Analysis of a robust implicit scheme for space–time fractional stochastic nonlinear diffusion wave model

Abstract

The current paper develops and analyzes a numerical scheme for the space–time fractional stochastic nonlinear diffusion wave equations. The implicit scheme is based on the matrix transform technique for discretizing the Riesz-space fractional derivative, $(3-\alpha )$-order approximation to the Caputo-fractional derivative in time and Taylor's series method to linearize the nonlinear source term, and has been successfully applied to solve a class of nonlinear fractional diffusion wave equation. We prove that the implicit scheme is convergent with β-order in space and $(3-\alpha )$ order in time, respectively. The optimum error estimates in the temporal-spatial direction and unconditional stability of the implicit scheme have been theoretically investigated. Moreover, several specific numerical experiments confirm the consistency and high efficacy of the provided algorithms, which minimizes the computational costs.

Keywords:

• Stochastic space–time fractional model
• unconditional stability
• convergence analysis
• Caputo-fractional derivative
• Riesz fractional derivative

2020 Mathematics subject classifications:

• 65C30
• 65D15
• 65D30
• 65G50
• 65J15

Acknowledgments

The authors sincerely thank the Editor for taking time to handle the paper and two reviewers for carefully reading the paper and their constructive comments and suggestions that really improved the quality of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author greatly acknowledges the financial support from Department of Science and Technology (DST), Government of India (Fellowship no: DST/INSPIRE Fellowship/2019/IF190322) for this work. The third author (corresponding author) acknowledges the financial support provided by Science and Engineering Research Board (SERB), India, under the MATRICS project with sanction order no. MTR/2021/000124.