Solution of First kind Fredholm Integral Equation by Sinc Function

Authors: Khosrow Maleknejad, Reza Mollapourasl, Parvin Torabi, Mahdiyeh Alizadeh,

Abstract:

Sinc-collocation scheme is one of the new techniques used in solving numerical problems involving integral equations. This method has been shown to be a powerful numerical tool for finding fast and accurate solutions. So, in this paper, some properties of the Sinc-collocation method required for our subsequent development are given and are utilized to reduce integral equation of the first kind to some algebraic equations. Then convergence with exponential rate is proved by a theorem to guarantee applicability of numerical technique. Finally, numerical examples are included to demonstrate the validity and applicability of the technique.

Keywords: Integral equation, Fredholm type, Collocation method, Sinc approximation.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1334816

References:

[1] M. Muhammad, A. Nurmuhammad, M. Mori, M. Sugihara, Numerical solution of integral equations by means of the Sinc collocation method based on the double exponential transformation, Journal of Computational and Applied Mathematics 177 (2005) 269-286.
[2] K. Maleknejad, K. Nouri, M. Nosrati Sahlan, Convergence of approximate solution of nonlinear Fredholm-Hammerstein integral equations, Communications in Nonlinear Science and Numerical Simulation, (In Press).
[3] M. Rasty, M. Hadizadeh, A product integration approach based on new orthogonal polynomials for nonlinear weakly singular integral equations, Acta Appl. Math. (In Press).
[4] K. Maleknejad, M. Nosrati, The Method of Moments for Solution of Second Kind Fredholm Integral Equations Based on B-Spline Wavelets International Journal of Computer Mathematics, (In Press).
[5] Adel Mohsen and Mohamed El-Gamel, A Sinc-Collocation method for the linear Fredholm integro-differential equations, Z. Angew. Math. Phys. 58 (2007) 380-390.
[6] K. Maleknejad, K. Nouri, R. Mollapourasl, Existence of solutions for some nonlinear integral equations, Communications in Nonlinear Science and Numerical Simulation, 14 (2009), 2559-2564.
[7] W. Volk, The iterated Galerkin methods for linear integro-differential equations, J. Comp. Appl. Math. 21 (1988), 63-74.
[8] K. Maleknejad, F. Mirzaee, Using rationalized Haar wavelet for solving linear integral equations, Appl. Math. Comp., 160 (2005), 579-587.
[9] A. Avudainayagam , C. Vani, Wavelet Galerkin method for integrodifferential equations, Appl. Numer. Math. 32 (2000), 247-254.
[10] F. Stenger, Numerical Methods Based on Sinc and Analytic Functions, Springer, New York, 1993.
[11] M. Sugihara, T. Matsuo, Recent developments of the Sinc numerical methods, J. Comput. Appl. Math. 164-165 (2004) 673-689.
[12] M.Sugihara, Near optimality of the Sinc approximation, Math. Comput. 72 (2003) 768-786.
[13] X. Shang, D. Han, Numerical solution of Fredholm integral equations of the first kind by using linear Legendre multi-wavelets, Applied Mathematics and Computation, 191, (2007), 440-444.
[14] E. Babolian, Z. Masouri, Direct method to solve Volterra integral equation of the first kind using operational matrix with block-pulse functions, Journal of Computational and Applied Mathematics, 220, (2008), 51-57.
[15] E. Babolian, T. Lotfi, M. Paripour, Wavelet moment method for solving Fredholm integral equations of the first kind, Applied Mathematics and Computation, 186, (2007), 1467-1471.
[16] C.T.H. Baker, The numerical treatment of integral equations , Clarendon Press, Oxford, 1969.
[17] Kendall E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, 1997.
[18] R. Kress, Linear Integral Equation, Springer-Verlag, New York, 1998.
[19] A. Karoui, Wavelets: Properties and approximate solution of a second kind integral equation, Computers & Mathematics with Applications, 46 (2003), 263-277