A novel collocation technique for parabolic partial differential equations

Abstract

The present study focuses on an accurate, convergent, stable and efficient method for solving the parabolic Fisher’s type equation with three different cases. The method makes use of the modified quintic trigonometric B-spline collocation technique over the Crank-Nicolson scheme for spatial discretization while the time derivative is discretized by the usual finite difference method. The nonlinear terms are linearized using the Rubin-Graves type linearization process. The efficiency and accuracy of the method are examined by taking three numerical examples which show that the method presents more accurate solutions than existing solutions. The convergence analysis and von Neumann stability of the method show that the method is fourth-order convergent in space and the discretized system is unconditionally stable. Computational efficiency of the method is investigated by small values of CPU-time. The proposed method can be implemented to solve the fourth-order partial differential equations in higher dimensional including physical, mechanical, or biophysical effects.

Keywords

• Fisher’s type equation
• Collocation technique
• QTB-spline basis functions
• Convergence analysis
• Stability analysis.