q-homotopy analysis method for solving nonlinear Fredholm integral equation of the second kind

Document Type : Research Paper

Authors

Faculty of Mathematics and Computer Science, University of Kufa, Iraq

Abstract

Several scientific and engineering applications are usually described as integral equations. We propose a numerical approach for solving the type of Fredholm nonlinear boundary value problems in a finite domain. The paper aims to use the q-homotopy analysis method to estimate the solution to test the efficiency of the proposed method. Comparison with updated work is exacted. The obtained results show that the proposed method is very effective and convenient for nonlinear Fredholm integral equations. The interval of convergence of homotopy analysis method, if exists, is increased when using q-homotopy analysis method is more to converge. The result reveals that the q- homotopy analysis method is considered a good method for solving NFIES.

Keywords

[1] S. Abbasbandy, Numerical solution of integral equation: Homotopy perturbation method and Adomian’s decomposition method, Appl. Math. Comput. 173 (2006) 493–500.
[2] K.E. Atkinson, A survey of numerical methods for the solution of Fredholm integral equations of the second kind,
J. Integ. Equ. Appl. 4(1) (1992) 15–46.
[3] K.E. Atkinson and H. Weimin. Theoretical Numerical Analysis: A Functional Analysis Framework, Springer,
2009.
[4] C.T.H. Baker, H.Geoffrey and F. Miller, Treatment of Integral Equations by Numerical Methods, Acad. Press,
1982.
[5] J. Bana, Integrable solutions of Hammerstein and Urysohn integral equations, J. Aust. Math. Soc. 46(1) (1989)
61–68.
[6] A.S. Bataineh, M.S.M. Noorani and I. Hashim, On a new reliable modification of homotopy analysis method,
Commun. Nonlinear Sci. Numerical Sim. 14 (2009) 409–423.
[7] N. Bildik and A. Konuralp, The use of variational iteration method, differential transform method and Adomian
decomposition method for solving different types of nonlinear partial differential equations, Int. J. Nonlinear Sci.
Numerical Sim. 7 (1) (2006) 65–70.
[8] H. Brunner, M.R. Crisci, E. Russo and A. Recchio, A family of methods for Abel integral equations of the second
kind, J. Comput. Appl. Math. 34 (1991) 211–219.[9] D. Bugajewski, On BV-solutions of some nonlinear integral equations, Integ. Equ. Ope. Theory 46 (2003) 387—
398.
[10] L.M. Delves and J. Walsh, Numerical Solution of Integral Equations, Clarendon Press, Oxford, 1974.
[11] M. El-Shahed, Application of He’s homotopy perturbation method to Volterra’s integro-differential equation, Int.
J. Nonlinear Sci. Numerical Sim. 6 (2005) 163–168.
[12] M.A. El-Tawil and S.N. Huseen, The q-homotopy analysis method (q-HAM), Int. J. Appl. Math. Mech. 8(15)
(2012) 51–75.
[13] M.A. El-Tawil and S.N. Huseen, On convergence of the q-homotopy analysis method, Int. J. Contemp. Math. Sci.
8(10) (2013) 481–497.
[14] A. Golbabai and B. Keramati, Modified homotopy perturbation method for solving Fredholm integral equations,
Chaos Solitons Fract. (2006), doi: 10.1016/j.chaos.2006.10.037.
[15] Y. Ikebe, The Galerkin method for the numerical solution of Fredholm integral equations of the Second Kind.
SIAM Rev. 14(3) (1972) 465–491.
[16] J.P. Kauthen, A survey of singular perturbed Volterra equations, Appl. Num. Math. 24 (1997) 95–114.
[17] A.A. Kilbas and M. Saigo, On solution of nonlinear Abel–Volterra integral equation, J. Math. Anal. Appl. 229
(1999) 41-–60.
[18] S.J. Liao, The proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems. PhD dissertation,
Shanghai Jiao Tong University, 1992.
[19] S.J. Liao. An approximate solution technique not depending on small parameters: A special example. Int. J.
Nonlinear Mech. 30(2) (1995) 371–380.
[20] S.J. Liao, What’s the common ground of all numerical and analytical tichniques for nonlinear problems, Commun.
Nonlinear Sci. Numerical Sim. 1(4) (1996) 26–30.
[21] S.J. Liao, Homotopy analysis method: A new analytical technique for nonlinear problems, Commun. Nonlinear
Sci. Numerical Sim. 2(2) (1997) 95–100.
[22] S.J. Liao, A direct boundary element approach for unsteady non-linear heat transfer problems, Engin. Anal.
Boundary Element Method 26 (2002) 55–59.
[23] S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, CRC Press, Chapman & Hall,
2003.
[24] S.J. Liao, On the homotopy analysis method: for nonlinear problems, Appl. Math. Comput. 147 (2004) 499–513.
[25] S.J. Liao, Notes on the homotopy analysis method: Some definitions and theorems, Commun. Nonlinear Sci.
Numerical Sim. 14 (2009) 983–997.
[26] B. Mandal and S. Bhattacharya, Numerical solution of some classes of integral equations using Bernstein polynomials, Appl. Math. Comput. 190(2) (2007) 1707–1716.
[27] D. Maturi, The successive approximation method for solving nonlinear Fredholm integral equation of the second
kind using Maple. Adv. Pure Math. 9 (2019) 832–843.
[28] R.A. Van Gorder and K. Vajravelu. On the selection of auxiliary functions, operators, and convergence control parameters in the application of the homotopy analysis method to nonlinear differential equations:A general
approach, Commun. Nonlinear Sci. Numerical Sim. 14 (2009) 4078–4089
[29] H. Wang, L. Zou and H. Zhang. Applying homotopy analysis method for solving differential difference equation,
Physics Letters A 369 (2007) 77–84.
[30] A.M. Wazwaz, Linear and Nonlinear Integral Equation: Methods and Applications, Springer, Berlin, 2011.
[31] J. A. Zarnan, On the numerical solution of Urysohn integral equation using Chebyshev polynomial, Int. J. Basic
Appl. Sci. 16(6) (2016) 23–27.
[32] J. A. Zarnan, A novel approach for the solution of a class of Urysohn integral equations using Bernstein polynomials, Int. J. Adv. Res. 5(1) (2017) 2156–2162.
Volume 12, Issue 2
November 2021
Pages 2145-2152
  • Receive Date: 12 April 2021
  • Revise Date: 08 June 2021
  • Accept Date: 04 July 2021