Solution of First kind Fredholm Integral Equation by Sinc Function

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.

Authors:

• Khosrow Maleknejad
• Reza Mollapourasl
• Parvin Torabi
• Mahdiyeh Alizadeh

Keywords:

• Integral equation
• Fredholm type
• Collocation method
• Sinc approximation.

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