Abstract
In this paper, two-dimensional Genocchi polynomials and the Ritz–Galerkin method were developed to investigate the Fractional Diffusion Wave Equation (FDWE) and the Fractional Klein–Gordon Equation (FKGE). A satisfier function that satisfies all the initial and boundary conditions was used. A linear system of algebraic equations was obtained for the considered equation with the help of two-dimensional Genocchi polynomials along with the Ritz–Galerkin method. The FDWE and FKGE, including the nonlinear case, were reduced to solve the linear system of the algebraic equation. Hence, the proposed method was able to greatly reduce the complexity of the problems and provide an accurate solution. The effectiveness of the proposed technique is demonstrated through several examples.
Highlights
This paper emphasises on solving two types of fractional partial differential equations, namelyFractional Diffusion Wave Equation (FDWE) and Fractional Klein–Gordon Equation (FKGE).The Fractional Diffusion Wave Equation (FDWE) is as follows:∂α u( x, t) ∂u( x, t) ∂2 u( x, t) + = + A( x, t), α ∂t ∂t20 < x < L, 0 < t < T
This proposed method is able to reduce the FDWE and FGKE to only solve the linear system of algebraic equations
This paper comprises the use of two-dimensional Genocchi polynomials along with the Ritz–Galerkin method and a satisfier function to provide the approximate solutions of FKGE and FDWE
Summary
This paper emphasises on solving two types of fractional partial differential equations, namely. In [9], fractional reduced differential transform method was applied to solve time-fractional order linear and nonlinear Klein–Gordon equations. The modified trial equation method had been applied to obtain the solution for some nonlinear fractional differential equations which include FKGE [11]. Some of the applications using Genocchi polynomials to solve differential equation problems are shown in [16], and [17,18,19] This proposed method is able to reduce the FDWE and FGKE to only solve the linear system of algebraic equations. This paper comprises the use of two-dimensional Genocchi polynomials along with the Ritz–Galerkin method and a satisfier function to provide the approximate solutions of FKGE and FDWE.
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