Least squares support vector regression for solving Volterra integral equations
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ORIGINAL ARTICLE
Least squares support vector regression for solving Volterra integral equations K. Parand1,2,3,4 · M. Razzaghi5 · R. Sahleh2 · M. Jani2 Received: 5 March 2020 / Accepted: 28 September 2020 © Springer-Verlag London Ltd., part of Springer Nature 2020
Abstract In this paper, a numerical approach is proposed based on least squares support vector regression for solving Volterra integral equations of the first and second kind. The proposed method is based on using a hybrid of support vector regression with an orthogonal kernel and Galerkin and collocation spectral methods. An optimization problem is derived and transformed to solving a system of algebraic equations. The resulting system is discussed in terms of the structure of the involving matrices and the error propagation. Numerical results are presented to show the sparsity of resulting system as well as the efficiency of the method. Keywords Volterra integral equations · Legendre kernel · Least squares support vector regression · Galerkin LS-SVR · Collocation LS-SVR
1 Introduction Many problems in physics and engineering are modeled in terms of integral equations. These include problems in elasticity, plasticity, heat and mass transfer, oscillation theory, fluid dynamics, filtration theory, electrostatics, electrodynamics, game theory, queueing theory, medicine, etc * K. Parand [email protected] M. Razzaghi [email protected] R. Sahleh [email protected] M. Jani [email protected] 1
School of Computer Science, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran
2
Department of Computer Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, G.C. Tehran, Iran
3
Department of Cognitive Modeling, Institute for Cognitive and Brain Sciences, Shahid Beheshti University, G.C. Tehran, Iran
4
Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Canada
5
Department of Mathematics and Statistics, Mississippi State University, Starkville, USA
[1–3]. Among two types of integral equations, Fredholm and Volterra, the Volterra integral equations are used in many physical applications like heat conduction problem, unsteady poiseuille flow in a pipe [4], diffusion problems [5], electroelastic [6], contact problems [7], etc. and have a close connection to differential equations. Numerical differentiation may suffer instability issues when the mesh size is decreased for the sake of getting higher accuracy. This may affect the methods based on the associated approximations. Some differential equations can be written as an equivalent Volterra integral equation in which the numerical approximations are used for integration instead of differentiation and so less computational instabilities are encountered. It is easy to see that an nth-order initial value problem u(n) (x) = f (x, u, u� , … , u(n−1) ),
u(i) (x0 ) = ui,0 , i = 0, … , n − 1, (1)
where u(i) stands for the ith derivative of u, may be reformulated as an equivalent system of first-order differential equations
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