Numerical algorithms for multidimensional time-fractional wave equation of distributed-order with a nonlinear source term

Jiahui Hu,Yufeng Nie,Jungang Wang

doi:10.1186/s13662-018-1817-2

Abstract

Fractional differential equations (FDEs) of distributed-order are important in depicting the models where the order of differentiation distributes over a certain range. Numerically solving this kind of FDEs requires not only discretizations of the temporal and spatial derivatives, but also approximation of the distributed-order integral, which brings much more difficulty. In this paper, based on the mid-point quadrature rule and composite two-point Gauss–Legendre quadrature rule, two finite difference schemes are established. Different from the previous works, which concerned only one- or two-dimensional problems with linear source terms, time-fractional wave equations of distributed-order whose source term is nonlinear in two and even three dimensions are considered. In addition, to improve the computational efficiency, the technique of alternating direction implicit (ADI) decomposition is also adopted. The unique solvability of the difference scheme is discussed, and the unconditional stability and convergence are analyzed. Finally, numerical experiments are carried out to verify the effectiveness and accuracy of the algorithms for both the two- and three-dimensional cases.

Highlights

The idea of distributed-order differential equation was first presented by Caputo for modeling the stress–strain behavior of an anelastic medium in [5] in the 1960s
Unlike the differential equations with the single-order fractional derivative and those with sums of fractional derivatives, i.e., multi-term Fractional differential equations (FDEs), the distributed-order differential equations are derived by integrating the order of differentiation over a given range [1]
In [35], considering the optimal control problems with dynamics described by ordinary distributed-order fractional differential equations, the generalized necessary conditions were derived

Summary

Introduction

The idea of distributed-order differential equation was first presented by Caputo for modeling the stress–strain behavior of an anelastic medium in [5] in the 1960s. Ye et al derived and analyzed a compact difference scheme for a distributed-order time-fractional wave equation in [32]. We propose efficient finite difference schemes for solving the twoand three-dimensional time-fractional wave equations of distributed-order, respectively, where a nonlinear source term is considered. Based on the weighted and shifted Grünwald–Letnikov formula, they developed two ADI difference schemes for solving the two-dimensional time distributed-order wave equations [15]. Abbaszadeh et al solved the two-dimensional distributed order time-fractional diffusion-wave equation by combining the ADI approach with the interpolating element-free Galerkin method, where the time derivatives was discretized by a finite difference scheme [1]. Numerical approximations for multidimensional time distributed-order diffusion-wave equations with nonlinear source terms have not been considered yet, including two- and three-dimensional cases.

Convergence

Description of the ADI scheme

Findings

Conclusion