Using Newton basis functions for solving diffusion equation with nonlocal boundary condition

Abstract

In this paper we introduce a meshless method based on spatial trial spaces spanned by the Newton basis functions. The Newton basis arising from a pivoted Cholesky factorization turns out to be stable and computationally cheap while being orthonormal in the “native” Hilbert space of the kernel. This method can overcome on some of challenges of the full meshless kernel-based methods such as ill-conditioning associated with the dense linear systems that arise. This approach is used to obtain the numerical solution of time-dependent partial differential equations (PDEs), diffusion equation with two types of nonlocal boundary condition. The time-dependent PDE is discretized in space by the collocation method based on Newton basis functions as trial functions and it is converted to a system of ordinary differential equations in time. Stability of the scheme is discussed. Numerical examples are provided to show the accuracy and easy implementation of the novel proposed method.