Solution of Boundary Value Obstacle Problems Using MQ-RBF and IMQ-RBF

Feng Gao,Chunmei Chi

doi:10.1155/2013/490639

Abstract

A kind of numerical method which is based on multiquadric RBF, inverse multiquadric RBF, and Wu-Schaback operators is presented for solving second-order and third-order boundary value problems associated with obstacle, unilateral, and contact problems. The algorithms are proved to be highly accurate and easy to implement. Some numerical tests are also presented to show the efficiency of the algorithms.

Highlights

We consider the numerical solutions of the following secondorder boundary value problems: u󸀠󸀠 (x) = {{f (x) {{g (x), u +f r, a ≤ x ≤ c, c ≤ x ≤ d, (1){f (x), d ≤ x ≤ b, with the boundary conditions u (a) = α, u (b) = β (2)and the continuity conditions of u and u󸀠 at c and d
A kind of numerical method which is based on multiquadric radial base functions (RBF), inverse multiquadric RBF, and Wu-Schaback operators is presented for solving second-order and third-order boundary value problems associated with obstacle, unilateral, and contact problems
In this paper, based on the multiquadric RBF and inverse multiquadric RBF, numerical schemes are presented for solving system of second-order and third-order boundary value problems associated with obstacle

Summary

Introduction

We consider the numerical solutions of the following secondorder boundary value problems: u󸀠󸀠. Some techniques were used to solve the previous two systems of second-order and third-order boundary value problems associated with obstacles. Noor and Khalifa [16] applied quartic spline method for odd-order obstacle problems. Gao and Chi [17] applied quartic B-spline method for third-order obstacle problems and Siraj-ul-Islam et al [12] proposed nonpolynomial spline methods Some of these methods require huge computational work for finding the numerical solutions. Among Radial Basis Functions, MQs and IMQs are especially suitable for solving second- and third-order boundary value problems associated with obstacles because we can integrate IMQs to get MQs. Our technique is that we first use IMQs or MQs quasiinterpolation to approximate the derivatives of the solution of problems of (1) and (3); we get their integrals to approximate the solutions.

RBF and MQ Quasi-Interpolation

Numerical Algorithms

Numerical Experiment

Conclusions