title: A bivariate Chebyshev spectral collocation quasilinearization method for nonlinear evolution parabolic equations.
abstract: This paper presents a new method for solving higher order nonlinear evolution partial differential equations (NPDEs). The method combines quasilinearisation, the Chebyshev spectral collocation method, and bivariate Lagrange interpolation. In this paper, we use the method to solve several nonlinear evolution equations, such as the modified KdV-Burgers equation, highly nonlinear modified KdV equation, Fisher's equation, Burgers-Fisher equation, Burgers-Huxley equation, and the Fitzhugh-Nagumo equation. The results are compared with known exact analytical solutions from literature to confirm accuracy, convergence, and effectiveness of the method. There is congruence between the numerical results and the exact solutions to a high order of accuracy. Tables were generated to present the order of accuracy of the method; convergence graphs to verify convergence of the method and error graphs are presented to show the excellent agreement between the results from this study and the known results from literature.
authors.affiliation: School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Scottsville, Pietermaritzburg 3209, South Africa.
authors.forename: S S
authors.initials: SS
authors.lastname: Motsa
authors.affiliation: School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Scottsville, Pietermaritzburg 3209, South Africa.
authors.forename: V M
authors.initials: VM
authors.lastname: Magagula
authors.affiliation: School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Scottsville, Pietermaritzburg 3209, South Africa.
authors.forename: P
authors.initials: P
authors.lastname: Sibanda
mesh_terms: D000465:Algorithms; D003198:Computer Simulation; D008432:Mathematical Computing; D008433:Mathematics / Q000379:methods; D017711:Nonlinear Dynamics; D009716:Numerical Analysis, Computer-Assisted; D015203:Reproducibility of Results