A trigonometric quintic B-spline collocation technique for the fifth-order KdV–Burgers–Fisher equation

Abstract The paper investigates numerical solutions to the KdV–Burgers–Fisher (KBF) equation, which models a dispersion–dissipation–reaction phenomenon. The stated equation is a mathematical structure for describing physical, chemical, or biological systems in which the dynamics of the system are shaped by the interaction of dispersion, dissipation, and reaction processes. To solve the KBF equation, a collocation method based on the finite element approach is utilized. In order to construct the approximate solutions satisfying the governing equation at collocation points, the finite element shape functions have been selected as quintic trigonometric B-spline basis functions. The application of the collocation method to the equation yields an algebraic equation system that has a well-known penta-diagonal coefficient matrix. The resulting system allows us to calculate the error norms $$L_{2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> and $$L_{\infty }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> and simulate space-time graphics of numerical solutions. As numerical examples of the KdV–Burgers–Fisher (KBF) equation, two test problems are presented to show the performance of the collocation method, while the error norms and graphs including comparison with exact solutions are used to prove the correctness and applicability of the method. Moreover, existence and uniqueness of the solutions are discussed via fixed-point theory, stability analysis which is investigated via von-Neumann technique are presented in this paper as well.