Homotopy Analysis And Homotopy Perturbation Method For Solving Fractional Acousticswave Equation

Abstract

This study focused on solving fractional acoustic wave equations, which are essential for understanding systems characterized by long-range interactions or anomalous diffusion. Fractional differential equations (FDEs), which generalize ordinary differential equations by incorporating fractional derivatives or integrals, were the primary mathematical framework used. The Homotopy Analysis Method (HAM) and Homotopy Perturbation Method(HPM) are semi-analytical technique renowned for its capability in solving nonlinear differential equations, were employed to derive exact solutions for these FDEs. By applying both HAM and HPM exact solutions for the fractional acoustic wave equations were successfully obtained and illustrated graphically. The results demonstrated the effectiveness and efficiency of HAM and HPM in solving fractional differential equations. Homotopy Perturbation Method is only a special case of the homotopy analysis method. Both methods are in principle based on Taylor series with respect to an embedding parameter. Besides, both can give very good approximations by means of a few terms, if initial guess and auxiliary linear operator are good enough. This underscores the methods are potential and powerful tools for researchers addressing complex systems that exhibit non-standard dynamic behaviors, thereby broadening the scope of analytical techniques available for studying such phenomena. Comparative analysis is conducted between the results obtained through HAM and HPM. This comparison helps in validating the methods and showcasing their respective strengths and potential limitations