Modified Galerkin algorithm for solving multitype fractional differential equations

Abstract

The primary point of this manuscript is to dissect and execute a new modified Galerkin algorithm based on the shifted Jacobi polynomials for solving fractional differential equations (FDEs) and system of FDEs (SFDEs) governed by homogeneous and nonhomogeneous initial and boundary conditions. In addition, we apply the new algorithm for solving fractional partial differential equations (FPDEs) with Robin boundary conditions and time‐fractional telegraph equation. The key thought for obtaining such algorithm depends on choosing trial functions satisfying the underlying initial and boundary conditions of such problems. Some illustrative examples are discussed to ascertain the validity and efficiency of the proposed algorithm. Also, some comparisons with some other existing spectral methods in the literature are made to highlight the superiority of the new algorithm.

Application of the modified operational matrices in multiterm variable‐order time‐fractional partial differential equations

“…where u(x, t) is the unknown function, f 0 (x), f 1 (x), 0 (t), 1 (t) are known functions defined over the interval = [0, 1] × [0, ∞), and F is a nonlinear function. Also, the symbol D (x,t) t denotes the variable-order Caputo fractional derivative operator with respect to variable t, which is defined by the following formula 10,[26][27][28] :…”

A new mixed discontinuous Galerkin method for the electrostatic field

“…In this section, we present numerical experiments obtained for the three-dimensional problem (2). For simplicity, we suppose that the coefficients ε and μ are constants, and so μ ≡ ε ≡ 1.…”

“…Numerical and analytical methods play an important role for solving many mathematical models arising in physics and applied sciences. Indeed, several researchers use numerical and analytical methods for solving some scientific problems (see, e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13]). As in [1], the authors have generalized F ξ -calculus for fractals embedding in R 3 , and in [12] an efficient computational technique for fractal vehicular traffic flow is presented.…”

Jacobi Spectral Galerkin Method for Distributed-Order Fractional Rayleigh–Stokes Problem for a Generalized Second Grade Fluid

“…In this paper, we use a non-local representation of the solution of the distributed-order time-fractional Rayleigh-Stokes problem to introduce spectral solutions. The spectral and pseudospectral methods are well-known for their high accuracy and have been used extensively in scientific computation, see [33][34][35][36][37][38][39][40][41] and the references therein. The main contribution of this paper is to develop Jacobi-Galerkin algorithms for solving the multidimensional distributed-order time-fractional Rayleigh-Stokes problem (1).…”