Application of Newton–Cotes quadrature rule for nonlinear Hammerstein integral equations

Iranian Journal of Numerical Analysis and Optimization

مقاله 8، دوره 11، شماره 2 - شماره پیاپی 20، آذر 2021، صفحه 385-399   

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.2021.69824.1024

نویسنده

A. Shahsavaran^*

Department of Mathematics, Borujerd Branch, Islamic Azad University, Borujerd, Iran.

چکیده

A numerical method for solving Fredholm and Volterra integral equations of the second kind is presented. The method is based on the use of  the Newton–Cotes quadrature rule and Lagrange interpolation polynomials. By the proposed method, the main problem is reduced to solve some nonlinear algebraic equations that can be solved by Newton’s method. Also, we prove some statements about the convergence of the method. It is shown that the approximated solution is uniformly convergent to the exact solution. In addition, to demonstrate the efficiency and applicability of the proposed method, several numerical examples are included, which confirms the convergence results.

کلیدواژه‌ها

Fredholm integral equation؛ Volterra integral equation؛ Newton–Cotes quadrature rule؛ Lagrange interpolation؛ Convergence

مراجع

1. Abdul Sathar, M.H., Rasedee, A.F.N., Ahmedov, A.A. and Bachok, N. Numerical solution of nonlinear Fredholm and Volterra integral equations by Newton–Kantorovich and Haar wavelets methods, Symmetry, 12 (2020) 1–13.
2. Babolian, E. and Mordad, M. A numerical method for solving system of linear and nonlinear integral equations of the second kind by hat basis functions, Comput. Math. Appl., 62 (2011) 187–198.
3. Babolian, E. and Shahsavaran, A. Numerical solution of nonlinear Fredholm integral equations of the second kind using Haar wavelet, J. Comput. Appl. Math. 225 (2009) 87–95.
4. Basirat, B., Maleknejad, K. and Hashemizadeh, E. Operational matrix approach for the nonlinear Volterra-Fredholm integral equations: Arising in physics and engineering, Int. J. Phys. Sci. 7 (2012) 226–233.
5. Bazm, S. Solution of nonlinear Volterra-Hammerstein integral equations using alternative Legendre collocation method, Sahand Communications in Mathematical Analysis, 4 (2016) 57–77.

6. Bellour, A., Sbibih, D. and Zidna, A. Two cubic spline methods for solving Fredholm integral equations, Appl. Math. Comput. 276 (2016) 1–11.
7. Brunner, H. Implicitly linear collocation method for nonlinear Volterra equations, J. Appl. Num. Math. 9 (1982) 235–247.
8. Delves, L.M. and Mohamed, J.L. Computational methods for integral equations, Cambridge University Press, 1985.
9. Fattahzadeh, F. Numerical solution of general nonlinear Fredholm Volterra integral equations using Chebyshev approximation,Int. J. Ind. Math. 8 (2016) 81–86.
10. He, G., Xiang, S. and Xu, Z. A Chebyshev collocation method for a class of Fredholm integral equations with highly oscillatory kernels, J. Comput. Appl. Math. 300 (2016) 354–368.
11. Ibrahim, I.A. On the existence of solutions of functional integral equations of Urysohn type, Comput. Math. with Appl. 57 (2009) 1609–1614.
12. Kumar, S. and Sloan, I.H. A New collocation-type method for Hammer stein integral equations, Math. Comput. 48 (1987) 585–593.
13. Maleknejad, K., Almasieh, H. and Roodaki, M. Triangular functions (TF) method for the solution of nonlinear Volterra–Fredholm integral equations, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 3293–3298.
14. Maleknejad, K., Lotfi, T. and Mahdiani, K. Numerical solution of first kind Fredholm integral equations with wavelets-Galerkin method and wavelets precondition, Appl. Math. Comput. 186 (2007) 794–800.
15. Maleknejad, K., Mollapourasl, R. and Ostadi, A. Convergence analysis of Sinc-collocation method for nonlinear Fredholm integral equations with a weakly singular kernel,J. Comput. Appl. Math. 278 (2015) 1–11.
16. Rahmoune, A. Spectral collocation method for solving Fredholm integral equations on the half-line, Appl. Math. Comput. 219 (2013) 9254–9260.
17. Sahu, P.K. and Ray, S.S. Hybrid Legendre Block-Pulse functions for the numerical solutions of system of nonlinear Fredholm–Hammerstein integral equations, Appl. Math. Comput. 270 (2015) 871–878.
18. Sepehrian, B. and Razzaghi, M.A new method for solving nonlinear Volterra-Hammerstein integral equations via single-term Walsh series, Mathematical Analysis and Convex Optimization, 1 (2020) 59–69.
19. Shahsavaran, A. Numerical solution of nonlinear Fredholm-Volterra integral equations via piecewise constant functions by collocation method, Am. J. Comput. Math. 1 (2011) 134–138.

20. Stoer, J. and Bulirsch, R. Introduction to numerical analysis, Springer Verlag, 1991.
21. Wazwaz, A.M. Linear and nonlinear integral equations: Methods and applications, Higher education, Springer, 2011.