Legendre Polynomials Operational Matrix Method for Solving Fractional Partial Differential Equations with Variable Coefficients

Abstract

A numerical method for solving a class of fractional partial differential equations with variable coefficients based on Legendre polynomials is proposed. A fractional order operational matrix of Legendre polynomials is also derived. The initial equations are transformed into the products of several matrixes by using the operational matrix. A system of linear equations is obtained by dispersing the coefficients and the products of matrixes. Only a small number of Legendre polynomials are needed to acquire a satisfactory result. Results obtained using the scheme presented here show that the numerical method is very effective and convenient for solving fractional partial differential equations with variable coefficients.

Existence theory to a class of boundary value problems of hybrid fractional sequential integro-differential equations

“…This section provides some important definitions of fractional calculus [2,35,36] and results of fixed point theory [1,20,21], which is base for the forthcoming sections.…”

Existence Results to a Class of Hybrid Fractional Differential Equations

“…In this section, we recall some definitions and results of fractional calculus and hybrid fixed point theory, that are necessary for further investigation [28,29,37]. Definition 2.1.…”

“…One of the important aspect which has been greatly developed and well explored by different researchers is known as existence theory. The respective aspects have been explored for BVPs of FODEs, see for some detail [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32].…”

“…FDEs has gained considerable attention of researchers, due to its large number of applications in diverse discipline of engineering and science, such as physics, chemistry, aerodynamics, electrodynamics of complex medium, economics, control theory, signal and image processing, biophysics, blood flow phenomena, etc [33][34][35][36]. Mostly significant phenomena in fluid flow, fluid dynamic like diffusion, traffic model, electromagnetic, solid mechanics, statistical mechanics, colored noise, polarization, bioengineering, electrochemical processes and modeling of frequency dependent damping behavior of viscoelastic flow are well described by fractional differential equations [28][29][30][31][32].…”

Numerical treatment of coupled system of fractional order partial differential equations

“…[5,6,22]). If u(t, x) is a continuous function defined over a region [0, 1] × [0, 1] has bounded mixed fourth order partial derivative ∂ 2 ∂t 2 ∂x 2 u(t, x), then the Legendre expansion of the function converges uniformly to the function.…”