### Solving system of first kind integral equations via the Chebyshev collocation approach

Document Type : Research Paper

Author

Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, P. O. Box 35195-363, Semnan, Iran.

Abstract

This paper discusses a numerical method for solving a first-kind Volterra integral equations system. Because of the ill-posedness of these equations, we need to apply an efficient computational method to discrete them to the system of algebraic equations. An expansion method known as the Chebyshev collocation method, based on the Chebyshev polynomials of the third kind, is employed to convert the system of integral equations to the linear algebraic system of equations. By solving the algebraic system, we conclude an approximate solution. Some numerical results support the accuracy and efficiency of the stated method.

Keywords

[1] K.E. Atkinson, An Introduction to Numerical Analysis, 2nd edition, John Wiley and Sons, New York, 1989.
[2] K.E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, Cambridge, 1997.
[3] D. Barrera, M. Barton, I. Chiarella and S. Remogna, On numerical solution of Fredholm and Hammerstein integral equations via Nystrom method and Gaussian quadrature rules for splines, Appl. Numer. Math. 174 (2022), 71–88.
[4] J. Biazar and H. Ghazvini, He’s homotopy perturbation method for solving systems of Volterra integral equations of the second kind, Chaos Solitons Fractals 39 (2009), no. 2, 770–777.
[5] A. Golbabai and B. Keramati, Easy computational approach to solution of system of linear Fredholm integral equations, Chaos Solitons Fractals 38 (2008), no. 2, 568–574.
[6] M. Javidi and A. Golbabai, A numerical solution for solving system of Fredholm integral equations by using homotopy perturbation method, Appl. Math. Comput. 189 (2007), no. 2, 1921–1928.
[7] G.H. Kazemi Gelian, R. Ghoochani Shirvan and M.A. Fariborzi Araghi, Comparison between Sinc approximation and differential transform methods for nonlinear Hammerstein integral equations, Int. J. Nonlinear Anal. Appl. 13 (2022), no. 1, 1291–1301.
[8] R. Kress, Linear Integral Equations, Springer-Verlag, New York, 1998.
[9] K. Maleknejad, N. Aghazadeh and M. Rabbani, Numerical solution of second kind Fredholm integral equations system by using a Taylor-series expansion method, Appl. Math. Comput. 175 (2006), no. 2, 1229–1234.
[10] K. Maleknejad, K. Nouri and M. Nosrati Sahlan, Convergence of approximate solution of nonlinear Fredholm-Hammerstein integral equations, Commun. Nonlinear Sci. Numer. Simul. 15 (2010), no. 2, 1432–1443.
[11] K. Maleknejad, M. Shahrezaee and H. Khatami, Numerical solution of integral equations system of the second kind by Block-Pulse functions, Appl. Math. Comput. 166 (2005), no. 1, 15-24.
[12] M. Mandal, K. Kant and G. Nelakanti, Projection and multi-projection methods for second kind Volterra-Hammerstein integral equation, Int. J. Nonlinear Anal. Appl. 12 (2021), 275-291.
[13] J.C. Mason and D.C. Handscomb, Chebyshev Polynomials, Chapman and Hall/CRC, New York, 2002.
[14] N. Negarchi and K. Nouri, Numerical solution of Volterra–Fredholm integral equations using the collocation method based on a special form of the Muntz–Legendre polynomials, J. Comput. Appl. Math. 344 (2018), 15–24.
[15] K. Nouri, An efficient method for solving system of Volterra integral equations, Kybernetes 41 (2012), no. 3, 501–507.
[16] R. Qiu, X. Duan, Q. Huangpeng and L. Yan, The best approximate solution of Fredholm integral equations of the first kind via Gaussian process regression, Appl. Math. Lett. 133 (2022), Article ID 108272.
[17] R. Qiu, L. Yan and X. Duan, Solving Fredholm integral equation of the first kind using Gaussian process regression, Appl. Math. Comput. 425 (2022), Article ID 127032.
[18] B.G. Spencer Doman, The Classical Orthogonal Polynomials, World Scientific Publishing, Singapore, 2015.
[19] D. Yuan and X. Zhang, An overview of numerical methods for the first kind Fredholm integral equation, SN Appl. Sci. 1 (2019), Article ID 1178.
###### Volume 14, Issue 9September 2023Pages 145-152
• Receive Date: 05 August 2022
• Accept Date: 28 December 2022