A Numerical Approach for Solving Quadratic Integral Equations of Urysohn’s Type using Radial Basis Function

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.

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Free Vibration Study of Anti-Symmetric Angle-Ply Laminated Plates under Clamped Boundary Conditions

Curved and Layered Structures ◽

10.1515/cls-2016-0020 ◽

2016 ◽

Vol 3 (1) ◽

Author(s):

K.K. Viswanathan ◽

K. Karthik ◽

Y.V.S.S. Sanyasiraju ◽

Z.A. Aziz

Keyword(s):

Free Vibration ◽

Boundary Conditions ◽

Radial Basis Function ◽

Basis Function ◽

Equations Of Motion ◽

Spline Approximation ◽

Numerical Approach ◽

Laminated Plates ◽

Radial Basis ◽

Plane Displacement

AbstractTwo type of numerical approach namely, Radial Basis Function and Spline approximation, used to analyse the free vibration of anti-symmetric angle-ply laminated plates under clamped boundary conditions. The equations of motion are derived using YNS theory under first order shear deformation. By assuming the solution in separable form, coupled differential equations obtained in term of mid-plane displacement and rotational functions. The coupled differential is then approximated using Spline function and radial basis function to obtain the generalize eigenvalue problem and parametric studies are made to investigate the effect of aspect ratio, length-to-thickness ratio, number of layers, fibre orientation and material properties with respect to the frequency parameter. Some results are compared with the existing literature and other new results are given in tables and graphs.

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Application of Radial Basis Function Method for Solving Nonlinear Integral Equations

Journal of Applied Mathematics ◽

10.1155/2014/381908 ◽

2014 ◽

Vol 2014 ◽

pp. 1-8 ◽

Cited By ~ 1

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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