An approach of extrapolation methods for the solution of nonlinear Volterra–Fredholm integral equations of the second kind

Iranian Journal of Numerical Analysis and Optimization

مقاله 3، دوره 15، Issue 1 - شماره پیاپی 32، خرداد 2025، صفحه 54-78   

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.2024.85903.1361

نویسندگان

H. Safdari¹؛ M. Moradkhani^{* 1}؛ M.T. Dastjerdi²

¹Faculty of Sciences, Teacher Training Shahid Rajaei University, Lavizan, Tehran, Iran.

²Faculty of Mathematics, Zanjan University, University Boulevard, Zanjan, Iran.

چکیده

This study presents the process of using extrapolation methods to solve the nonlinear Volterra–Fredholm integral equations of the second kind. To do this, by approximating the integral terms contained in equations by a quadrature rule, the nonlinear Volterra–Fredholm integral equations of the second kind are reduced to a set of nonlinear algebraic equations. Then, the solution of the corresponding system of nonlinear equations is approxi-mated by an iterative method, and finally, these iterations are accelerated by an extrapolation method. We demonstrate the effectiveness of the pro-posed approach by solving some numerical examples.

کلیدواژه‌ها

Volterra–Fredholm integral equations؛ Extrapolation methods؛ Iterative methods

مراجع

[1] Babolian, E., Bazm, S. and Lima, P. Numerical solution of nonlinear two-dimensional integral equations using rationalized Haar functions, Commun. Nonlinear. Sci. Numer. Simul. 16(3) (2011), 1164–1175.
[2] Borzabadi, A.H. and Heidari, M. A Successive numerical scheme for some classes of Volterra–Fredholm integral equations, Iranian J. Math. Sci. Inf. 10(2) (2015), 1–10.
[3] Brezinski, C. Numerical stability of a quadratic method for solving sys-tems of non linear equations, Computing 14 (1975), 205–211.
[4] Brezinski, C. Generalisations de la transformation de shanks, de la table de Pade et de l’ε-algorithme, Calcolo 12 (1975), 317–360.

[5] Brezinski, C. Padé-type approximation and general orthogonal polyno-mials, Birkhauser-Verlag, 1980.
[6] Brezinski, C. and Chehab, J.-P. Nonlinear hybrid procedures and fixed point iterations, Numer. Funct. Anal. Optim. 19 (1998), 465–487.
[7] Brezinski, C. and Redivo-Zaglia, M. Extrapolation methods: Theory and practice, North-Holland, 1991.
[8] Brezinski, C. and Redivo-Zaglia, M. The simplified topological ε-algorithms for accelerating sequences in a vector space, Siam J. Sci. Comput. 36(5) (2014), A2227–A2247.
[9] Brezinski, C. and Radivo-Zagila, M. Extrapolation methods for the nu-merical solution of nonlinear Fredholm integral equations, J. Integral Equ. Appl. 31 (2019), 29–57.
[10] Brezinski, C. and Sadok, H. Lanczos-type algorithms for solving systems of linear equations, Appl. Numer. Math. 11 (1993), 443–473.
[11] Brunner, H. On the numerical solution of nonlinear Volterra–Fredholm integral equation by collocation methods, SIAM J. Numer. Anal. 27(4) (1990), 987–1000.
[12] Delves, L.M. and Mohamed, J.M. Computational methods for integral equations, Cambridge University Press, 1985.
[13] Ghasemi, M., Tavassoli Kajani, M. and Babolian, E. Numerical solutions of the nonlinear Volterra–Fredholm integral equations by using Homotopy Perturbation Method, Appl. Math. Comput. 188(1) (2007), 446–449.
[14] Jbilou, K., Messaoudi, A. and Tabaa, K. On some matrix extrapolation methods, C. R. Math. Acad. Sci. Paris 341 (2005), 781–786.
[15] Jbilou, K. and Sadok, H. Vector extrapolation methods. Applications and numerical comparison, J. Comput. Appl. Math. 122 (2000), 149–165.
[16] Kuttler, U. and Wall, W.A. Fixed-point fluid-structure interaction solvers with dynamic relaxation, Comput. Mech. 43 (2008), 61–72.

[17] Laeli Dastjerdi, H. and Nili Ahmadabadi, M. The numerical solution of nonlinear two-dimensional Volterra–Fredholm integral equations of the second kind based on the radial basis functions approximation with error analysis, Appl. Math. Comput. 293 (2017), 545–554.
[18] MacLeod, A.J. Acceleration of vector sequences by multi-dimensional ∆2 methods, Commun. Appl. Numer. Meth. 2 (1986), 385–392.
[19] Maleknejad, K., Almasieh, H. and Roodaki, M. Triangular functions (TF) method for the solution of nonlinear Volterra–Fredholm integral equations, Commun. Nolin. Sci. Numer. Simul. 15(11) (2010), 3293–3298.
[20] Mann, W.R. Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506–510.
[21] Mirzaee, F. and Hadadiyan, E. Numerical solution of Volterra–Fredholm integral equations via modification of hat functions, Appl. Math. Com-put. 280 (2016), 110–123.
[22] Mosa, G.A., Abdou, M.A. and Rahby, A.S. Numerical solutions for non-linear Volterra–Fredholm integral equations of the second kind with a phase lag, AIMS Math. 6 (2021), 8525–8543.
[23] Ordokhani, Y. Solution of nonlinear Volterra–Fredholm–Hammerstein integral equations via rationalized Haar functions, Appl. Math. Comput. 180(2) (2006), 436–443.
[24] Patchpatte, B.G. On a class of Volterra and Fredholm non-linear integral equations, Sarajevo J. Math. 16 (2008), 61–71.
[25] Salam A. and Graves-Morris, P.R. On the vector ε-algorithm for solving linear systems of equations, Numer. Algorithms 29 (2002), 229–247.
[26] Sedogbo, G.A. Some convergence acceleration processes for a class of vector sequences, Appl. Math. (Warsaw) 24 (1997), 299–306
[27] Shali, J.A. and Ebadi, G. Approximate solutions of nonlinear Volterra–Fredholm integral equations, Int. J. Non. Sci. 14(4) (2012), 425–433.

[28] Shanks, D. Nonlinear transformations of divergent and slowly convergent sequences, J. Math. Phys. 34 (1955), 1–42.
[29] Sidi, A. Practical extrapolation methods: Theory and applications, Cam-bridge University Press, 2003.
[30] Wazwaz, A.M. Linear and nonlinear integral equations: Methods and applications, Springer-Verlag, 2011.
[31] Wynn, P. On a device for computing the em(Sn) transformation, Math.Tables Aids Comput. 10 (1956), 91–96.
[32] Wynn, P. Acceleration techniques for iterated vector and matrix problems, Math. Comp. 16 (1962), 301–322.
[33] Yalcinbas, S. Taylor polynomial solutions of nonlinear Volterra–Fredholm integral equations, Appl. Math. Comput. 127 (2002), 195–206.
[34] Yousefi, S. and Razzaghi, M. Legendre wavelets method for the nonlin-ear Volterra–Fredholm integral equations, Math. Comput. Simul. 70(1) (2005), 1–8.