Caputo Fractional Operator and Aboodh Transform Technique of WBK Equations: Some Comparisons of Analytical and Numerical Solutions with Other Techniques

Caputo Fractional Operator and Aboodh Transform Technique of WBK Equations

Özet

Among the most important equations in the theory of shallow water waves are the Whitham–Broer–Kaup equations (WBK equations), which form a coupled system of nonlinear partial differential equations describing the propagation of shallow water waves with different dispersion relations and diffusion effects. In this paper, we combine the Aboodh transform with the residual power series method to develop a new technique called the Aboodh Residual Power Series Technique (ARPST). This technique, together with the Caputo fractional operator, is used to obtain analytical and numerical solutions of the WBK equations by calculating the coefficients of the corresponding power series. The proposed ARPST is applied to solve the WBK equations in two cases: exact initial conditions and approximate initial conditions. To demonstrate the capability, efficiency, and reliability of the proposed technique, the numerical results of the analytical solutions are presented graphically and compared with results obtained by other established methods, including the Adomian decomposition method, the optimal homotopy asymptotic method, and the Sumudu decomposition method.