Analysis of New RBF-FD Weights, Calculated Based on Inverse Quadratic Functions

Abstract

Local radial basis functions (RBFs) have many advantages for solution of differential equations. In some of these radial functions, there is a parameter that has a special effect on the accuracy of the answer and is known as the shape parameter. In this article, first of all, we derive inverse quadratic (IQ)-based RBF-generated finite difference coefficients for some derivatives in one dimension (1D). Then, to evaluate the efficiency of these new weights and also the effect of the shape parameter on the accuracy of the resulting approximations, we will test them with a suitable function. After that, we focus on solving some boundary value problems (BVPs), using IQ-based RBF-FD method. There is a range for the shape parameter in which the approximation error is less than other areas. We use an efficient algorithm to find the best value of the RBF parameter for the problem domain. Our studies show that IQ-based RBF-FD weights could be derived analytically easier than multiquadrics (MQs) which were previously presented in the literature. Besides, the results of numerical examples confirm the high accuracy of these new formulas. For better comparison, we revisit some previously studied illustrative examples.