An analytic approach to a class of fractional differential‐difference equations of rational type via symbolic computation

Abstract

Communicated by S. G. GeorgievFractional derivatives are powerful tools in solving the problems of science and engineering. In this paper, an analytical algorithm for solving fractional differential-difference equations in the sense of Jumarie's modified Riemann-Liouville derivative has been described and demonstrated. The algorithm has been tested against time-fractional differentialdifference equations of rational type via symbolic computation. Three examples are given to elucidate the solution procedure. Our analyses lead to closed form exact solutions in terms of hyperbolic, trigonometric, and rational functions, which might be subject to some adequate physical interpretations in the future.

Variational approach, soliton solutions and singular solitons for new coupled ZK system

“…The domain restriction that is given by (25) must also hold in order for the singular soliton to exist. The 3D graphs in Fig.…”

“…So we should search for a mathematical algorithm to discover the exact solutions of nonlinear partial differential equations. In recent years, powerful and efficient methods explored to find analytic solutions of nonlinear equations have drawn a lot of interest by a variety of scientists, such as Adomian decomposition method [2], the homotopy perturbation method [3,4], some new asymptotic methods searching for solitary solutions of nonlinear differential equations, nonlinear differential-difference equations and nonlinear fractional differential equations using the parameter-expansion method, the Yang-Laplace transform, the Yang-Fourier transform and ancient Chinese mathematics [4], the variational iteration method [5,6] which is used to introduce the definition of fractional derivatives [7,4], the He's variational approach [8], the extended homoclinic test approach [9,10], homogeneous balance method [11][12][13][14], Jacobi elliptic function method [15][16][17][18], Băclund transformation [19,20], G ′ /G expansion method for nonlinear partial differential equation [21,22], and fractional differential-difference equations of rational type [23][24][25] It is important to point out that a new constrained variational principle for heat conduction is obtained recently by the semi-inverse method combined with separation of variables [26], which is exactly the same with He-Lee's variational principle [27]. A short remark on the history of the semi-inverse method for establishment of a generalized variational principle is given in [28].…”

Analytical Insights into a Generalized Semidiscrete System with Time-Varying Coefficients: Derivation, Exact Solutions, and Nonlinear Soliton Dynamics

“…In many research fields, more and more attention has been paid to fractional-order models [60][61][62][63][64][65][66][67][68][69][70][71][72][73][74][75][76]. Recently, Aslan [77][78][79][80] has successfully extended analytical methods-combined symbolic computation to solve fractional semidiscrete equations. How to extend the method used in this paper to such fractional semidiscrete equations is also worthy of studying.…”

Exact solutions of the space-time fractional equal width equation

“…As a result, many effective methods have been established to solve PDE exactly. For example, lumped Galerkin methods based on B-splines, and also these methods were implemented for fractional differential-difference equations in quoted equation [8][9][10].…”