American Journal of Computational Mathematics, 2012, 2, 312315 http://dx.doi.org/10.4236/ajcm.2012.24042 Published Online December 2012 (http://www.SciRP.org/journal/ajcm) Numerical Solution of Generalized Abel’s Integral Equation by Variational Iteration Method R. N. Prajapati1, Rakesh Mohan1, Pankaj Kumar2 1Department of Mathematics, Dehradun Institute of Technology, Dehradun, India 2Kishori Lal Sharma Institue of Engineering & Technology, Chandok, Bijnor, India Email: narain.prajapati@gmail.com, rkmohan2k4@yahoo.com, Pankaj.rssb@gmail.com Received June 1, 2012; revised September 13, 2012; accepted October 8, 2012 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 yn(x) converges to the exact solution irrespective of the initial choice y0(x). Illustrative numerical examples are given to demonstrate the effi ciency and simplicity of the method in solving these types of singular integral equations. Keywords: Variational Iteration Method; Singular Integral Equation; Abel’s Kernel 1. Introduction The real world problems in scientific fields such as solid state physics, plasma physics, fluid mechanics, chemical kinetics and mathematical biology are nonlinear in gen eral when formulated as partial differential equations or integral equations. In the last two decades, many power ful techniques have been proposed to solve the singular integral equations and differential equation by using VIM [115]. The generalized Abel’s integral equation of the second kind is given by 0 d, 01, xyt yxf xtx xt (1) where C is a parameter and 01. Theorem 1. For each complex , 0 d, x yxf xRxtftt (2) is the unique solution of Equation (1), where 1 1 1 , 1 n n x Rx xn (3) The closed form solution (2) is not very useful in many cases where it is difficult to evaluate the integral appearing in (2). So, it is desirable to have numerical solution for the generalized Abel’s integral Equation (1). In the present paper, we have proposed an algorithm based on the variational iteration method to solve the generalized Abel’s integral Equation (1). It is observed that the choice of the initial approximation 0 x has a small effect on the efficiency of the method. The appro ximate solutions n x will always converge to the exact solution. 2. Basic Idea of Variational Iteration Method Variational Iteration method was first proposed by He [26] and has been successfully used by many research ers to solve various linear and nonlinear models [715]. The idea of the method is based on constructing a correc tion functional by a general Lagrange multiplier and the multiplier is chosen in such a way that its correction so lution is improved with respect to the initial approxima tion or to the trial function. Now, to illustrate the basic concept of the variational iteration method, we consider the following general non linear system: ,Lyx Nyxfx (4) where L is a linear operator, N is a nonlinear operator and x is a known analytic function. The basic character of the method is to construct a correction functional for the system, which reads 1 0 d, 0 nn x nn yxyx LysN ysfss n (5) C opyright © 2012 SciRes. AJCM
R. N. PRAJAPATI ET AL. 313 where is a general Lagrange multiplier, which can be identified optimally via variational theory, n is the nth approximate solution, and n is considered as a re stricted variation, i.e. namely y y 0. n y Successive ap proximations, 1, n x 0 will be obtained by applying the obtained Lagrange multiplier and a properly chosen initial approximation . x 3. Variational Iteration Method of Solution We consider the following iteration formula for Equation (1) in the following form 1 0 d,0,1,2,3,; xn n yt yxfx tn xt (6) where n x is the nth approximate solution of (1) and 0 x is an appropriately chosen initial guess. The value of is found to be −1. 4. Numerical Examples The simplicity and accuracy of the proposed method are illustrated by the following numerical examples by com puting the absolute error , nn xyxyx where x is the exact solution and n x is the nth appro ximate solution of the problem. The absolute error has evaluated with examples 1 and 2 for value of and also the absolute error has calculated with examples 3 and 4 for different values of n Ex 20 n , 30n and respectively. 14n Example 1. Consider the following generalized Abel’s integral equation of second kind 24 74 34 12 0 Fresnel 16131115 1923π ,,1,, , ,, 214416 16 16 1616 d, x yx Cx xF yt t xt (7) where 2 0 π Fresnelcosd, 2 xt Cx t with exact solution Fresnel xCx solve the above integral equation by taking 4 different choices of the initial guess 0. x It is observed that the method always converges to the exact solution. Case 1a. Taking the initial guess 0. x as 0 24 74 34 Fresnel 16131115 1923π ,,1,, , ,,, 214416 16 16 1616 yx Cx xF the various approximate solutions , n x obtained from Equation (6), are given as 1 2 52 24 34 Fresnel 3 84137911 13π ,,1,,, ,, 4488 8816 15 π yx Cx x F 2 3 13 4 24 34 Fresnel 3 41 317212529π ,,1,, ,,, 17 4416 16161616 4 yx Cx x F Case 1b. Now we take a value of the initial gu different ess 0 xerfcx and we get 34 4x 1 74 2 22 24 74 34 Fresnel 3 32111 15 ,1 ,,, 288 21 π 161311 15 1923π ,,1,, , ,,, 214416 16 16 1616 yx Cx xFx x xF 2 32 2 2 52 2 22 2 52 24 34 3 44 Fresnel 3π 3 16 4179 ,1 ,,, 15π244 3 84137911 13π ,,1,,, ,, 4488 8816 15 π x yx Cx x Fx x F Case 1c. Taking a different value of the initial guess 01 xx and solving we obtain 34 416x74 1 24 74 34 Fresnel 321 161311 15 1923π ,,1,, , ,,, 214416 16 16 1616 yx Cxx x xF 2 32 2 24 34 3 44 Fresnel 15 π 137911 13π 522,,1,,, ,,, 44 8888 16 x yx Cx x xxF Case 1d. In this case, we are taking the initial guess 0cos xx lowing approximati and using Equation (6) we get the fol ons of the solution Copyright © 2012 SciRes. AJCM
R. N. PRAJAPATI ET AL. 314 1 34 2 12 Fresnel 4711 1, ,, x Cx xx F 24 74 34 38 84 161311 15 1923π ,,1,, , ,,, 214416 16 16 1616 y x xF 2 2 32 2 12 2 52 24 34 Fresnel 3 4457 1, , , 44 4 3π 3 84137911 13π ,,1,,, ,, 4488 8816 15 π yx Cx xx F x F Figures 14 show the errors between the exact solution x and the approximate solutions 20 x for the different initials choices of 0 x foove four paper, the variational iteration method has been a user friendly algorithm nder the variations of initial guess r the ab cases. 5. Conclusion In this successfully used to obtain which is stable u 0 x to solve the generalized Abel’s integral equations. The variational iteration method yields solutions in the forms of a convergent series with easily calculable terms. Figure 1. The absolute error E(x) for example 1 (case a). Figure 3. The absolute error E(x) for example 1 (case c). Figure 4. The absolute error for E(x) example 1 (case d). It is shown that the variational iteration method is a promising tool for such types of singular integral equa tions. REFERENCES [1] S. Abbasbandy and E Shivanian, “Applications Varia tional Iteration Method for nth Order Integro—Differen tial Equations,” Zeitschrift für Naturforschung A, Vol 64a, 2009, pp. 439444. f i:10.1016/S00207462(98)000481 . [2] J. H. He, “Variational Iteration Method—A Kind o Nonlinear Analytical Technique: Some Examples,” In ternational Journal of NonLinear Mechanics, Vol. 34, No. 4, 1999, pp. 699708. do [3] J. H. He, “Some Asymptotic Methods for Strongly NonLinear EJournal of Modern Physics B, Vo1411199. quations,” International l. 20, No. 10, 2006, pp. 1 doi:10.1142/S0217979206033796 [4] J. H. He, “Variational Iteration MethodSome Recent Results and New Interpretations,” Journal of Computa tional and Applied Mathematics, Vol. 27, No. 1, 2007, pp. 317. doi:10.1016/j.cam.2006.07.009 [5] J. H. He, “NonPerturbative Methods for Strongly Non linear Problems, Dissertation,” Deverlag im Internet GmbH, Berlin, 2006. [6] J. H. He and X. H. Wu, “Constructio tio n of Solitary Solu “He’s Variational n and ComptonLike Solution by Variational Iteration Method,” Chaos, Solitons and Fractals, Vol. 29, 2006, pp. 108113. [7] S. A. Yousefi, A. Lotfi and M. Dehgan, Figure 2. The absolute error E(x) for example 1 (case b). Copyright © 2012 SciRes. AJCM
R. N. PRAJAPATI ET AL. Copyright © 2012 SciRes. AJCM 315 1112, 2009, pp Iteration Method for Solving Nonlinear Mixed Volterra Fredholm Integral Equations,” Computers & Mathematics with Applications, Vol. 58, No. . 21722176. doi:10.1016/j.camwa. 2009.03.083 [8] M. Tatari and M. Dehghan, “Improvement of He’s Varia tional Iteration Method for Solving Systems of Differen tial Equations,” Computers & Mathematics with Applica tions, Vol. 58, No. 1112, 2009, pp. 21602166. 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 & Mathe matics with Applications, Vol. 58, No. 1112, 2009, pp. 21672171. doi:10.1016/j.camwa.2009.03.008 [10] T. Ozis and A. Yildirim, “A Study of Nonlinear Oscilla tors with u1/3 Force by He’s Variational Iteration Method,” Journal of Sound and Vibration, Vol. 306, No. 12, 2007, pp. 372376. doi:10.1016/j.jsv.2007.05.021 [11] S. Momani and Z. M. Odibat, “Numerical Comp Methods for Solving Linear Diff arison of erential Equations of Fractional Order,” Chaos, Solitons and Fractals, Vol. 31, No. 5, 2007, pp. 12481255. doi:10.1016/j.chaos.2005.10.068 [12] S. Abbasbandy and E. Shivanian, “Application of Varia tion Solution of a Non uville’s Fractional De tional Iteration Method for nthOrder IntegroDifferential Equations,” Zeitschrift für Naturforschung A, Vol. 64a, 2009, pp. 439444. [13] S. Abbasbandy, “An Approxima linear Equation with RiemannLio rivatives by He’s Variational Iteration Method,” Journal of Computational and Applied Mathematics, Vol. 207, No. 1, 2007, pp. 5358. doi:10.1016/j.cam.2006.07.011 [14] S. Abbasbandy, “A New Application of He’s Variational Iteration Method for Quadratic Riccati Differential Equa tion by Adomian Polynomials,” Journal of Computa tional and Applied Mathematics, Vol. 207, No. 1, 2007, pp. 5963. doi:10.1016/j.cam.2006.07.012 [15] N. H. Sweilam and M. M. khader, “Variational Iteration Method for One Dimensional Nonlinear Thermo Elastic ity,” Chaos, Solitons and Fractals, Vol. 32, No. 1, 2007, pp. 145149. doi:10.1016/j.chaos.2005.11.028
