Abstract
A new numerical method for Fredholm functional integral equations is proposed. The method combines the fixed point technique with numerical integration and cubic spline interpolation. The convergence and the numerical stability of the method are proved and tested on some numerical examples.
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Abbasbandy, S.: Application of He’s homotopy perturbation method to functional integral equations. Chaos, Solitons Fractals 31, 1243–1247 (2007)
Abbasbandy, S., Shivanian, E.: A new analytical technique to solve Fredholm integral equations. Numer. Algorithms 56(1), 27–43 (2011)
Abbasbandy, S.: Numerical solutions of the integral equations: homotopy perturbation method and Adomian decomposition method. Appl. Math. Comput. 173, 493–500 (2006)
Agarwall, R.P., O’Regan, D., Wong, P.J.Y.: Positive Solutions of Differential, Difference and Integral Equations. Kluwer Academic, Dordrecht (1999)
Allouch, C., Sablonniere, P., Sbibih, D.: Solving Fredholm integral equations by approximating kernels by spline quasi-interpolants. Numer. Algorithms (2010). doi:10.1007/s11075-010-9396-7
Anselone, P. (ed.): Nonlinear Integral Equations. University of Wisconsin, Madison (1964)
Argyros, I.K.: Quadratic equations and applications to Chandrasekhar’s and related equations. Bull. Aust. Math. Soc. 32, 275–292 (1985)
Atkinson, K.: A survey of numerical methods for solving nonlinear integral equations. J. Integral Equ. Appl. 4, 15–46 (1992)
Atkinson, K., Han, W.: Theoretical Numerical Analysis. A Functional Analysis Framework, Texts in Applied Mathematics 39. Springer, New York (2007)
Atkinson, K.: The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press, Cambridge (1997)
Atkinson, K., Graham, I., Sloan, I.: Piecewise continuous collocation for integral equations. SIAM J. Numer. Anal. 20, 172–186 (1983)
Atkinson, K., Potra, F.: Projection and iterated projection methods for nonlinear integral equations. SIAM J. Numer. Anal. 24, 1352–1373 (1987)
Atkinson, K., Flores, J.: The discrete collocation method for nonlinear integral equations. IMA J. Numer. Anal. 13, 195–213 (1993)
Atkinson, K., Potra F.: The discrete Galerkin method for nonlinear integral equations. J. Integral Equ. Appl. 1, 17–54 (1988)
Atkinson, K.: Iterative variants of the Nyström method for the numerical solution of integral equations. Numer. Math. 22, 17–31 (1973)
Atkinson, K.: The numerical evaluation of fixed points for completely continuous operators. SIAM J. Numer. Anal. 10, 799–807 (1973)
Babolian, E., Abbasbandy, S., Fattahzadeh, F.: A numerical method for solving a class of functional and two dimensional integral equations. Appl. Math. Comput. 198, 35–43 (2008)
Babolian, E., Biazar, J., Vahidi, A.R.: The decomposition method applied to systems of Fredholm integral equations of the second kind. Appl. Math. Comput. 148, 443–452 (2004)
Banaś, J., Sadarangani, K.: Solutions of some functional-integral equations in Banach algebra. Math. Comput. Model. 38, 245–250 (2003)
Bica, A.M.: New numerical method for Hammerstein integral equations with modified argument. An. Univ. Oradea, Fasc. Mat. 17, 33–44 (2010)
de Boor, C.: A Practical Guide to Splines. Springer, New York (2001)
Borzabadi, A.H., Kamyad, A.V., Mehne, H.H.: A different approach for solving the nonlinear Fredholm integral equations of the second kind. Appl. Math. Comput. 173, 724–735 (2006)
Borzabadi, A.H., Fard, O.S.: Numerical scheme for a class of nonlinear Fredholm integral equations of the second kind. J. Comput. Appl. Math. 232, 449–454 (2009)
Brutman, L.: An application of the generalized alternating polynomials to the numerical solution of Fredholm integral equations. Numer. Algorithms 5, 437–442 (1993)
Cerone, P., Dragomir, S.: Trapezoidal and midpoint-type rules from inequalities point of view. In: Anastassiou, G.A. (ed.) Handbook of Analytic Computational Methods in Applied Mathematics. Chapman & Hall/CRC Press, Boca Raton (2000)
Han, G.Q.: Extrapolation of a discrete collocation-type method of Hammerstein equations. J. Comput. Applied Math. 61, 73–86 (1995)
Han, G.Q., Zhang, L.: Asymptotic error expansion of a collocation-type method for Hammerstein equations. Appl. Math. Comput. 72, 1–19 (1995)
Han, G.Q.: Asymptotic expansion for the Nystrom method for nonlinear Fredholm integral equations of the second kind. BIT Numer. Math. 34, 254–262 (1994)
de Hoog, F., Weiss, R.: Asymptotic expansions for product integration. Math. Comput. 27, 295–306 (1973)
Hübner, O.: The Newton method for solving the Theodorsen integral equation. J. Comput. Appl. Math. 14, 19–30 (1986)
Ibrahim, I.A.: On the existence of solutions of functional integral equation of Urysohn type. Comput. Math. Appl. 57, 1609–1614 (2009)
Jankowski, T.: Positive solutions for fourth-order differential equations with deviating arguments and integral boundary conditions. Nonlinear Anal. 73, 1289–1299 (2010)
Kantorovich, L., Akilov, G.: Functional Analysis in Normed Spaces. Pergamon, London (1964)
Kaneko, H., Noren, R.D., Padilla, P.A.: Superconvergence of the iterated collocation methods for Hammerstein equations. J. Comput. Appl. Math. 80, 335–349 (1997)
Maleknejad, K., Almasieh, H., Roodaki, M.: Triangular functions (TF) method for the solution of nonlinear Volterra–Fredholm integral equations. Commun. Nonlinear Sci. Numer. Simul. 15, 3293–3298 (2010)
Kelley, C.T.: Approximation of solutions of some quadratic integral equations in transport theory. J. Integral Equ. 4, 221–237 (1982)
Kelley, C.T., Northrup, J.: A pointwise quasi-Newton method for integral equations. SIAM J. Numer. Anal. 25, 1138–1155 (1988)
Kelley, C.T., Sachs, E.: Broyden’s method for approximate solution of nonlinear integral equations. J. Integral Equ. 9, 25–43 (1985)
Kelley, C.T.: A fast two-grid method for matrix H-equations. Trans. Theory Stat. Phys. 18, 185–204 (1989)
Krasnoselskii, M.: Topological Methods in the Theory of Nonlinear Integral Equations. Macmillan, New York (1964)
Kress, R.: Linear Integral Equations. Springer, Berlin (1989)
Kumar, S., Sloan, I.: A new collocation-type method for Hammerstein equations. Math. Comput. 48, 585–593 (1987)
Kumar, S.: A discrete collocation-type method for Hammerstein equations. SIAM J. Numer. Anal. 25, 328–341 (1988)
Maleknejad, K., Mollapourasl, R., Nouri, K.: Study on existence of solutions for some nonlinear functional-integral equations. Nonlinear Anal. 69, 2582–2588 (2008)
Maleknejad, K., Nouri, K., Mollapourasl, R.: Investigation of the existence of solutions for some nonlinear functional–integral equations. Nonlinear Anal. 71, 1575–1578 (2009)
Maleknejad, K., Karami, M.: Numerical solution of non-linear Fredholm integral equations by using multiwavelets in the Petrov–Galerkin method. Appl. Math. Comput. 168, 102–110 (2005)
Maleknejad, K., Nouri, K., Sahlan, M.N.: Convergence of approximate solution of nonlinear Fredholm–Hammerstein integral equations. Commun. Nonlinear Sci. Numer. Simul. 15, 1432–1443 (2010)
Mastroiani, G., Monegato, G.: Some new applications of truncated Gauss–Laguerre quadrature formulas. Numer. Algorithms 49, 283–297 (2008)
Mennicken, R., Wagenfuhrer, E.: Numerische Mathematik, vol. 2. Vieweg, Braunschweig/ Wiesbaden (1977)
Micula, G., Micula, S.: Handbook of Splines. Mathematics and its Applications, vol. 462, Kluwer Academic, Dordrecht (1999)
Milne-Thomson, L.: Theoretical Hydrodynamics, 5th edn. Macmillan, New York (1968)
Plato, R.: Concise Numerical Mathematics, Graduate Studies in Mathematics, vol. 57. AMS Providence, Rhode Island (2003)
Petryshyn, W.: Projection methods in nonlinear numerical functional analysis. J. Math. Mech. 17, 353–372 (1967)
Rashed, M.T.: Numerical solution of functional differential, integral and integro-differential equations. Appl. Math. Comput. 156, 485–492 (2004)
Rashed, M.T.: Numerical solution of functional integral equations. Appl. Math. Comput. 156, 507–512 (2004)
O’Regan, D., Meehan, M.: Existence Theory for Nonlinear Integral and Integrodifferential Equations. Kluwer Academic, Dordrecht (1998)
Shimasaki, M., Kiyono, T.: Numerical solution of integral equations by Chebyshev series. Numer. Math. 21, 373–380 (1973)
Sommariva, A.: A fast Nyström-Broyden solver by Chebyshev compression. Numer. Algorithms 38, 47–60 (2005)
Stibbs, D., Weir, R.: On the H-function for isotropic scattering. Mon. Not. R. Astron. Soc. 119, 512–525 (1959)
Yao, Q.: Positive solutions for eigenvalue problems of fourth order elastic beam equations. Appl. Math. Lett. 17, 237–243 (2004)
Yao, Q.: Solvability of a fourth-order beam equation with all-order derivatives. Southeast Asian Bull. Math. 32, 563–571 (2008)
Zeidler, E.: Nonlinear Functional Analysis and its Applications, vol. I: Fixed Points Theorems. Springer, Berlin (1986)
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Bica, A.M. The numerical method of successive interpolations for Fredholm functional integral equations. Numer Algor 58, 351–377 (2011). https://doi.org/10.1007/s11075-011-9459-4
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DOI: https://doi.org/10.1007/s11075-011-9459-4