 American Journal of Computational Mathematics, 2012, 2, 312-315 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 [1-15]. The generalized Abel’s integral equation of the second kind is given by  0d, 01,xytyxf xtxxt  (1) where C is a parameter and 01. Theorem 1. For each complex ,   0d,xyxf xRxtftt (2) is the unique solution of Equation (1), where  111,1nnxRx 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 0yx has a small effect on the efficiency of the method. The appro- ximate solutions nyx will always converge to the exact solution. 2. Basic Idea of Variational Iteration Method Variational Iteration method was first proposed by He [2-6] and has been successfully used by many research- ers to solve various linear and nonlinear models [7-15]. 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 fx is a known analytic function. The basic character of the method is to construct a correction functional for the system, which reads  10d,0nnxnnyxyxsLysN ysfssn (5) Copyright © 2012 SciRes. AJCM R. N. PRAJAPATI ET AL. 313where  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 yy0.nySuccessive ap- proximations, 1,nyx0 will be obtained by applying the obtained Lagrange multiplier and a properly chosen initial approximation .yx 3. Variational Iteration Method of Solution We consider the following iteration formula for Equation (1) in the following form  10d,0,1,2,3,;xnnytyxfx tnxt  (6) where nyx is the nth approximate solution of (1) and 0yx 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 ,nnExyxyx where yx is the exact solution and nyx 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 nEx20n,30n and respectively. 14nExample 1. Consider the following generalized Abel’s integral equation of second kind  247434120Fresnel16131115 1923π,,1,, , ,,214416 16 16 1616d,xyx CxxxFyt txt    (7) where20πFresnelcosd,2xtCx t with exact solution  FresnelyxCx solve the above integral equation by taking 4 different choices of the initial guess 0.yx It is observed that the method always converges to the exact solution. Case 1a. Taking the initial guess 0.yx as  0247434Fresnel16131115 1923π,,1,, , ,,,214416 16 16 1616yx CxxxF    the various approximate solutions ,nyx obtained from Equation (6), are given as 12522434Fresnel384137911 13π,,1,,, ,,4488 881615 πyx CxxxF    2313 42434Fresnel341 317212529π,,1,, ,,,17 4416 161616164yx CxxxF     Case 1b. Now we take a value of the initial gudifferent ess 0yxerfcx and we get  344x174222247434Fresnel 332111 15,1 ,,,28821 π161311 15 1923π,,1,, , ,,,214416 16 16 1616yx CxxFxxxF        23222522222522434344Fresnel 3π316 4179,1 ,,,15π244384137911 13π,,1,,, ,,4488 881615 πxyx CxxFxxxF        Case 1c. Taking a different value of the initial guess 01yxx and solving we obtain  34416x741247434Fresnel 321161311 15 1923π,,1,, , ,,,214416 16 16 1616yx CxxxxF     23222434344Fresnel 15 π137911 13π522,,1,,, ,,,44 8888 16xyx CxxxxF     Case 1d. In this case, we are taking the initial guess 0cosyxx lowing approximatiand using Equation (6) we get the fol- ons of the solution Copyright © 2012 SciRes. AJCM R. N. PRAJAPATI ET AL. 314  134 212Fresnel47111, ,,x CxxxF2474343884161311 15 1923π,,1,, , ,,,214416 16 16 1616yxxF     22322122522434Fresnel344571, , ,44 43π384137911 13π,,1,,, ,,4488 881615 πyx CxxxFxxF      Figures 1-4 show the errors between the exact solution yx and the approximate solutions 20yx for the different initials choices of 0yx foove four paper, the variational iteration method has been a user friendly algorithm nder the variations of initial guess r the abcases. 5. Conclusion In this successfully used to obtainwhich is stable u0yx 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. 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