Abstract
The purpose of this article is to present a technique for the numerical solution of Caputo time fractional superdiffusion equation. The central difference approximation is used to discretize the time derivative, while non-polynomial quintic spline is employed as an interpolating function in the spatial direction. The proposed method is shown to be unconditionally stable and O(h^{4}+Delta t^{2}) accurate. In order to check the feasibility of the proposed technique, some test examples have been considered and the simulation results are compared with those available in the existing literature.
Highlights
In this article, we consider the following time fractional fourth order superdiffusion equation [1]:∂αy ∂4y ∂tα + γ ∂z4 = f (z, t),0 ≤ z ≤ L, 0 ≤ t ≤ T, (1)with the initial/end conditions y(z, 0) = φ(z), yt(z, 0) = ψ(z), y(0, t) = y(L, t) = 0, yzz(0, t) = yzz(L, t) = 0, where α ∈ (1, 2] denotes the order of time fractional derivative, γ is a constant, and φ(z) is continuous on [0, L].Fourth order time fractional partial differential equations (PDEs) arise in mathematical modeling of several plate-like objects [2]
Most of the analytical techniques for solving fractional order PDEs are based on Laplace and Fourier transforms, while others involve the separation of variables technique [3, 4]
Spline functions have been frequently employed for the numerical solution of fractional order PDEs
Summary
We consider the following time fractional fourth order superdiffusion equation [1]:. Siddiqi and Arshed [13] used the quintic B-spline collocation method for numerical solution of time fractional fourth order PDEs. In [14], the authors introduced new fractional order spline functions to study the numerical solution of fractional Bagely–Torvik equation. Khalid et al [18] utilized the non-polynomial quintic spline collocation method to explore the numerical solution of fourth order fractional boundary value problem, involving product terms. In [19], Amin et al employed the quintic non-polynomial spline collocation scheme for solving time fractional fourth order PDEs. There are several techniques to deal with the fractional differentiation but Riemann–. This paper aims to develop a spline collocation approach for numerical solution of fourth order time fractional superdiffusion problem.
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