A new two-level implicit scheme based on cubic spline approximations for the 1D time-dependent quasilinear biharmonic problems

Abstract

In this article, we present a new two-level implicit cubic spline numerical method of accuracy 2 in time and 4 in spatial direction for the numerical solution of 1D time-dependent quasilinear biharmonic equation subject to appropriate initial and natural boundary conditions prescribed. The easiness of the proposed numerical method lies in their 3-point discretization in which we use two points $$ x \pm (h/2) $$ and a central point ‘x’ in spatial direction. Using the continuity of the first-order derivative of cubic spline function, we derive the fourth-order accurate numerical method for the time-dependent biharmonic equation on a uniform mesh. The stability consideration of the proposed method is discussed using a model linear problem. The proposed cubic spline method successfully implements on generalized Kuramoto–Sivashinsky and extended Fisher–Kolmogorov equations. From the numerical experiments, we obtain better computational results compared to the results discussed in earlier research work.