Abstract
In this paper, we construct and analyze a linearized finite difference/Galerkin–Legendre spectral scheme for the nonlinear multiterm Caputo time fractional-order reaction-diffusion equation with time delay and Riesz space fractional derivatives. The temporal fractional orders in the considered model are taken as 0 < β 0 < β 1 < β 2 < ⋯ < β m < 1 . The problem is first approximated by the L 1 difference method on the temporal direction, and then, the Galerkin–Legendre spectral method is applied on the spatial discretization. Armed by an appropriate form of discrete fractional Grönwall inequalities, the stability and convergence of the fully discrete scheme are investigated by discrete energy estimates. We show that the proposed method is stable and has a convergent order of 2 − β m in time and an exponential rate of convergence in space. We finally provide some numerical experiments to show the efficacy of the theoretical results.
Highlights
Fractional-order partial differential equations have evolved into powerful tools for describing a wide range of anomalous behavior and complex systems in natural science and engineering [1,2,3,4,5,6,7,8]
Ding and Jiang [23] used the technique of spectral representation of the fractional Laplacian operator in order to provide the analytical solutions for the multiterm time-space fractional advectiondiffusion equations with mixed boundary conditions
Zaky et al [29] presented a discrete fractional Grönwall inequality that is consistent with the L2 − 1σ to cope with the analysis of multiterm time-fractional partial differential equations
Summary
Fractional-order partial differential equations have evolved into powerful tools for describing a wide range of anomalous behavior and complex systems in natural science and engineering [1,2,3,4,5,6,7,8]. Through the use of a Journal of Function Spaces domain decomposition technique, they were able to derive the linear and nonlinear diffusion-wave equations of the fractional order. We consider the numerical approximations to the following generalized nonlinear multiterm time-space fractional reaction-diffusion equations with delay: m. The main aim of this work is to construct and analyze an efficient linearized numerical scheme for the nonlinear multiterm Riesz space and Caputo time fractional reactiondiffusion problem with fixed delay. The main challenges of the considered work are represented in how to numerically approximate the time Caputo fractional derivative, Riesz space fractional derivatives, and the time delay to produce an easy-to-implement and consistent numerical scheme Overcoming all of these challenges to yield a hybrid linear numerical scheme is a first target.
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