Three-level techniques for one-dimensional parabolic equation with nonlinear initial condition

Abstract

In this paper a boundary value problem for one-dimensional heat equation is considered under the constraint of a nonlocal initial condition. In place of the classical specification of initial data, we impose the nonlocal initial condition. Efficient higher-order algorithms are developed for solving this parabolic partial differential equation with nonstandard initial condition. Several three-level finite difference schemes are presented. These schemes are based on the modification of the centred-time centred-space explicit formula, the three-level (1,3,1) explicit technique, the sixth-order (1,3,3) explicit method and the Dufort–Frankel finite difference scheme. Numerical computations combined with a simple iteration procedure of some examples are given to demonstrate the efficiency and accuracy of the new algorithms.