scholarly journals Application of radial basis function to approximate functional integral equations

2016 ◽  
Vol 2016 (2) ◽  
pp. 77-86
Author(s):  
Reza Firouzdor ◽  
Shukooh Sadat Asari ◽  
Majid Amirfakhrian
Keyword(s):  
Radial Basis Function ◽  
Integral Equations ◽  
Basis Function ◽  
Functional Integral ◽  
Radial Basis ◽  
Functional Integral Equations

scholarly journals Solving the Linear Integral Equations Based on Radial Basis Function Interpolation

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Huaiqing Zhang ◽  
Yu Chen ◽  
Xin Nie
Keyword(s):  
Radial Basis Function ◽  
Integral Equations ◽  
Basis Function ◽  
Approximation Solution ◽  
Integration Schemes ◽  
Radial Basis ◽  
The Matrix ◽  
Linear Integral ◽  
Split Technique ◽  
Linear Integral Equations

The radial basis function (RBF) method, especially the multiquadric (MQ) function, was introduced in solving linear integral equations. The procedure of MQ method includes that the unknown function was firstly expressed in linear combination forms of RBFs, then the integral equation was transformed into collocation matrix of RBFs, and finally, solving the matrix equation and an approximation solution was obtained. Because of the superior interpolation performance of MQ, the method can acquire higher precision with fewer nodes and low computations which takes obvious advantages over thin plate splines (TPS) method. In implementation, two types of integration schemes as the Gauss quadrature formula and regional split technique were put forward. Numerical results showed that the MQ solution can achieve accuracy of1E-5. So, the MQ method is suitable and promising for integral equations.

scholarly journals Application of Radial Basis Function Method for Solving Nonlinear Integral Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Huaiqing Zhang ◽  
Yu Chen ◽  
Chunxian Guo ◽  
Zhihong Fu
Keyword(s):  
Radial Basis Function ◽  
Integral Equations ◽  
Basis Function ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Nonlinear Integral Equations ◽  
Mesh Free ◽  
Radial Basis ◽  
Background Mesh ◽  
Mesh Free Methods

The radial basis function (RBF) method, especially the multiquadric (MQ) function, was proposed for one- and two-dimensional nonlinear integral equations. The unknown function was firstly interpolated by MQ functions and then by forming the nonlinear algebraic equations by the collocation method. Finally, the coefficients of RBFs were determined by Newton’s iteration method and an approximate solution was obtained. In implementation, the Gauss quadrature formula was employed in one-dimensional and two-dimensional regular domain problems, while the quadrature background mesh technique originated in mesh-free methods was introduced for irregular situation. Due to the superior interpolation performance of MQ function, the method can acquire higher accuracy with fewer nodes, so it takes obvious advantage over the Gaussian RBF method which can be revealed from the numerical results.

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