The Local Fractional Mamadu Decomposition Method for Solving Singularly Perturbed Fractional Telegraph Equations

Abstract

Fractional partial differential equations have gained significant attention in recent years due to their ability to model complex physical phenomena involving memory effects and anomalous transport processes. The Mamadu Transform provides an effective analytical framework for solving fractional partial differential equations, including the fractional telegraph equation. This article discusses an efficient computational scheme based on the Local Fractional Mamadu Decomposition Method (LFMDM) for solving singularly perturbed fractional telegraph equations involving Caputo fractional derivatives. The proposed method is based on the application of the Mamadu transform for simplifying the temporal component of the equation and eigen function expansion in space, which reduces the equation into a set of decoupled fractional ordinary differential equations. The analytical solutions of the fractional ordinary differential equations are obtained using Mittag-Leffler functions and series solutions for the particular solutions. The numerical results for various values of perturbation and fractional parameters are obtained and compared with the exact solutions and results obtained by using the Laplace Fractional Decomposition (LFD) method. It is found that the results obtained by using the proposed method have high accuracy with absolute errors of order 10-4 , making it more efficient compared to the LFD method, particularly for large time values and memory effects. The proposed method is found to be more efficient and effective for solving singularly perturbed fractional telegraph equations and is applicable for simulating complex phenomena involving wave-diffusion and memory effects.