An Innovative Approach to Nonlinear Fractional Shock Wave Equations Using Two Numerical Methods

Abstract

In this research, we propose a combined approach to solving nonlinear fractional shock wave equations using an Elzaki transform, the homotopy perturbation method, and the Adomian decomposition method. The nonlinear fractional shock wave equation is first transformed into an equivalent integral equation using the Elzaki transform. The homotopy perturbation method and Adomian decomposition method are then utilized to approximate the solution of the integral equation. To evaluate the effectiveness of the proposed method, we conduct several numerical experiments and compare the results with existing methods. The numerical results show that the combined method provides accurate and efficient solutions for nonlinear fractional shock wave equations. Overall, this research contributes to the development of a powerful tool for solving nonlinear fractional shock wave equations, which has potential applications in many fields of science and engineering. This study presents a solution approach for nonlinear fractional shock wave equations using a combination of an Elzaki transform, the homotopy perturbation method, and the Adomian decomposition method. The Elzaki transform is utilized to transform the nonlinear fractional shock wave equation into an equivalent integral equation. The homotopy perturbation method and Adomian decomposition method are then employed to approximate the solution of the integral equation. The effectiveness of the combined method is demonstrated through several numerical examples and compared with other existing methods. The results show that the proposed method provides accurate and efficient solutions for nonlinear fractional shock wave equations.