Numerical solution of Volterra-Fredholm integral equations based on ε-SVR method

In this paper, we try to study the numerical methods for solving integral equations from a new perspective-machine learning method. By means of the idea of kernel e -support vector regression machine ( e -SVR), we construct an optimization modeling for a class of Volterra-Fredholm integral equations and propose a novel numerical method for solving them. The proposed method has a certain versatility and can be used to solve some other kinds of integral equations. In order to verify the effectiveness of the proposed method, we perform a series of comparative experiments with six specific Volterra-Fredholm integral equations and a method proposed in Wang et?al. (2014). Experimental results show that the proposed method has a good approximation property.

[1]  Keyan Wang,et al.  Least squares approximation method for the solution of Volterra-Fredholm integral equations , 2014, J. Comput. Appl. Math..

[2]  Jay I. Frankel,et al.  A Galerkin solution to a regularized Cauchy singular integro-differential equation , 1995 .

[3]  M. A. Abdou.  Fredholm-Volterra integral equation of the first kind and contact problem , 2002, Appl. Math. Comput..

[4]  Min Wang,et al.  Seeking multi-thresholds directly from support vectors for image segmentation , 2005, Neurocomputing.

[5]  Mahdi Heidari,et al.  A quadrature method with variable step for solving linear Volterra integral equations of the second kind , 2007, Appl. Math. Comput..

[6]  Johan A. K. Suykens,et al.  Optimal control by least squares support vector machines , 2001, Neural Networks.

[7]  Vladimir N. Vapnik,et al.  The Nature of Statistical Learning Theory , 2000, Statistics for Engineering and Information Science.

[8]  Saeid Abbasbandy,et al.  Artificial neural networks based modeling for solving Volterra integral equations system , 2015, Appl. Soft Comput..

[9]  Somayeh Nemati,et al.  Numerical solution of Volterra-Fredholm integral equations using Legendre collocation method , 2015, J. Comput. Appl. Math..

[10]  Hwajoon Kim,et al.  The solution of Volterra integral equation of the second kind by using the Elzaki transform , 2014 .

[11]  Mahdi Heidari,et al.  Solving linear integral equations of the second kind with repeated modified trapezoid quadrature method , 2007, Appl. Math. Comput..

[12]  Vladimir Vapnik,et al.  The Nature of Statistical Learning , 1995 .

[13]  X. C. Guo,et al.  LS-SVR-based solving Volterra integral equations , 2012, Appl. Math. Comput..

[14]  Q. Henry Wu,et al.  Local prediction of non-linear time series using support vector regression , 2008, Pattern Recognit..

[15]  Samuel A. Isaacson,et al.  Numerical solution of linear Volterra integral equations of the second kind with sharp gradients , 2011, J. Comput. Appl. Math..

[16]  Nello Cristianini,et al.  An Introduction to Support Vector Machines and Other Kernel-based Learning Methods , 2000 .