Communications in Numerical Analysis

Volume 2012 (2012), Article ID cna-00108, 20 Pages

doi: 10.5899/2012/cna-00108


Research Article


The Use of Iterative Methods to Solve Two-Dimensional Nonlinear Volterra-Fredholm Integro-Differential Equations


Sh. Sadigh Behzadi *


Department of Mathematics, Islamic Azad University, Central Tehran Branch, Tehran, Iran.


* Corresponding author. Email address: shadan_behzadi@yahoo.com; Tel: +989123409593


Received: 18 October 2011; Accepted: 16 Februar 2012


Copyright © 2012 Sh. Sadigh Behzadi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this present paper, we solve a two-dimensional nonlinear Volterra-Fredholm integro-differential equation by using the following powerful, efficient but simple methods:


(i) Modified Adomian decomposition method (MADM),


(ii) Variational iteration method (VIM),


(iii) Homotopy analysis method (HAM) and


(iv) Modified homotopy perturbation method (MHPM).


The uniqueness of the solution and the convergence of the proposed methods are proved in detail. Numerical examples are studied to demonstrate the accuracy of the presented methods.


Keywords: Two-dimensional Volterra and Fredholm integral equations; Integro-differential equations; Modified Adomian decomposition method; Variational iteration method; Homotopy analysis method; Modified homotopy perturbation method.

References

  1. S. Abbasbany, Homptopy analysis method for generalized Benjamin-Bona-Mahony equation, Zeitschriff fur angewandte Mathematik und Physik ( ZAMP) 59 (2008) 51-62.


  2. S. Abbasbandy, Numerical method for non-linear wave and diffusion equations by the variational iteration method, Q1 Int. J. Numer. Methods Eng. 73 (2008) 1836-1843.


  3. S. Abbasbandy andA. Shirzadi, The variational iteration method for a class of eight-order boundary value differential equations, Z. Naturforsch. A 63(a) (2008) 745-751.


  4. S. Abbasbandy and E. Shivanian, Application of the variational iteration method for nonlinear Volterra's integrodifferential equations, Z. Naturforsch A 63(a) (2008) 538-542.


  5. G. Adomian, A review of the decomposition method in applied mathematics, J. Math. Anal.Appl, 135 (1988) 501 - 544.


  6. N. Bildik, M. Inc, Modified decomposition method for nonlinear Volterra-Fredholm integral equations, Chaos, Solitons and Fractals 33 (2007) 308-313.


  7. P. Darania, E. Abadian, Development of the Taylor expansion approach for nonlinear integro-differential equations, Int. J. Contemp. Math. Sci. 14 (2006) 651-664.


  8. P. Darania, K. Ivaz, Numerical solution of nonlinear Volterra-Fredholm integro-differential equations, Appl. Math. Comput. 56 (2008) 2197-2209.


  9. P. Darania, E. Ebadian, Numerical solution of the nonlinear two-dimensional Volterra integral equations, New Zealand Journal of Mathematics 36 (2007) 163-174.


  10. V. Didenko, B. Silbermann, On the approximate solution of some two-dimensional singular integral equations, Mathematics Methods in the Applied Sciences 24 (2001) 1125-1138.


  11. M. Dehghan, Application of the Adomian decomposition method for two-dimensional parabolic equation subject to nonstandard boundary specifications, Applied mathematics and computation 157 (2004) 549-560.


  12. I.L. El-Kalaa, Convergence of the Adomian method applied to a class of nonlinear integral equations, App.Math.Comput. 21 (2008) 327-376.


  13. D. Eyre, Cubic spline-projection method for two-dimensional equations of scattering thery, Journal of Computational Physics 114 (1994) 1-8.


  14. M.A. Fariborzi Araghi, Sh. S. Behzadi, Solving nonlinear Volterra-Fredholm integro-differential equations using the modified Adomian decomposition method, Comput. Methods in Appl. Math. 9 (2009) 1-11.


  15. M.A. Fariborzi Araghi, Sh.S. Behzadi, Numerical solution of nonlinear Volterra-Fredholm integro-differential equations using Homotopy analysis method, Journal of Applied Mathematics and Computing, 37 (2011), 1-12.


  16. M.A. Fariborzi Araghi, Sh.S. Behzadi, Solving nonlinear Volterra-Fredholm integro-differential equations using He's variational iteration method, International Journal of Computer Mathematics, 88 (2010), 829-838.


  17. H. Guoqiang, W. Jiong, Extrapolation of nystrom solution for two dimensional nonlinear Fredholm integral equations, J. Comput. Apll. Math. 134 (2001) 259-268.


  18. H. Guoqiang, W. Ruifang, Richardson extrapolation of iterated discrete Galerkin solution for two dimensional nonlinear Fredholm integral equations, J. Comput. Apll. Math. 139 (2002) 49-63.


  19. A. Golbabai , B. Keramati , Solution of non-linear Fredholm integral equations of the first kind using modified homotopy perturbation method, Chaos, Solitons and Fractals 5 (2009) 2316-2321.


  20. M. Ghasemi, M. Tavassoli Kajani, A. Davari, Numerical solution of two-dimensional nonlinear differential equation by homotopy perturbation method, Applied Mathematics and Computation 189 (2007) 341-345.


  21. A. Golbabai, B. Keramati, Modified homotopy perturbation method for solving Fredholm integral equations, Chaos, Solitons and Fractals 37 (2008) 1528 - 1537.


  22. J. H. He, Variational iteration method for autonomous ordinary differential system, Appl. Math. Comput. 114 (2000) 115-123.


  23. J. H. He, Approximate analytical solution for seepage folw with fractional derivatives in porous media, Comput. Methods. Appl. Mech. Eng. 167 (1998) 57-68.


  24. J. H. He, Shu-Qiang Wang, Variational iteration method for solving integro-differential equations, Physics Letters A 367 (2007) 188-191.


  25. J. H. He, Variational principle for some nonlinear partial differential equations with variable cofficients, Chaos, Solitons and Fractals 19 (2004) 847-851.


  26. M. Hadizadeh, N. Moatamedi, A new differential transformation approach for two-dimensional Volterra integral equations, International Journal of Computer Mathematics, 84 (2007), 515-526.


  27. J.H. He, Variational iteration method-a kind of nonlinear analytical technique: Some examples, International Journal of Nonlinear Mechanics 34 (1999) 699-708.


  28. J.H. He, Variational iteration method for autonomous ordinary differential systems, Appl. Math. Comput. 114 (2000) 115 - 123.


  29. J.H. He, Variational iteration method-Some recent results and new interpretations, Journal of Computational and Applied Mathematics 207 (2007) 3-17.


  30. J.H. He, Homotopy perturbation method for solving boundary value problems, Phys. Lett. A. 350 (2006) 87-98.


  31. M. Inokuti, General use of the Lagrange multiplier in non-linear mathematical physics, in: S. Nemat-Nasser (Ed.), Variational Method in the Mechanics of Solids, Pergamon Press, Oxford, (1978) 156-162.


  32. M. Javidi, Modified homotopy perturbation method for solving non-linear Fredholm integral equations, Chaos, Solitons and Fractals 50 (2009) 159-165.


  33. M. Jalaal, D. Ganji, F. Mohammadi, He's homotopy perturbation method for two-dimensional heat conduction equation: Comparison with finite element method, Heat Transfer-Asian Research 39 (2010) 232-245.


  34. M. Javidi, A. Golbabai, Modified homotopy perturbation method for solving nonlinear Fredholm integral equations, Chaos, Solitons and Fractals 4 (2009)1408 -1412.


  35. D. Kaya, An application of the modified decomposition method for two dimensional sine-Gordon equation, Applied Mathematics and Computation 159 (2004) 1-9.


  36. S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman and Hall/CRC Press, Boca Raton,2003.


  37. S.J. Liao, Notes on the homotopy analysis method:some definitions and theorems, Communication in Nonlinear Science and Numerical Simulation 14 (2009) 983-997.


  38. S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman and Hall/CRC Press, Boca Raton, 2003.


  39. J. Lin, Application of the Modified Homotopy Perturbation Method to the Two Dimensional sine-Gordon Equation, Int. J. Contemp. Math. Sciences 5 (2010) 985 - 990.


  40. K. Maleknejad, Y. Mahmoudi, Taylor polynomial solution of high-order nonlinear Volterra-Fredholm integro-differential equations, Appl. Math. Comput. 145 (2003) 641-653.


  41. W. Rudin, Principles of Mathematical Analysis, 3re ed. McGraw-Hill, New York, 1976.


  42. A. Rahman, Adomian decomposition method for two-dimensional nonlinear volterra integral equations of the second kind, Far East Journal of Applied Mathematics 34 (2009) 169-179.


  43. Sh. Sadigh Behzadi, The convergence of homotopy methods for solving nonlinear Klein-Gordon equation, J. Appl. Math. Informatics 28 (2010) 1227-1237.


  44. A. Tari, Modified Homotopy Perturbation Method for Solving two-dimensional Fredholm Integral Equations, International Journal of Computational and Applied Mathematics 5 (2010) 585- 593.


  45. F. Tian-You, S. Zhu-Feng, A class of two-dimensional dual integral equations and its application, Applied Mathematics and Mechanics 28 (2007) 247-252.


  46. A. Tari, M.Y. Rahimi, S. Shahmorad, F. Talati, Solving a class of two-dimensional linear and nonlinear Volterra integral equations by the differential transform method, Appl. Math. Comput. 228 (2009) 70-76.


  47. A.R. Vahidi, Solution of a system of nonlinear equations by Adomian decomposition method, Journal of Applied Mathematics and Computation 150 (2004) 847-854.


  48. A.M. Wazwaz, A first course in integral equations, WSPC, New Jersey, 1997.


  49. A.M. Wazwaz, S.M. El-Sayed, A new modification of the Adomian decomposition method for linear and nonlinear operators, Appl. Math. Comput. 122 (2001) 393-404.


  50. A.M. Wazwaz, Construction of solitary wave solution and rational solutions for the KdV equation by ADM, Chaos, Solution and fractals 12 (2001) 2283-2293.


  51. W.J. Xie, F.R. Lin, A fast numerical solution method for two-dimensional Fredholm integral equations of the second kind, Applied Numerical Mathematics 59 (2009) 1709-1419.


  52. S. Yalcinbas, M. Sezar, The approximate solution of high-order linear Volterra-Fredholm integro-differential equations in terms of Taylor polynomials, Appl. Math. Comput. 112 (2000) 291-308.


  53. H. Zhua, H. Shub, M. Ding, Numerical solutions of two-dimensional Burgers' equations by discrete Adomian decomposition method, Computers and Mathematics with Applications 60 (2010) 840-848.


  54. T.T. Zhang, L. Jia, Z.C. Wanga, X. Lia, The application of homotopy analysis method for 2-dimensional steady slip flow in microchannels, Physics Letters A. 372 ( 2008) 3223-3227.