Generalized Abel Inversion Using Extended Hat Functions Operational Matrix

Abel type integral equations play a vital role in the study of compressible flows around axially symmetric bodies. The relationship between emissivity and the measured intensity, as measured from the outside cylindrically symmetric, optically thin extended radiation source, is given by this equation as well. The aim of the present paper is to propose a stable algorithm for the numerical inversion of the following generalized Abel integral equation: I(y) = a(y) ∫y α ((r μ−1 ε(r))/(y μ − r μ ) γ )dr + b(y) ∫ β


Introduction
Abel integral equation [1] occurs in many branches of science and technology, such as plasma diagnostics and flame studies, where the most common problem of deduction of radial distributions of some important physical quantity from measurement of line-of-sight projected values is encountered. For a cylindrically symmetric, optically thin plasma source, the relation between radial distribution of the emission coefficient and the intensity measured from outside of the radial source is described by Abel transform. The challenging task of reconstruction of emission coefficient from its projection is known as Abel inversion. The earliest application, due to Mach [2], arose in the study of compressible flows around axially symmetric bodies.
The Abel integral equation is given by where ( ) and ( ) represent, respectively, the emissivity and measured intensity, as measured from outside the source [3].
Singh et al. [19] constructed an operational matrix of integration based on orthonormal Bernstein polynomials and used it to propose a stable algorithm to invert the following form of Abel integral equation: In 2010, Singh et al. [20] constructed yet another operational matrix of integration based on orthonormal Bernstein polynomials and used it to propose an algorithm to invert the Abel integral equation (1).

International Journal of Analysis
In 2008, Chakrabarti [21] employed a direct function theoretic method to determine the closed form solution of the following generalized Abel integral equation: where the coefficients ( ) and ( ) do not vanish simultaneously. But the numerical inversion is still needed for its application in physical models since the experimental data for the intensity ( ) is available only at a discrete set of points, and it may also be distorted by the noise. This motivated us to look for a stable algorithm which can be used for numerical inversion of the Abel integral equation (4) obtained by joining the two integrals (1) and (3). In this paper, we construct extended hat functions operational matrix of integration to invert the generalized Abel integral equation (4). Using hat functions for approximation of emissivity and intensity profiles has an edge over the earlier works of Singh et al. [19,20], where they have used orthonormal Bernstein polynomials to approximate those physical quantities in the sense that a general formula for × operational matrix of integration is obtained in the earlier case whereas no such formula is available for the latter case. In Sections 3 and 4, we give the error estimate and the stability analysis followed by numerical examples to illustrate the efficiency and stability of the proposed algorithm.
Mostly for = 1,2 and = 1/2 the generalized Abel integral equation models the physical problems but the integral equation for = 2 can be reduced to the case = 1, by change of variables. So we restrict ourselves to = 1 only.
A function ∈ 2 [0, 1] may be approximated in vector form as where +1 ≜ [ 0 , 1 , 2 , . . . , ] , The important aspect of using extended hat functions in the approximation of function ( ) lies in the fact that the coefficients in (11) are given by Taking = 0, = 1, and = 2 and by change of variables, the Abel integral equation (4) reduces to which may be written as Instead of considering (15), we consider the more general equation of the form: Using (11), the functions 1 ( ) and ( ) may be approximated as Thus the problem of Abel inversion is reduced to finding the unknown matrix +1 . Substituting (17) into (16), we get The integrals in (18) involve, evaluating integrals of the type ∫ 1+ ((Ψ +1 ( ))/( − ) ) and ∫ − (( ( ))/( − ) ) . Let and compute the two operational matrices of integration to evaluate these integrals. The scheme of derivation of these two operational matrices is based on the following theorems. Proof. We prove the theorem for = 1, 2 . . . , − 1. The proofs for = 0 and = are skipped as they may be proved on the same pattern. Based on subdivision of interval [− , 1 + ], we calculate Φ ( ) by considering the following cases.
(ii) For < , From (10), we have Similarly, Similar arguments prove the following theorem.
From Theorem 3 and (11), we have The coefficients 's are given as follows.
Remark 4. It is evident from (40) that when = 0, then 0 = 0, and so the lower triangular matrix +1 becomes a singular matrix. In this case, the singularity of the matrix +1 makes it redundant for numerical computation since the invertibility of the matrix is required to obtain the solution. To make the matrix +1 invertible, we introduced a positive parameter and extend the traditional hat function to the interval [− , 1+ ].
If we partition the matrix +1 in four blocks as Using (38) and (43), (18) may be written as Solving the above system of linear equations, we obtain Substituting the value of +1 from (47) into (17), the approximate emissivity ( ) is given by

Numerical Results and Stability Analysis
In this section, we discuss the implementation of our proposed algorithm and investigate its accuracy and stability by applying it on test functions with known analytical Abel inverse. For, it is always desirable to test the behaviour of a numerical inversion method using simulated data for which the exact results are known, and thus making a comparison between inverted results and theoretical data is possible. We have tested our algorithm on several well-known test profiles that are commonly encountered in experimental data and widely used by researchers [7,15,20]. The accuracy of the proposed algorithm is demonstrated by calculating the parameters of absolute error Δ ( ), average deviation also known as root mean square error (RMS). They are calculated using the following equations: where ( ) is the emission coefficient calculated at point using (48) and ( ) is the exact analytical emissivity at the corresponding point. Note that , henceforth, denoted by +1 (for computational convenience) is the discrete 2 norm of the absolute error Δ denoted by ‖Δ ‖ 2 . Note that the calculation of +1 in (55) is performed by taking different values of . In all the test profiles, the exact and noisy intensity profiles are denoted by 1 ( ) and 1 ( ), respectively, where 1 ( ) is obtained by adding a noise to 1 ( ) such that 1 ( ) = 1 ( ) + , where = ℎ, = 0, 1, 2, . . . , , ℎ = 1, and is the uniform random variable with values in [−1, 1] such that Max 0≤ ≤ | 1 ( ) − 1 ( )| ≤ .
The following test problems are solved with and without noise to illustrate the efficiency and stability of our method by choosing three different values of the noises as 0 = 0, 1 = +1 , and 2 = % of +1 , where we mean +1 = (1/( + 1)) ∑ =0 1 ( ). In each of the test problems given in this section we have taken positive parameter = 0.0001 and = 0.1, except for Example 10, where = 0 has been used. The absolute errors between exact and approximate emissivities, corresponding to different noises , = 0, 1, 2 have been denoted by 0 ( ), 1 ( ), and 2 ( ), respectively. In the text boxes of the figures, the notations 0( ), 1( ), and 2( ) have been used for 0 ( ), 1 ( ), and 2 ( ), respectively. Though the stability of the algorithm is illustrated by various numerical experiments performed in this section, we analyze it also mathematically as follows.
The reconstructed emissivities ( ) (with noise) and 0 ( ) (without noise) are obtained with and without noise term in the intensity profile 1 ( ), and using (48) these are given by International Journal of Analysis 7 where +1 and +1 are known matrices, and they are obtained from the following equations: Hence Writing +1 = +1 − +1 and replacing random noise by its maximum value , we get then ℎ( ) reflects the noise reduction capability of the algorithm and its values at various points, and its graph is shown in Table 1 and Figure 1, respectively. Table 1 demonstrates the noise filtering capability of the algorithm for three different noise outputs. From Table 1 and Figure 1
Example 7. In this example, we consider the following Abel's integral equation [7,20]: The absolute errors corresponding to different noises are given in Table 3. The values of various parameters are given as:     In Figure 3, the exact and reconstructed emissivities (with 2 noise) have been shown for = 50, and the two emissivities match very well even for higher noise 2 introduced in the intensity profile. For = 100, Figures 4 and 5 show the absolute errors 0 ( ), 1 ( ) and 0 ( ), 2 ( ), respectively.
Example 8. In this example we consider the following Abel's integral equation [22]: The exact solution of the integral equation (62) is given by In Table 4, the absolute errors for different noises have been shown. Various parameters used for Table 4 are International Journal of Analysis 9   Figure 6 shows the graph of exact and approximate emissivities ( ) (without noise) for = 50. Absolute errors 0 ( ) and 1 ( ), for = 50 and = 100, are shown in Figures  7 and 8, respectively. Example 9. Consider the generalized Abel integral equation: The exact solution of (65) is ( ) = 10 /9. In Figure 9, the comparison between 0 ( ) and 1 ( ) is shown, for = 100.  with ( ) = ( ) = 1, and the various parameters are as follows:  The absolute errors corresponding to different noises are given in Table 5. Figure 10 shows the exact and approximate emissivities (without noise and with noise 2 = 0.0036) whereas, in Figure 11, a comparison between 0 ( ) and 1 ( ) is shown for 1 = 1.8232 × 10 −4 , = 100.

Conclusions
We have constructed operational matrices of integration based on extended hat functions and used them to propose a stable algorithm for the numerical inversion of the generalized Abel integral equation. The earlier numerical    these matrices of any order extremely easy whereas in case of Bernstein operational matrices no such formula was available [19,20]. The stability with respect to the data is restored and good accuracy is obtained even for high noise levels in the data. An error analysis and stability analysis are also given.