Communications in Numerical Analysis
Volume 2015, No. 1 (2015), Pages 51-61
Article ID cna-00231, 11 Pages
Analytic approximate solution for some integral equations by optimal homotopy analysis transform method
Mohamed S. Mohamed1,2 *, Muteb R. Alharthi1, Refah A. Alotabi1
1Mathematics Department, Faculty of Science,Taif University, Taif, Saudi Arabia
2Mathematics Department, Faculty of Science, Al-Azhar University, Cairo, Egypt
* Corresponding author. Email address: email@example.com
Received: 07 February 2015; Accepted: 03 March 2015
Copyright © 2015 Mohamed S. Mohamed, Muteb R. Alharthi and Refah A. Alotabi. 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.
The main aim of this paper is to propose a new and simple algorithm namely homotopy analysis transform method (HATM), to obtain approximate analytical solutions of integral equations. Integral equation occurs in the mathematical modeling of several models in physics, astrophysics, solid mechanics and applied sciences. The numerical solutions obtained by proposed method indicate that the approach is easy to implement and computationally very attractive. Finally, several numerical examples are given to illustrate the accuracy and stability of this method. Comparison of the approximate solution with the exact solutions also we show that the proposed method is very efficient and computationally attractive. A new efficient approach is proposed to obtain the optimal value of convergence controller parameter $\hslash$ to guarantee the convergence of the obtained series solution.
Keywords: Integeral equation; Optimal homotopy analysis transform method; Laplace transform.
A. M. Wazwaz, Linear and Nonlinear Integral Equations Methods and Applications, Springer, Heidelberg, Dordrecht, London, New York, (2011).
A. M. Wazwaz, A First Course in Integral Equations, World Scientific, New Jersey, (1997).
R. Gorenflo, S. Vessella, Abel Integral Equations, Springer, Berlin, (1991).
A. M. Wazwaz, M. S. Mehanna, The combined Laplace- Adomian method for handling singular integral equation of heat transfer, Int. J. Nonlinear Sci, 10 (2010) 248-252.
N. Zeilon, Sur quelques points de la theorie de l'equationintegraled'Abel, Arkiv. Mat. Astr. Fysik, 18 (1924) 1-19.
R. K. Pandey, Om P. Singh, V. K. Singh, Efficient algorithms to solve singular integral equations of Abel type, Comput. Math. Appl, 57 (2009) 664-676.
S. Kumar, Om P. Singh, Numerical inversion of Abel integral equation using homotopy perturbation method, Z. Naturforsch. 65 (2010) 677-682.
S. Kumar, Om P. Singh, S. Dixit, Homotopy perturbation method for solving system of generalized Abel's integral equations, Appl. Appl. Math, 6 (2009) 268-283.
S. Dixit, Om P. Singh, S. Kumar, A stable numerical inversion of generalized Abel's integral equation, Appl. Numer. Math, 62 (2012) 567-579.
S. A. Yousefi, Numerical solution of Abel's integral equation by using Legendre wavelets, Appl. Math. Comput, 175 (2006) 574- 580.
M. Khan, M. A. Gondal, A reliable treatment of Abel's second kind singular integral equations, Appl. Math. Lett, 25 (11) (2012) 1666-1670.
M. Li, W. Zhao, Solving Abel's type integral equation with Mikusinski's operator of fractional order, Adv. Math. Phys, Article ID 806984, 2013 (2013) 4 pages.
S. J. Liao, On the homotopy analysis method for nonlinear problems, Appl. Math. Comput, 147 (2004) 499-513.
S. J. Liao, Comparison between the homotopy analysis method and homotopy perturbation method, Appl. Math. Comput, 169 (2005) 1186-1194.
S. Abbasbandy, T. Hayat, A. Alsaedi, M. M. Rashidi, Numerical and analytical solutions for Falkner-Skan flow of MHD Oldroyd-B fluid, Internat. J. Numer. Methods Heat Fluid Flow, 24 (2) (2014) 390-401.
K. Hemida, M. S. Mohamed, Numerical simulation of the generalized Huxley equation by homotopy analysis method, Journal of applied functional analysis, 5 (4) (2010) 344-350.
S. Abbasbandy, R. Naz, T. Hayat, A. Alsaedi, Numerical and analytical solutions for Falkner-Skan flow of MHD Maxwell fluid, Appl. Math. Comput, 242 (2014) 569-575.
K. A. Gepreel, M. S. Mohamed, Analytical approximate solution for nonlinear space-time fractional Klein Gordon equation, Chinese physics B, 22 (1) (2013) 010201-6.
S. A. Khuri, A Laplace decomposition algorithm applied to a class of nonlinear differential equations, Journal of Applied Mathematics, 1 (2001) 141-155.
E. Yusufoglu, Numerical solution of Duffing equation by the Laplace decomposition algorithm, Applied Mathematics and Computation, 177 (2006) 572-580.
K. Yasir, An effective modication of the Laplace decomposition method for nonlinear equations, International Journal of Nonlinear Sciences and Numerical Simulation, 10 (2009) 1373-1376.
Y. Khan, N. Faraz, S. Kumar, A. A.Yildirim, A coupling method of homotopy method and Laplace transform for fractional modells, UPB Sci Bull Ser A Appl Math Phys, 74 (1) (2012) 57-68.
Sunil Kumar, Jagdev Singh, Devendra Kumar, Saurabh Kapoor, New homotopy analysis transform algorithm to solve Volterra integral equation, Ain Shams Eng J, 5 (2014) 243-246.
D. Kumar, J. Singh, Sunil Kumar, Analytical modeling for fractional multi-dimensional diffusion equations by using Laplace transform, Communications in Numerical Analysis, 1 (2015) 16-29.
A. S. Arife, S. K. Vanani, F. Soleymani, The Laplace homotopy analysis method for solving a general fractional difusion equation arising in nano-hydrodynamics, J Comput Theor Nanosci, 10 (2012) 1-4.
Mohamed S. Mohamed, Khaled A. Gepreel, Faisal Al-Malki, Maha Al-humyani, Approximate solutions of the generalized Abel's integral equations using the extension Khan's homotopy analysis transformation method, Journal of Applied Mathematics, Article ID 357861, 2015 (2015) 9 pages.
ZM. Odibat, Differential transform method for solving Volterra integral equation with seperable kernel, Math Comput Modell, 48 (2008) 1144-1149.
M. S. Mohamed, Analytical treatment of Abel integral equations by optimal homotopy analysis transform method, Journal of Information and Computing Science, 10 (1) (2015) 19-28.
S. M. Abo-Dahab, Mohamed S. Mohamed, T. A. Nofal, A One Step Optimal Homotopy Analysis Method for propagation of harmonic waves in nonlinear generalized magneto-thermoelasticity with two relaxation times under inuence of rotation, Journal of in Abstract and Applied Analysis, (2013) 1-14.
Khaled A. Gepreel, Mohamed S. Mohamed, An optimal homotopy analysis method onlinear fractional differential equation, Journal of Advanced Research Dynamical and Control Systems, 6 (2014) 1-10.
H. N. Hassan, M. S. Semary, Analytic approximate solution for the Bratu's problem by optimal homotopy analysis method, Communications in Numerical Analysis, (2013) 1-14.
S. Kumar, A. Kumar, D. Kumar, J. Singh, A. Singh, Analytical solution of Abel integral equation arising in astrophysics via Laplace transform, Journal of the Egyptian Mathematical Society, 23 (1) (2015) 102-107.
L. H. Yang, H. Y. Li, J. R. Wang, Solving a system of linear Volterra Integral equations using the modified reproducing kernel method, Abstract and Applied Analysis, Article ID 196308, 2013 (2013) 5 pages.