Solving fractional physical evolutionary wave equations using advanced techniques

In this study, two high-accuracy approaches, namely Mohand new iterative method (MNIM) and Mohand residual power series method (MRPSM) are used to analyze various types of linear and nonlinear fractional differential equations (FDEs) with the help of Caputo-fractional operator. These approaches are created by combining the Mohand transform (MT) with the traditional new iterative method (NIM) and residual power series method (RPSM). These methods derive highly accurate approximations for different linear and nonlinear FDEs. To evaluate the efficiency, accuracy, stability, and convergence of the derived approximations using the given approaches, we numerically compare all derived approximations to the exact results for the integer cases (i.e., at fractional parameter equates to one) and calculate the absolute error of these approximations. Furthermore, we derive the residual error function for all the derived approximations and calculate this error’s numerical value along the whole study domain. The proposed methods exhibit enhanced computational efficiency and accuracy relative to conventional techniques, substantiated by comprehensive comparisons illustrated in tables and figures. This study highlights the efficacy of MNIM and MRPSM in resolving the intricacies of fractional calculus and nonlinear dynamics through explicit functional solutions. The results markedly enhance the utilization of fractional calculus in addressing nonlinear fractional partial differential equations (FPDEs) and create new opportunities for both theoretical and applied study in this field.