Computational study of time-fractional non-linear Kawahara equations using Quintic B-spline and Galerkin’s method

This study presents two numerical methods focused on Quintic B-spline (QBS) and Galerkin finite element method (GFEM) for solving time-fractional Kawahara equations. The QBS is utilized as both the basis and test function in the FEM approach. We apply Caputo formula with quadrature rule for evaluation of temporal fractional part. The QBS and GFEM formulation are used to approximate the space functions and their derivatives. Furthermore, a four-point Gauss Legendre quadrature is employed to evaluate the source term in the GFEM. The efficiency and accuracy of the proposed scheme are evaluated using the $E_2$ and $E_{\infty }$ norms. Additionally, Fourier stability analysis is conducted, and it is revealed that the method exhibits unconditional stability. The results, presented in the form of tables and graphs to demonstrate the effectiveness of the scheme.