Numerical solution of the second kind integral equations using radial basis function networks

Golbabai, A.; Seifollahi, Sattar

doi:10.1016/j.amc.2005.05.034

Cited by 41 publications

(32 citation statements)

References 6 publications

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“…The loading is remote shear. [28,29,[33][34][35][36][37][38][39], the representation of the solution in Equation (5) for the multiple-inclusion problem as shown in Figure 1, can be approximated in terms of the strengths j of the singularities as s j as…”

Section: Governing Equation and Boundary Conditionsmentioning

confidence: 99%

Regularized meshless method for antiplane shear problems with multiple inclusions

Chen

Kao

2007

Numerical Meth Engineering

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SUMMARYIn this paper, we employ the regularized meshless method to solve antiplane shear problems with multiple inclusions. The solution is represented by a distribution of double-layer potentials. The singularities of kernels are regularized by using a subtracting and adding-back technique. Therefore, the troublesome singularity in the method of fundamental solutions (MFS) is avoided and the diagonal terms of influence matrices are determined. An inclusion problem is decomposed into two parts: one is the exterior problem for a matrix with holes subjected to remote shear, the other is the interior problem for each inclusion. The two boundary densities, essential and natural data, along the interface between the inclusion and matrix satisfy the continuity and equilibrium conditions. A linear algebraic system is obtained by matching boundary conditions and interface conditions. Finally, numerical results demonstrate the accuracy of the present solution. Good agreements are obtained and compare well with analytical solutions and Gong's results.

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Section: Governing Equation and Boundary Conditionsmentioning

confidence: 99%

Regularized meshless method for antiplane shear problems with multiple inclusions

Chen

Kao

2007

Numerical Meth Engineering

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“…However, the RBF method in solving integral equations was initially proposed in 2006. Golbabai applied the RBF networks for solving the linear integral equations [14], the linear integrodifferential equations [15], and the nonlinear integral 2 Journal of Applied Mathematics equations [16]. Parand and Rad [17] presented the RBF collocation method for one-dimensional Volterra-Fredholm-Hammerstein integral equations.…”

Section: Introductionmentioning

confidence: 99%

Application of Radial Basis Function Method for Solving Nonlinear Integral Equations

Zhang

Guo

et al. 2014

Journal of Applied Mathematics

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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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“…Nowadays, the artificial neural networks have widely been applied in variety of fields, such as biology, mechanical engineering, electrical and computer engineering, computer science, and physics, etc. 1,3,5,17 The application problems have been occasionally converted into the problems of utilizing an underlying artificial neural network as an approximation function. 14,15,4,3,8,16,10 Although the approximation ability of artificial neural networks has been sufficiently discussed in some earlier articles, 3,8,16,10 the works of related quantitative analysis is recently gave rise to the strong attention of the people, especially on the topic of relationship between the converge rate of approximation and the structural topology of hidden layer.…”

Section: Introductionmentioning

confidence: 99%

Constructive Estimation of Approximation for Trigonometric Neural Networks

Wang

Zou

2012

Int. J. Wavelets Multiresolut Inf. Process.

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For the three-layer artificial neural networks with trigonometric weights coefficients, the upper bound and lower bound of approximating 2π-periodic pth-order Lebesgue integrable functions L p 2π are obtained in this paper. Theorems we obtained provide explicit equational representations of these approximating networks, the specification for their numbers of hidden-layer units, the lower bound estimation of approximation, and the essential order of approximation. The obtained results not only characterize the intrinsic property of approximation of neural networks, but also uncover the implicit relationship between the precision (speed) and the number of hidden neurons of neural networks.

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Numerical solution of the second kind integral equations using radial basis function networks

Cited by 41 publications

References 6 publications

Regularized meshless method for antiplane shear problems with multiple inclusions

Regularized meshless method for antiplane shear problems with multiple inclusions

Application of Radial Basis Function Method for Solving Nonlinear Integral Equations

Constructive Estimation of Approximation for Trigonometric Neural Networks

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