; Bilateral trade; Lotka-Volterra model; commensalism; competition
Online: 11 November 2022 (04:05:04 CET)
The bilateral trade relationship between PRC and ASEAN is close. As early as 2012, China has been the largest trading partner of ASEAN, and ASEAN has also taken the place of the United States as PRC’s second largest trading partner in 2019.This paper uses the theory of ecological population evolution, selects the total import and export volume of PRC and the ten ASEAN countries from 2000 to 2019 as the data sample, studies the bilateral trade relations between PRC and the ten ASEAN countries, analyzes the competition and symbiotic relationship of bilateral trade between PRC and the ten ASEAN countries and the possibility of trade friction according to the results, and proposes corresponding improvement strategies. According to the results of the population evolution competition model, China has a competitive relationship with Indonesia, Malaysia, the Philippines, Singapore and Thailand, which is most likely to cause trade friction. China has a biased relationship with Brunei, Cambodia, Laos, Myanmar and Vietnam, which is more likely to cause trade friction. On this basis, in order to further strengthen the trade relations between PRC and the ten ASEAN countries, this paper puts forward policy suggestions on improving infrastructure, strengthening independent innovation, seeking cooperation fields and actively investing abroad.
ARTICLE | doi:10.20944/preprints202305.0674.v1
Subject: Engineering, Energy And Fuel Technology Keywords: error estimation; Korhonen network; machine learning; extraction algorithm; Volterra kernels; wind speed prediction
Online: 10 May 2023 (03:10:09 CEST)
This study aims to focus on using the Volterra series and machine learning for forecasting random and chaotic wind speed regimes, since the calm weather is mostly noticed at the local site, making the dataset selection difficult. A novel method is proposed to predict Volterra kernels up to the third order, using a forward-back propagation neural network with 12-month measurements at Fujairah site (UAE). Both daily and monthly wind speed data sets are investigated for forecasting. The three dominant hourly and daily kernels are extracted for each day and each month. Predicted future Volterra kernels are estimated from past values using both statistical analysis and individual neuro networks for each of the Volterra kernel coefficients. Due to the random nature of wind speed at the local site, a two-layer with 4 neurons per layer neuro network is used to locate the most variable and intense speed during 8-hours in the day. Forecasted wind speed is determined with errors arising from different sources such as the utilization of only 3rd-order Volterra kernels and the difficulty of machine training of the employed shallow network. Nevertheless, this work depicts a useful algorithm to forecast chaotic and random wind speed regimes. Computational time is a trade of the complexity of Volterra mathematical analysis.
ARTICLE | doi:10.20944/preprints201901.0121.v2
Subject: Computer Science And Mathematics, Applied Mathematics Keywords: ruin probability; jump-diffusion; HJB equation; Volterra equation; block-by-block method; proportional reinsurance; investments
Online: 24 January 2019 (08:52:20 CET)
In this paper, we work with a diffusion-perturbed risk model comprising a surplus generating process and an investment return process. The investment return process is of standard Black-Scholes type, that is, it comprises a single risk-free asset that earns interest at a constant rate and a single risky asset whose price process is modelled by a geometric Brownian motion. Additionally, the company is allowed to purchase noncheap proportional reinsurance priced via the expected value principle. Using the Hamilton-Jacobi-Bellman approach, we derive a second-order Volterra integrodifferential equation which we transform into a linear Volterra integral equation of the second kind. We proceed to solve this integral equation numerically using the block-by-block method for the optimal reinsurance retention level that minimizes the ultimate ruin probability. The numerical results based on light- and heavy-tailed distributions show that proportional reinsurance and investments play a vital role in enhancing the survival of insurance companies. But the ruin probability exhibits sensitivity to the volatility of the stock price.
REVIEW | doi:10.20944/preprints201809.0321.v1
Subject: Computer Science And Mathematics, Mathematics Keywords: Kirchhoff’s voltage law, Volterra integral equation, Mixed quadrature, Initial value problem, Error bound, Maclaurin's theorem
Online: 17 September 2018 (15:13:41 CEST)
In this study, a mixed rule of degree of precision nine has been developed and implemented in the field of electrical sciences to obtain the instantaneous current in the RLC- circuit for particular value .The linearity has been performed with the Volterra’s integral equation of second kind with particular kernel . Then the definite integral has been evaluated through the mixed quadrature to obtain the numerical result which is very effective. A polynomial has been used to evaluate Volterra’s integral equation in the place of unknown functions. The accuracy of the proposed method has been tested taking different electromotive force in the circuit and absolute error has been estimated.
ARTICLE | doi:10.20944/preprints202311.0107.v1
Subject: Engineering, Electrical And Electronic Engineering Keywords: Inverter-based resources; Measure-based method; Model identification; Non-linear dynamics; Power system; SINDy; Synchronous generators; System-level nonlinearity; Volterra-based nonlinearity index
Online: 1 November 2023 (17:22:36 CET)
The complexity of modern power grids, exacerbated by integrating diverse energy sources, espe-cially inverter-based resources (IBRs), presents a significant challenge to grid operation and plan-ning since linear models fail to capture the intricate IBR dynamics. This study employs the Sparse Identification of Nonlinear Dynamics (SINDy) method to bridge the gap between theoretical un-derstanding and practical implementation in power system analysis. It introduces the novel Volterra-based Nonlinearity Index (VNI) to examine system-level nonlinearity comprehensively. The distinction of dynamics into first-order linearizable terms, second-order nonlinear dynamics, and third-order noise elucidates the intricacy of power systems. The findings demonstrate a fundamental shift in system dynamics as power sources transit to IBRs, revealing system-level nonlinearity compared to module-level nonlinearity in conventional syn-chronous generators. The VNI quantifies nonlinear-to-linear relationships, enriching our comprehension of power system behavior and offering a versatile tool for distinguishing between different nonlinearities and visualizing their distinct patterns through the proposed VIN profile.