Radial Basis Function Methods for Solving Partial Differential Equations-A Review

Abstract

Background/Objectives: The approximation using radial basis function (RBF) is an extremely powerful method to solve partial differential equations (PDEs). This paper presents different types of RBF methods to solve PDEs. Methods/ Statistical Analysis: Due to their meshfree nature, ease of implementation and independence of dimension, RBF methods are popular to solve PDEs. In this paper we examine different generalized RBF methods, including Kansa method, Hermite symmetric approach, localized and hybrid methods. We also discussed the preference of using meshfree methods like RBF over the mesh based methods. Findings: This paper presents a state-of-the-art review of the RBF methods. Some recent development of RBF approximation in solving PDEs is also discussed. The mathematical formulation of different RBF methods are discussed for better understanding. RBF methods have been actively developed over the years from global to local approximation and then to hybrid methods. Hybrid RBF methods help in reduction of computational cost and become very effective in solving large scale problems. Application/Improvements: RBF methods have been applied to various diverse fields like image processing, geo-modeling, pricing option and neural network etc.