Numerical methods play an important role in modern mathematical research, especially studying the symmetry analysis and obtaining the numerical solutions of fractional differential equation. In the current work, we use two numerical schemes to deal with fractional differential equations. In the first case, a combination of the group preserving scheme and fictitious time integration method (FTIM) is considered to solve the problem. Firstly, we applied the FTIM role, and then the GPS came to integrate the obtained new system using initial conditions. Figure and tables containing the solutions are provided. The tabulated numerical simulations are compared with the reproducing kernel Hilbert space method (RKHSM) as well as the exact solution. The methodology of RKHSM mainly relies on the right choice of the reproducing kernel functions. The results confirm that the FTIM finds the true solution. Additionally, these numerical results indicate the effectiveness of the proposed methods.

The partial differential equations (PDEs) describe several phenomena in wide fields of engineering and physics. The purpose of this paper is to employ the reproducing kernel Hilbert space method (RKHSM) in obtaining effective numerical solutions to nonlinear PDEs, which are arising in acoustic problems for a fluid flow. In this paper, the RKHSM is used to construct numerical solutions for PDEs which are found in physical problems such as sediment waves in plasma, sediment transport in rivers, shock waves, electric signals’ transmission along a cable, acoustic problems for a fluid flow, vibrating membrane, and vibrating string. The RKHSM systematically produces analytic and approximate solutions in the form of series. The convergence analysis and error estimations are discussed to prove the applicability theoretically. Three applications are tested to show the performance and efficiency of the used method. Computational results indicated a good agreement between the exact and numerical solutions.

Recently, a new fractional derivative operator has been introduced so that it presents the combination of the Riemann–Liouville integral and Caputo derivative. This paper aims to enhance the reproducing kernel Hilbert space method (RKHSM, for short) for solving certain fractional differential equations involving this new derivative. This is the first time that the application of the RKHSM is employed for solving some differential equations with the new operator. We illustrate the convergence analysis of the applicability and reliability of the suggested approaches. The results confirm that the RKHSM finds the true solution. Additionally, these numerical results indicate the effectiveness of the proposed method.