A Method for Numerical Solution of Third-Kind Volterra Integral Equations Using Krall-Laguerre Polynomials

Subject Areas : International Journal of Industrial Mathematics

پروانه جمی ¹ , الهام هاشمی زاده ²

1 - Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran.
2 - Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran.

Received: 2021-04-12 Accepted : 2021-06-26 Published : 2022-01-01

Keywords: Krall-Laguerre ‎Polynomials, ‎Approximation, ‎ Krall-Hahn Orthogonal Polynomials, Third-Kind Integral ‎Equations, Analytical solution‎‎. Fredholm Integral Equations,

Abstract :

The numerical solution of linear integral equations of third kind is discussed in various studies, but in the previous researches on this kind of equations only the analytical solution was investigated. Due to some limitations for this kind of solutions, in this paper we propose a new method for numerical solution of linear integral equations of third kind. The proposed method is based on the approximation of the unknown function with Krall-Laguerre polynomials. This method has a simple computation with a quite acceptable approximate solution. Moreover, we obtain an estimate of the error bound for suggested method. Two examples are also presented to show the efficiency of the proposed method.

References:

 [1] L. M. Delves, J. Mohamed, Computational methods for integral equations, 1988, CUP Archive.
[2] E. Hashemizadeh, B. Basirat, An efficient computational method for the system of linear Volterra integral equations by means of hybrid functions, Math. Sci. 4 (2011) 355-368.
[3] K. Maleknejad, E. Hashemizadeh, A numerical approach for Hammerstein integral equations of mixed type using operational matrices of hybrid functions, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 73 (2011) 95-104.
[4] G. Bal, Inverse problems for homogeneous transport equations: I. The one-dimensional case, Inverse Problems 16 (2000) 997-1007.
[5] G. Bart, R. Warnock, Linear integral equations of the third kind, SIAM Journal on Mathematical Analysis 4 (1973) 609-622.
[6] J. A. Ellison, A. V. Sobol, M. Vogt, A new model for the collective beam beam interaction, New Journal of Physics 9 (2007) 32-45.
[7] S. S. Allaei, Z. W. Yang, H. Brunner, Existence, uniqueness and regularity of solutions to a class of third-kind Volterra integral equations, Journal of Integral Equations and Applications 27 (2015) 325-342.
[8] M. Imanaliev, A. Asanov, Solutions to systems of linear Fredholm integral equations of the third kind, Doklady Mathematics, 2010. Springer.
[9] G. Vainikko, Cordial Volterra integral equations 1, Numerical Functional Analysis and Optimization 30 (2009) 1145-1172.
[10] G. Vainikko, Cordial Volterra integral equations 2, Numerical functional analysis and optimization 31 (2010) 191-219.
[11] S. S. Allaei, Z. W. Yang, H. Brunner, Collocation methods for third-kind VIEs, IMA Journal of Numerical Analysis 27 (2016) 1104-1124.
[12] A. M. Krall, Orthogonal polynomials satisfying fourth order differential equations, Proceedings of the Royal Society of Edinburgh Section A: Mathematics 87 (1981) 271-288.
[13] A. Zhedanov, A method of constructing Krall’s polynomials, Journal of computational and applied mathematics 107 (1999) 1-20.
[14] N. S. Gabbasov, New variants of the collocation method for integral equations of the third kind, Mathematical Notes 50 (1991) 802-806.
[15] N. S. Gabbasov, A new direct method for solving integral equations of the third kind, Mathematical Notes 49 (1991) 29-33.
[16] N. S. Gabbasov, A special version of the collocation method for integral equations of the third kind, Differential Equations 41 (2005) 1768-1774.
[17] D. Shulaia, Solution of a linear integral equation of third kind, Georgian Mathematical Journal 9 (2002) 179-196.
[18] D. Shulaia, Linear integral equations of the third kind arising from neutron transport theory, Mathematical Methods in the Applied Sciences 30 (2007) 1941-1964.
[19] S. Pozza, M. S. Prani, Z. Strako, Gauss quadrature for quasi-definite linear functionals, IMA Journal of Numerical Analysis 37 (2016) 1468-1495.
[20] S. Bochner, ber Sturm-Liouvillesche Polynomsysteme, Mathematische Zeitschrift 29 (1929) 730-736.
[21] J. Koekoek, R. Koekoek, On a differential equation for Koornwinders generalized Laguerre polynomials, Proceedings of the American Mathematical Society 11 (1991) 1045-1054.
[22] R. Koekoek, P. A. Lesky, R. F. Swarttouw, Hypergeometric orthogonal polynomials and their q-analogues, Springer Science & Business Media (2010).

[23] O. Bihun, New properties of the zeros of Krall polynomials, Journal of nonlinear mathematical physics 24 (2017) 495-515.
[24] R. lvarez-Nodarse, R. S. Adgzel, On the Krall-type polynomials on q-quadratic lattices, Indagationes Mathematicae 21 (2011) 181-203.
[25] K. Kwon, G. Yoon, L. Littlejohn, BochnerKrall orthogonal polynomials, World Scientific, Singapore 12 (2000) 181-193.
[26] A. M. Krall, Hilbert Spaces, in Hilbert Space, Boundary Value Problems and Orthogonal Polynomials 29 (2002) 1-16.
[27] L. Haine, The Bochner-Krall problem: some new perspectives, Special functions 2000: current perspective and future directions 43 (2001) 141-178.
[28] G. Mastroianni, G. Milovanovic, Interpolation processes: Basic theory and applications, Science & Business Media. 21 (2008) 982-999.
[29] M. J. D. Powell, Approximation theory and methods, Cambridge university press, 1981.