Galerkin approximation for multi-term time-fractional differential equations

Abstract

Fractional differential equations (FDEs) are utilized as a precise model for describing a wide range of biological and physical processes, benefiting from the inherent symmetry feature in natural phenomena. The introduction of multi-term time-fractional order equations serves to tackle intricate phenomena spanning various physical domains. This study solves multi-term time-fractional DEs using the quadratic B-spline Galerkin method. In Galerkin approach, the quadratic B-spline is utilized as both a test and a basis function. The fractional portion of time is evaluated using the Caputo definition. In addition, the Gauss quadrature formula is used to evaluate the integration of complex function which enhance the accuracy. Lax-Richtmyer's stability criterion is employed to evaluate the stability of the proposed scheme. The E2 and E∞ norms are computed and displayed in the tables to illustrate the robustness and efficiency of the method. We compute simulation for various nodal points as well as various mesh sizes to analyze the solution behavior. The obtained results are compared with the exact and available results, indicating that the proposed scheme is more efficient. The graphical solution of each example is also demonstrated which verifies that exact and approximate solutions graphs closed with each other.

Keywords

• Galerkin method
• Quadratic B-spline
• Gauss quadrature