Numerical Simulation for a Multidimensional Fourth-Order Nonlinear Fractional Subdiffusion Model with Time Delay

Sarita Nandal,Ahmed S Hendy,Rob H De Staelen,Mahmoud A Zaky

doi:10.3390/math9233050

Abstract

The purpose of this paper is to develop a numerical scheme for the two-dimensional fourth-order fractional subdiffusion equation with variable coefficients and delay. Using the L2−1σ approximation of the time Caputo derivative, a finite difference method with second-order accuracy in the temporal direction is achieved. The novelty of this paper is to introduce a numerical scheme for the problem under consideration with variable coefficients, nonlinear source term, and delay time constant. The numerical results show that the global convergence orders for spatial and time dimensions are approximately fourth order in space and second-order in time.

Highlights

Troca Cabella and Carlo CattaniThe theory of fractional integrals and derivatives of arbitrary real or complex orders dates back approximately three centuries
There are many difficulties to build numerical schemes for fourth-order fractional differential equations with delay due to non-locality of the problem, variable coefficients, nonlinearity, and error depends upon the history of considered problem
The above problem has variable coefficients and the exact solution is not known to us; we check the accuracy of proposed numerical scheme by means of absolute residual error function, which is a measure of how well the approximation satisfies the original nonlinear fractional differential problem given in Example Section 2.1

Summary

Introduction

The theory of fractional integrals and derivatives of arbitrary real or complex orders dates back approximately three centuries. Efficient numerical procedure, the local radial basis function created by the finite difference method, for computing the approximation solution of the time-fractional fourthorder reaction-diffusion equation in terms of the Riemann–Liouville derivative. Liu et al [38] studied and analyzed a Galerkin mixed finite element method combined with time second-order discrete scheme for solving nonlinear time fractional diffusion equation with fourth-order derivative term. There are many difficulties to build numerical schemes for fourth-order fractional differential equations with delay due to non-locality of the problem, variable coefficients, nonlinearity, and error depends upon the history of considered problem. Our target is to present high ordered difference scheme to solve the following two-dimensional time-fractional subdiffusion equation of fourth-order with variable coefficients and nonlinear source term having a delay constant.

Numerical Simulations

A Final Example

Conclusions