Numerical approximation of the time-fractional regularized long-wave equation emerging in ion acoustic waves in plasma

This work accomplished a novel approximate solution of the time-fractional regularized long-wave (TFRLW) equation. This equation is an appropriate mathematical model in physical sciences that designates the nature of ion acoustic waves in plasma and waves of shallow water. A cubic B-spline (CBS) collocation procedure was used for the spatial discretization, offering greater flexibility and accuracy compared to traditional spline methods. For time discretization, the finite difference method was used, ensuring computational efficiency, while the time-fractional derivative was settled by Caputo's definition. The Rubin-Graves linearization procedure was involved to handle the nonlinear term. To demonstrate the possessions of different constraints and variables on the displacement, the approximate solutions were shown in tabular as well as graphical forms. The method's unconditional stability was confirmed through a detailed von Neumann stability analysis, making it particularly robust for long-term simulations. The order of convergence was also estimated numerically. Three invariant capacities analogous to mass, momentum, and energy were assessed for further justification. Obtained solutions established the exactitude and efficiency of the anticipated method. Furthermore, unlike many existing methods, this approach can be tailored to handle the complexity of higher-order equations while maintaining stability and accuracy over large-scale problems.