A collocation method for the solution of nonlinear one-dimensional parabolic equations

In this paper, we develop a collocation method based on cubic B-spline to the solution of nonlinear parabolic equation $varepsilon u_{xx}=a(x,t)u_{t}+phi(x,t,u,u_{x})$ subject to appropriate initial, and Dirichlet boundary conditions, where $varepsilon >0$ is a small constant. We developed a new two-level three-point scheme of order $O(k^2+h^2)$. The convergence analysis of the method is proved. Numerical results are given to illustrate the efficiency of our method computationally.