Efficient numerical approaches for time fractional good Boussinesq equation

Purpose The purpose of this study is to develop and evaluate the effectiveness of Radial Basis Function (RBF) collocation and quintic B-spline methods for solving the fractionally differentiable Good Boussinesq equation in the Caputo sense. By comparing numerical results with the exact solution and analyzing error norms, the study aims to demonstrate the accuracy and computational efficiency of the proposed methods for solving complex fractional differential equations (FDEs). Design/methodology/approach This study employs RBF collocation and quintic B-spline methods as numerical techniques to solve the Good Boussinesq equation with fractional derivatives defined in the Caputo sense. The accuracy and efficiency of these methods are systematically evaluated using $L_2$ and $L_\infty$ error norms. Numerical solutions are computed and compared against the exact solution, with results presented through detailed tables and graphical illustrations to validate the methods' performance. Findings The study finds that both the RBF collocation and quintic B-spline methods deliver highly accurate solutions for the fractionally differentiable Good Boussinesq equation. Error analyses using $L_2$ and $L_\infty$ norms confirm their precision. Additionally, the methods demonstrate strong computational efficiency, making them practical for solving complex FDEs. Comparative results show close agreement with the exact solution, validating the reliability of both approaches. Overall, these methods offer robust and effective numerical tools for fractional PDE problems. Originality/value This study is the first to apply both the RBF collocation and quintic B-spline methods specifically to the fractionally differentiable Good Boussinesq equation in the Caputo sense. By introducing these novel numerical approaches to this class of FDEs, the research expands the available toolkit for solving such complex problems. The demonstrated accuracy and computational efficiency highlight the value of these methods as reliable and practical solutions, contributing original insights and techniques to the field of fractional partial differential equations.