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International Journal of Analysis
Volume 2013 (2013), Article ID 652541, 12 pages
http://dx.doi.org/10.1155/2013/652541
Research Article

Generalized Abel Inversion Using Extended Hat Functions Operational Matrix

1Department of Applied Mathematics, Indian Institute of Technology, Banaras Hindu University, Varanasi 221005, India
2Department of Mathematics, Udai Pratap Autonomous College, Varanasi 221002, India

Received 24 September 2012; Accepted 2 April 2013

Academic Editor: Frédéric Robert

Copyright © 2013 Manoj P. Tripathi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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: , , , using our newly constructed extended hat functions operational matrix of integration, and give an error analysis of the algorithm. The earlier numerical inversions available for the above equation assumed either or .

1. 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].

The analytical inversion formula for (1) is given as [4]

There are several analytic and numerical inversion formulae available in the literature [1, 520].

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).

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.

The above two forms (1) and (3) of Abel integral equations are obtained by taking and(i);(ii), respectively, in (4).

Mostly for and the generalized Abel integral equation models the physical problems but the integral equation for can be reduced to the case , by change of variables. So we restrict ourselves to only.

2. Extended Hat Functions and Their Operational Matrices for Abel Inversion

Hat functions are defined on the domain . These are continuous functions with shape of hats, when plotted on two-dimensional planes. The interval is divided into subintervals , , of equal lengths where . The hat function’s family of first hat functions is defined as follows: We modify these functions by adding characteristics functions and to and , respectively, to yield a new class of extended hat functions defined over for , and these are given by Thus, the supports of and are extended to and , respectively. These extended hat functions are continuous, linearly independent and are in . As , obviously, the extended hat functions will converge to the traditional hat functions.

A function may be approximated in vector form as where

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 , , and and by change of variables, the Abel integral equation (4) reduces to which may be written as where , , , and .

Instead of considering (15), we consider the more general equation of the form: Using (11), the functions and may be approximated as Thus the problem of Abel inversion is reduced to finding the unknown matrix . Substituting (17) into (16), we get The integrals in (18) involve, evaluating integrals of the type 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.

Theorem 1. The functions for .

Proof. We prove the theorem for . The proofs for and are skipped as they may be proved on the same pattern. Based on subdivision of interval , we calculate by considering the following cases. (i)When , then Changing the variable, , we get  (ii) When , then Adopting the same procedure as in (i), we get  (iii) When , then  (iv) When , then . HenceThus, from (25), we see that and hence for and bounded. This completes the proof.

Therefore, from (11), we get

Theorem 2. The coefficients in (26) are given by (i) for , ,  (ii) for , ,
where .

Proof.  (i) When follows from (25).
When , which is equivalent to , we get since the support of lies in . Using (6) and (9), we get from change of variable Similarly, for , that is, , (it follows trivially from (19)).
 (ii) For , From (10), we have so Similarly, , thus, proving the theorem.

Similar arguments prove the following theorem.

Theorem 3. The functions for .

From Theorem 3 and (11), we have

The coefficients ’s are given as follows. (i) For ; , and (ii) for ; ,

Using (12) and (26), the following integral may be written as

Substituting the approximation of from (26) in (37), we get where is a matrix whose entry is , given by (27) and (28) for , . The matrix is given as where The matrix is called extended hat functions lower operational matrix for Abel’s inversion.

Remark 4. It is evident from (40) that when , then , and so the lower triangular matrix becomes a singular matrix. In this case, the singularity of the matrix makes it redundant for numerical computation since the invertibility of the matrix is required to obtain the solution. To make the matrix invertible, we introduced a positive parameter and extend the traditional hat function to the interval .
Similarly, using (19) and (34)–(36), we construct the extended hat functions upper operational matrix for Abel’s inversion, , such that where
The various entries , , and are given by (40)–(42), for and .
If we partition the matrix in four blocks as , then , where , is a null vector, and Using (38) and (43), (18) may be written as Solving the above system of linear equations, we obtain Substituting the value of from (47) into (17), the approximate emissivity is given by

3. Error Analysis

In this section, an error analysis of our proposed algorithm is given. Let , and then, using (11), it is approximated as The above approximation gives exact values at nodal points. We denote the right-hand side of (49) by , and then, for , , we have Expanding in Taylor series at , we obtain Thus, from (50) and (51), we get as , and we have as , thus proving the following theorem.

Theorem 5. The absolute error associated with the approximation (49) is of the order .

4. 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 , 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 (for computational convenience) is the discrete norm of the absolute error denoted by . Note that the calculation of in (55) is performed by taking different values of . In all the test profiles, the exact and noisy intensity profiles are denoted by and , respectively, where is obtained by adding a noise to such that , where , , , and is the uniform random variable with values in such that .

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 , , and of , where we mean . In each of the test problems given in this section we have taken positive parameter and , except for Example 10, where has been used. The absolute errors between exact and approximate emissivities, corresponding to different noises , have been denoted by , and , respectively. In the text boxes of the figures, the notations , , and have been used for , , and , 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 (without noise) are obtained with and without noise term in the intensity profile , and using (48) these are given by where and are known matrices, and they are obtained from the following equations: Hence Writing and replacing random noise by its maximum value , we get Let , then 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.

tab1
Table 1: Noise reduction for at different values of .
652541.fig.001
Figure 1: Noise reduction for , , , and .

Table 1 demonstrates the noise filtering capability of the algorithm for three different noise outputs. From Table 1 and Figure 1 we see that noise reduction is symmetric about the point , and the maximum reduction in noise is achieved at for all the three levels of noises , , and introduced in . The general behaviour of the noise reduction is the same irrespective of the value of . In the interval the algorithm is stable, whereas the noise filtering capability decreases continuously and then jumps symmetrically in .

Example 6. Consider the generalized Abel integral equation: where , with the exact analytical solution .
The absolute errors have been calculated for and are given in Table 2. The value of is , for . As , the absolute error is appreciably higher than and . The Figure 2 compares the absolute errors and for noise .

tab2
Table 2: The absolute errors , at different nodal points , for , in Example 6.
652541.fig.002
Figure 2: Absolute errors and for , .

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:
, , , , , and . Taking , the various values of respective are given in order as , , and .
In Figure 3, the exact and reconstructed emissivities (with noise) have been shown for , and the two emissivities match very well even for higher noise introduced in the intensity profile. For , Figures 4 and 5 show the absolute errors , and , , respectively.

tab3
Table 3: The absolute errors , at different nodal points , for , and .
652541.fig.003
Figure 3: Exact and approximate emissivities with noise for .
652541.fig.004
Figure 4: Absolute errors and for , .
652541.fig.005
Figure 5: Absolute errors and for , .

Example 8. In this example we consider the following Abel’s integral equation [22]: where 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 , and . Figure 6 shows the graph of exact and approximate emissivities (without noise) for . Absolute errors and , for and , are shown in Figures 7 and 8, respectively.

tab4
Table 4: The absolute errors , at different nodal points , for , in Example 8.
652541.fig.006
Figure 6: Exact and approximate solutions for .
652541.fig.007
Figure 7: Absolute errors and for , .
652541.fig.008
Figure 8: Absolute errors , for , .

Example 9. Consider the generalized Abel integral equation: where . The exact solution of (65) is . In Figure 9, the comparison between and is shown, for .

652541.fig.009
Figure 9: Comparison between and for , .

Example 10. For the pair [14, 15, 23]: with , and the various parameters are as follows:
, , , and , for all the three chosen values of . 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 ) whereas, in Figure 11, a comparison between and is shown for , .

tab5
Table 5: The absolute errors , at different nodal points , for , and .
652541.fig.0010
Figure 10: Exact and approximate emissivities (without noise and with noise ) for .
652541.fig.0011
Figure 11: Comparison between and for , .

Example 11. Consider the generalized Abel integral equation (15) with , , , for the pair and where and in (67) are called Fresnel integrals. These are defined as For, , different absolute errors are given in Table 6. The various parameters for are and . In Figure 12, the exact and approximate emissivities (without noise) are shown for .

tab6
Table 6: The absolute errors for , for Example 11.
652541.fig.0012
Figure 12: Exact and approximate emissivities (without noise) for .

5. 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 inversions were restricted to a part of the general Abel’s integral equation. We have extended the hat function beyond their domain to avoid the singularity of the matrix at . These operational matrices are given by the general formulae (39) and (44), thus making the evaluation of 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.

Acknowledgments

The authors are grateful to the learned reviewer for his valuable suggestions which have led to the improvement of the paper in the present form. Also, the first author acknowledges the financial support from UGC New-Delhi, India, under Faculty Improvement Program (FIP), whereas the second and the third authors acknowledge the financial support from UGC and CSIR New-Delhi, India, respectively, under JRF schemes.

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