Wavelet based algorithm for numerical study of (1+2)-dimensional time fractional diffusion problems

Abdul Ghafoor,Poom Kumam,Sirajul Haq,Manzoor Hussain

doi:10.1186/s13662-020-02861-0

Abstract

An effective and robust scheme is developed for solutions of two-dimensional time fractional heat flow problems. The proposed scheme is based on two-dimensional Haar wavelets coupled with finite differences. The time fractional derivative is approximated by an L_{1}-formula while spatial part is approximated by two-dimensional Haar wavelets. The proposed methodology first converts the problem to a discrete form and then with collocation approach to a system of linear equations which is easily solvable. To check the efficiency of the scheme, two error norms, E_{infty } an E_{mathrm{rms}}, have been computed. The stability of the scheme has been discussed which is an important part of the manuscript. It is also observed that the spectral radius of the amplification matrix satisfies a stability condition. From computation it is clear that computed results are comparable with the exact solution.

Highlights

1 Introduction Fractional differential equations (FDEs) play a crucial role in modeling of different physical phenomena such as heat and mass transfer [1, 2], fluid mechanics [3], and financial theory [4, 5]. These equations arise in the form of fractional partial differential equations (FPDEs)
The above discussion about Haar wavelets is limited to integer order differential equations, this approach works for solution of fractional differential equations
Saeed and Rehman [29] applied the same technique for numerical solution of fractional partial differential equations

Summary

Introduction

Fractional differential equations (FDEs) play a crucial role in modeling of different physical phenomena such as heat and mass transfer [1, 2], fluid mechanics [3], and financial theory [4, 5]. The above discussion about Haar wavelets is limited to integer order differential equations, this approach works for solution of fractional differential equations. Lepik [24] solved fractional integral equations with the help of Haar wavelets. Li [8] applied Haar wavelets for solution of fractional ordinary differential equations (FODEs).

Results

Conclusion