One-parameter orthogonal spline collocation methods for nonlinear two-dimensional Sobolev equations with time-variable delay

Abstract

This paper deals with the numerical solutions of initial–boundary value problems (IBVPs) of nonlinear two-dimensional Sobolev equations with time-variable delay. For solving this type of IBVPs, two classes of one-parameter orthogonal spline collocation (OSC) methods are constructed. For the obtained one-parameter OSC methods, their errors in L2- and H1-norm are analyzed and thus the corresponding error estimates are derived. With some numerical experiments, the computational effectiveness and theoretical accuracy of the methods are further confirmed. Moreover, a numerical comparison shows that the both kinds of one-parameter OSC methods have almost the same computational efficiency and accuracy under the same parameter and stepsizes.