A numerical method for solving distributed-order multi-term time-fractional telegraph equations involving Caputo and Riesz fractional derivatives

Abstract

This paper introduces a robust distributed-order time-fractional telegraph model, incorporating Caputo time- and Riesz space-fractional derivatives. The spatial Riesz derivative is discretized using an optimized finite difference method. For the distributed-order fractional operator, the midpoint rule was first used to approximate the integral with respect to the order distribution, followed by the application of a finite difference scheme to approximate the Caputo time-fractional derivative. The method’s flexibility and high accuracy make it a valuable tool for modeling and simulating these systems, providing insights into the behavior of fractional-order systems with both temporal and spatial fractional effects. Additionally, the proposed approach outperforms existing numerical methods in terms of both precision and computational efficiency, making it highly applicable for real-world problems requiring accurate and efficient solutions. A comprehensive analysis of convergence and stability was conducted to validate the proposed numerical method. To demonstrate its effectiveness, several numerical simulations were performed, revealing the method’s exceptional accuracy and computational efficiency. Furthermore, a comparison with existing numerical approaches from the literature is provided, highlighting the proposed method’s superior performance in both precision and practical applicability.

Keywords

Distributed-order / Finite difference method / Fractional derivative / Riesz fractional derivative / Stability analysis / Telegraph equations

Funding

The work was supported by the University of Tabriz, Iran (Grant No. 2070).

Conflict of interest

The authors declare that they have no competing interests.

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