Application of Cubic B-Spline Functions in Galerkin Finite Element Method for Solving Second Order Sub-Diffusion Equation

Abstract

 In this study, the Galerkin finite element method (FEM) is applied to find the numerical solutions of the second-order sub-diffusion equation. The proposed approach employs cubic B-spline functions as both the trial and test functions. The weak formulation of the governing equation is developed, and the connection between the global and local coordinate systems is established through a suitable transformation. For time discretization, the standard finite difference formulation is used for the time derivative, while the Crank-Nicolson scheme is applied to approximate the unknown functions. A stability analysis is performed to verify the robustness of the scheme and to ensure that no numerical errors grow during computations. Additionally, error estimates are derived to assess the accuracy of the proposed method. The efficiency and reliability of the developed scheme are demonstrated by solving several test problems. The computed results are presented in both graphical and tabular forms, showing good agreement with the exact or reference solutions.