New group iterative schemes in the numerical solution of the two-dimensional time fractional advection-diffusion equation

Alla Tareq Balasim,Norhashidah Hj Mohd Ali,Fawang Liu

doi:10.1080/23311835.2017.1412241

Abstract

Numerical schemes based on small fixed-size grouping strategies have been successfully researched over the last few decades in solving various types of partial differential equations where they have been proven to possess the ability to increase the convergence rates of the iteration processes involved. The formulation of these strategies on fractional differential equations, however, is still at its infancy. Appropriate discretization formula will need to be derived and applied to the time and spatial fractional derivatives in order to reduce the computational complexity of the schemes. In this paper, the design of new group iterative schemes applied to the solution of the 2D time fractional advection-diffusion equation are presented and discussed in detail. The Caputo fractional derivative is used in the discretization of the fractional group schemes in combination with the Crank–Nicolson difference approximations on the standard grid. Numerical experiments are conducted to determine the effectiveness of the proposed group methods with regard to execution times, number of iterations, and computational complexity. The stability and convergence properties are also presented using a matrix method with mathematical induction.The numerical results will be proven to agree with the theoretical claims.

Highlights

Fractional calculus, which is the calculus of integrals and derivatives in random order, dates back as far as the more popular integer calculus, and has been gaining significant interest over the past few years with its history and development being explored in detail by Oldham and Spanier (1974), Miller and Ross (1993), Samko, Kilbas, and Marichev (1993) and Podlubny (1998)
Since there are at most no exact solutions to the majority of fractional differential equations, it is necessary to resort to approximation and numerical methods (Abdelkawy, Zaky, Bhrawy, & Baleanu, 2015; Balasim & Ali, 2015; Baleanu, Agheli, & Al Qurashi, 2016; Bhrawy & Baleanu, 2013)
There has been an influx of numerical methods development for solving various types of Fractional differential equations (FDEs) (Agrawal, 2008; Ali, Abdullah, & Mohyud-Din, 2017; Chen, Deng, & Wu, 2013; Chen & Liu, 2008; Chen, Liu, Anh, & Turner, 2011; Chen, Liu, & Burrage, 2008; Leonenko, Meerschaert, & Sikorskii, 2013; Li, Zeng, & Liu, 2012; Liu, Zhuang, Anh, Turner, & Burrage, 2007; Shen, Liu, & Anh, 2011; Sousa & Li, 2015; Su, Wang, & Wang, 2011; Uddin & Haq, 2011; Zhang, Huang, Feng, & Wei, 2013; Zheng, Li, & Zhao, 2010; Zhuang, Gu, Liu, Turner, & Yarlagadda, 2011; Zhuang, Liu, Anh, & Turner, 2009)

Summary

Introduction

Fractional calculus, which is the calculus of integrals and derivatives in random order, dates back as far as the more popular integer calculus, and has been gaining significant interest over the past few years with its history and development being explored in detail by Oldham and Spanier (1974), Miller and Ross (1993), Samko, Kilbas, and Marichev (1993) and Podlubny (1998). In recent decades, grouping strategies have been proven to possess characteristics that are able to reduce the spectral radius of the generated matrix resulting from the finite difference discretization of the differential equation, and increase the convergence rates of the iterative algorithms They have been shown to reduce the computational timings compared to their point wise counterparts in solving several types of partial differential equation (Ali & Kew, 2012; Evans & Yousif, 1986; Kew & Ali, 2015; Ng & Ali, 2008; Othman & Abdullah, 2000; Tan, Ali, & Lai, 2012; Yousif & Evans, 1995). Using the Caputo time fractional approximation (3) at the left-hand side of (1) and the second order Crank–Nicolson difference scheme with 2h-spaced points at the right hand side of (1), the following approximation formula is obtained: ax uki−+21,j −2uki,+j 1 +uki++21,j. Direct h-spaced rotated points Direct h-spaced standard points Total direct points (m+1)

Convergence analysis

Findings

Conclusion