Generalized Abel Inversion Using Homotopy Perturbation Method

Many problems in physics like reconstruction of the radially distributed emissivity from the line-of-sight projected intensity, the 3-D image reconstruction from cone beam projections in computerized tomography, etc. lead naturally, in the case of radial symmetry, to the study of Abel’s type integral equation. Obtaining the physically relevant quantity from the measured one requires, therefore the inversion of the Abel’s integral equation. The aim of this letter is to present a user friendly algorithm to invert generalized Abel integral equation by using homotopy perturbation method. The stability of the algorithm is analysed. The validity and applicability of this powerful technique is illustrated through various particular cases which demonstrate its efficiency and simplicity in solving these types of integral equations.


Introduction
Since Abel formulated his integral equation [1] and presented its analytic solution, the equation has found application in many branches of physical science.The earliest application, due to Mach [2], arose in the study of compressible flows around axially symmetric bodies.Usually, physical quantities accessible to measurement are quite often related to physically important but experimentally inaccessible ones by Abel's integral equation, [3][4][5][6][7][8][9].Obtaining the physically relevant quantity from the measured one requires, therefore, the inversion of the Abel's integral equation, and in case the object does not have radial symmetry, it requires, in principal, the inversion of Random transform.
We consider the following generalized Abel's integral equation which is equivalent to Though, while the analytic solution to the Abel Equation (1) is given by (2), in practice we have only a point wise approximation to g, so the inversion must be carried out numerically.Since the integral transform (2) is equivalent to fractional differentiation of order 1   some amplification of data noise is inevitable [11].As the process of estimating the solution function  , y x if the data function   g x is given approximately and only at a discrete set of data points, is ill-posed since even very small, high frequency errors in the measured   g x , such as will arise from experimental errors, photon counting noise and noise in the electronics, might cause large errors in the reconstructed solution  . y x This is due to the fact that inversion formula (3) requires differentiating the measured data   g x .In 1982, an analytic but derivative free inversion formula was obtained by Deutsch and Beniaminy [12] to avoid this problem.In addition, many numerical inversion methods [13][14][15][16][17][18][19][20][21] have been developed with varying degree of success with the inherent limita-tions of all measured data.Consequently, the direct use of ( 2) and ( 3) are restricted and stable numerical methods become important.
The aim of the present letter is to propose an algorithm to invert the Abel's integral Equation (1) by using the homotopy perturbation method (HPM).We construct a convex homotopy by using HPM to obtain an iterative solution to (1) and analyze the stability of the algorithm.Some numerical examples are also presented to illustrate the accuracy of the algorithm.

Method of Solution
In this method, using the homotopy technique of topology, a homotopy is constructed with an embedding parameter which is considered as a "small parameter".When the homotopy theory is coupled with perturbation theory it provides a powerful mathematical tool.The details of the method can be found in [22][23][24].
We construct the following convex homotopy to develop a numerical inversion algorithm for the Abel integral Equation ( 1), where the embedding parameter   0,1 p  can be consider as an expanding parameter [23], to obtain where,  , 0,1,2,3, i L x i   are the functions to be determined.We use the following iterative scheme to evaluate   i L x .Substituting (5) in (4) and the equating the coefficients of p with the same power, we get .
Hence the solution of Equation ( 1) is given by Now, we consider the stability of the solution (7) to small changes in data.That is we are interested in what happens to y, when we replace   g x by      is unknown apart from some restriction on its magnitude relative to  .
g x For computational convenience, we write where   n L x is given by ( 6) and Thus, the new solution  , The effect of the noise  in the data deviates the solution by 10), we conclude that y  and g  are connected via the following generalized Abel integral equation Thus, we have proved the following theorem Theorem.The presence of a noise term


in the observable data   g x changes the solution   y x by an amount equivalent to the solution of the Abel integral Equation (11) with input equal to the noise term is not known before hand, we take an upper bound for which can be readily solved.

Numerical Examples
The simplicity and accuracy of the proposed algorithm is illustrated by the following numerical examples.We compute the error       is the exact solution and   y x is an approximate solution of the problem obtained by truncating equation (7).
Examples.To illustrate the method, we consider the following three pairs of generalized Abel integral equations with their inversions: with the exact solution 2 , with exact solution , where with exact solution .

Conclusions
kernel of the Abel's integral equation or Abel kernel.An Abel's integral equation belongs to the class of Volterra equation of the first kind.If   g x is a continuously differentiable function, then the Abel's integral Equation (1) has a unique solution erf x are the modified Bessel function of first kind and the error function respectively.

A
very simple but powerful and user friendly algorithm to invert generalized Abel integral equation is proposed by using homotopy perturbation method.It is proved that the change in the observable data  , g x is the solution of the generalized Abel integral equation with input data equal to the noise

Figure 1 .
Figure 1.The various errors