Space–time localized radial basis function collocation method for solving parabolic and hyperbolic equations

Abstract

A radial basis collocation method, to solve parabolic and hyperbolic equations, based on the local space–time domain formulation is developed and presented in this paper. The method is different from those that approximate the time derivative using different formulas such as the implicit, explicit, method of lines, or other numerical methods. Considering a partial differential equation with d spatial dimensions, our technique solves the problem as a (d+1)-dimensional one without distinguishing between space and time variables, and the collocation points have both space and time coordinates. The parabolic equation is solved using the governing domain equation as a condition on the boundary characterized by the final time T. The hyperbolic equation is solved using two different methods. The first one is based on adapting the technique used for solving parabolic equations. The second one is a new technique that looks at the problem as an ill-posed one with incomplete boundary condition data at the final time T of the space–time domain. The accuracy of our proposed method is demonstrated through different examples in one-, two- and three-dimensional spaces on regular and irregular domains.