An efficient higher-order trigonometric cubic B-spline collocation method for time-fractional Burgers equations

This manuscript is devoted to investigate the numerical solutions of the nonlinear time-fractional Burgers equation representing a significant extension of the classical Burgers equation to fractional derivative. For this purpose, an efficient higher-order trigonometric cubic B-spline collocation method, which is based on finite element analysis, is presented and used to achieve the aim of this work. While obtaining the numerical solutions of the mentioned equation, the discretization of the spatial part is performed via the Crank-Nicolson approach and the time derivative is performed in Caputo sense and the discretization of the time derivative is made by L1 algorithm. Also, the nonlinear term seen in the Burgers equation is linearized through the use of the Rubin-Graves linearization technique. Consequently, the performing of the collocation method is resulted to obtain a numerical scheme which is producing an algebraic system being solved by iteratively. The stability of the numerical scheme is investigated using the von-Neumann stability criteria. Three test problems are considered to confirm the validity, accuracy and efficiency of the method. The error between numerical solutions and exact ones is measured with the norms L₂ and L_∞ Comparison results are presented by tables, the behaviour of the numerical solutions and the harmony with the exact solutions are depicted with graphs as well.