An efficient numerical approach to solve a class of variable‐order fractional integro‐partial differential equations

Abstract

The main purpose of this work is to investigate an initial boundary value problem related to a suitable class of variable order fractional integro-partial differential equations with a weakly singular kernel. To discretize the problem in the time direction, a finite difference method will be used. Then, the Sinc-collocation approach combined with the double exponential transformation is employed to solve the problem in each time level. The proposed numerical algorithm is completely described and the convergence analysis of the numerical solution is presented. Finally, some illustrative examples are given to demonstrate the pertinent features of the proposed algorithm.

A novel higher-order numerical method for parabolic integro-fractional differential equations based on wavelets and $L2$-$1_\sigma$ scheme

“…Further, C 4,3 x,t ( Ω, R) represents the space of all real-valued functions whose fourth and third derivatives with respect to x and t, respectively, exist and are continuous. Then, for any V ∈ C 4,3 x,t ( Ω, R), we define…”

“…Then, from (6), we have U (x, t) = F F F U(x, t), for (x, t) ∈ Ω. In order to show F F F to be compact, we need to prove {F F F un} is bounded, and equicontinuous [18] on C 4,3 x,t ( Ω, R) for any bounded sequence {un} in C 4,3 x,t ( Ω, R). Then, the Arzela-Ascoli's theorem [27] confirms the existence of a uniformly convergent subsequence of {F F F un} with respect to the norm ∥ • ∥∞.…”

“…Notice that the functions p, q, r, g, f and the kernel K are sufficiently smooth function, and {Un} is bounded in C 4,3 x,t ( Ω, R). Hence, there exists a real number M 1 > 0 such that ∥F F F Un∥∞ ≤ M 1 for all n ∈ N, (x, t) ∈ Ω.…”

“…Further, let ϵ > 0 be given. For any Un ∈ C 4,3 x,t ( Ω, R), consider where x < θ i < y for i = 1, 2, . .…”

“…A few semi-analytical approaches, such as the Adomian decomposition method [31], the homotopy analysis method [1], and the variational iteration method [14], can be used to solve FOIDEs. In addition to these, many equivalent articles are available in the literature on the numerical solutions of FDEs as well as FOIDEs based on finite difference methods [3,40], and also on finite element methods [8]. A finite difference solution for an FDE/FOIDE at a particular time level depends on solutions at all previous time levels, however, this is not the case for integer order differential systems.…”