Numerical Solution of Generalized Abel’s Integral Equation by Variational Iteration Method

R. N. Prajapati, Rakesh Mohan, Pankaj Kumar

Department of Mathematics, Dehradun Institute of Technology, Dehradun, India.
Kishori Lal Sharma Institue of Engineering & Technology, Chandok, Bijnor, India.

DOI: 10.4236/ajcm.2012.24042                

Abstract

In this paper, a user friendly algorithm based on the variational iteration method (VIM) is proposed to solve singular integral equations with generalized Abel’s kernel. It is observed that an approximate solutions y_n(x) converges to the exact solution irrespective of the initial choice y₀ (x). Illustrative numerical examples are given to demonstrate the efficiency and simplicity of the method in solving these types of singular integral equations.

Keywords

Variational Iteration Method; Singular Integral Equation; Abel’s Kernel

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	S. Abbasbandy and E Shivanian, “Applications Variational Iteration Method for n-th Order Integro—Differential Equations,” Zeitschrift für Naturforschung A, Vol. 64a, 2009, pp. 439-444.
[2]	J. H. He, “Variational Iteration Method—A Kind of Nonlinear Analytical Technique: Some Examples,” International Journal of Non-Linear Mechanics, Vol. 34, No. 4, 1999, pp. 699-708. doi:10.1016/S0020-7462(98)00048-1
[3]	J. H. He, “Some Asymptotic Methods for Strongly Non-Linear Equations,” International Journal of Modern Physics B, Vol. 20, No. 10, 2006, pp. 1141-1199. doi:10.1142/S0217979206033796
[4]	J. H. He, “Variational Iteration Method-Some Recent Results and New Interpretations,” Journal of Computational and Applied Mathematics, Vol. 27, No. 1, 2007, pp. 3-17. doi:10.1016/j.cam.2006.07.009
[5]	J. H. He, “Non-Perturbative Methods for Strongly Nonlinear Problems, Dissertation,” Deverlag im Internet GmbH, Berlin, 2006.
[6]	J. H. He and X. H. Wu, “Construction of Solitary Solution and Compton-Like Solution by Variational Iteration Method,” Chaos, Solitons and Fractals, Vol. 29, 2006, pp. 108-113.
[7]	S. A. Yousefi, A. Lotfi and Mehdi Dehgan, “He’s Variational Iteration Method for Solving Nonlinear Mixed Volterra-Fredholm Integral Equations,” Computers & Mathematics with Applications, Vol. 58, No. 11-12, 2009, pp. 2172-2176. doi:10.1016/j.camwa. 2009.03.083
[8]	M. Tatari and M. Dehghan, “Improvement of He’s Variational Iteration Method for Solving Systems of Differential Equations,” Computers & Mathematics with Applications, Vol. 58, No. 11-12, 2009, pp. 2160-2166. doi:10.1016/jcamwa. 2009.03.081
[9]	R. Saadati, M. Dehghan, S. M. Vaezpour and M. Saravi, “The Convergence of He’s Variational Iteration Method for Solving Integral Equations,” Computers & Mathematics with Applications, Vol. 58, No. 11-12, 2009, pp. 2167-2171. doi:10.1016/j.camwa.2009.03.008
[10]	T. Ozis add A. Yildirim, “A Study of Nonlinear Oscillators with u1/3 Force by He’s Variational Iteration Method,” Journal of Sound and Vibration, Vol. 306, No. 1-2, 2007, pp. 372-376. doi:10.1016/j.jsv.2007.05.021
[11]	S. Momani and Z.M. Odibat, “Numerical Comparison of Methods for Solving Linear Differential Equations of Fractional Order,” Chaos, Solitons and Fractals, Vol. 31, No. 5, 2007, pp. 1248-1255. doi:10.1016/j.chaos.2005.10.068
[12]	S. Abbasbandy and E. Shivanian, “Application of Variational Iteration Method for nth-Order Integro-Differential Equations,” Zeitschrift für Naturforschung A, Vol. 64a, 2009, pp. 439-444.
[13]	S. Abbasbandy, “An Approximation Solution of a Nonlinear Equation with Riemann-Liouville’s Fractional Derivatives by He’s Variational Iteration Method”, Journal of Computational and Applied Mathematics, Vol. 207, No. 1, 2007, pp. 53-58. doi:10.1016/j.cam.2006.07.011
[14]	S. Abbasbandy, “A New Application of He’s Variational Iteration Method for Quadratic Riccati Differential Equation by Adomian Polynomials,” Journal of Computational and Applied Mathematics, Vol. 207, No. 1, 2007, pp. 59-63. doi:10.1016/j.cam.2006.07.012
[15]	N. H. Sweilam and M. M. khader, “Variational Iteration Method for One Dimensional Nonlinear Thermo Elasticity,” Chaos, Solitons and Fractals, Vol. 32, No. 1, 2007, pp. 145-149. doi:10.1016/j.chaos.2005.11.028