Document Type : Research Paper

**Authors**

Mathematics Department, Tabriz Branch, Islamic Azad University, Tabriz, Iran.

**Abstract**

This paper presents the development of a series of fractional multi-step linear finite difference methods (FLMMs) designed to address fractional multi-delay pantograph differential equations of order $0 < \alpha \leq 1$. These $p$ FLMMs are constructed using fractional backward differentiation formulas of first and second orders, thereby facilitating the numerical solution of fractional differential equations. Notably, we employ accurate approximations for the delayed components of the equation, guaranteeing the retention of stability and convergence characteristics in the proposed $p$-FLMMs. To substantiate our theoretical findings, we offer numerical examples that corroborate the efficacy and reliability of our approach.

**Keywords**

**Main Subjects**

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July 2024

Pages 511-522

**Receive Date:**16 August 2023**Revise Date:**15 November 2023**Accept Date:**20 November 2023