Solution of the Generalized Abel Integral Equation by Using Almost Bernstein Operational Matrix

A direct almost Bernstein operational matrix of integration is used to propose a stable algorithm for numerical inversion of the generalized Abel integral equation. The applicability of the earlier proposed methods was restricted to the numerical inversion of a part of the generalized Abel integral equation. The method is quite accurate and stable as illustrated by applying it to intensity data with and without random noise to invert and compare it with the known analytical inverse. Thus it is a good method for applying to experimental intensities distorted by noise.

Recently, Chakrabarti [20] 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.

  b x
Earlier the generalized Abel equation (1) was examined in Gakhov's book [21], under the special assumptions that the coefficients and satisfy Holder's condition in  The method of Gakhov's has a particular disadvantage in the sense that while solving a singular equation that involves integrals only with weak singularity of the type , occurrence of strongly singular integrals involving Cauchy type singularities of the type has to be permitted [20,21].Chakrabarti [20] obtained the solution involving only weakly singular integrals of the Abel type and thus Cauchy type singular integrals were avoided.But the numerical inversion is still needed for its application in physical models since the experimental data for the intensity   f x is available only at a discrete set of points and it may also be distorted by noise.
The aim of the present paper is to propose a new stable algorithm for the numerical inversion of Abel's integral equation (1), based on the newly constructed almost Bernstein operational matrices of integration.Numerical examples are given to illustrate the accuracy and stability of the proposed algorithm.

The Bernstein Polynomials
A Bernstein polynomial, named after Sergei Natanovich Bernstein, is a polynomial in the Bernstein form that is a linear combination of Bernstein basis polynomials.
The Bernstein basis polynomials of degree are defined by n , ( ) (1 ) , 0,1, 2, , There are degree Bernstein basis polynomials forming a basis for the linear space n consisting of all polynomials of degree less than or equal to in R[x]-the ring of polynomials over the field R. For mathematical convenience, we usually set   in may be written as Then is called a polynomial in Bernstein form or Bernstein polynomial of degree .The coefficients i  

B x n
 are called Bernstein or Bezier coefficients.But several mathematicians call Bernstein basis polynomials as the Bernstein polynomials.We will follow this convention as well.These polynomials have the following properties: , where  is the Kronecker delta function.
2) has one root, each of multiplicity and , at and respectively.
  5) The Bernstein polynomials form a partition of unity i.e. .

 
, 0 Using Gram-Schmidt orthonormalization process on , i n , we obtain a class of orthonormal polynomials.We call them orthonormal Bernstein polynomials of order and denote them by .

Function Approximation
and , is the standard inner product on If the series ( 5) is truncated at , then we have where, and C   B t are matrices given by   and

Solution of Generalized Abel Integral Equation
In this section we solve generalized Abel integral equation by orthonormal Bernstein polynomials.
Using Equation ( 8), we approximate where the matrix F is known.Then from equation ( 1) and ( 9) we have From Equation ( 8) and from the derived formulae, (1 ) 1, ; 2; , ( 2) it is obvious that where and are matrices, which we call as almost Bernstein operational matrix of integration for Abel integral equation with generalized kernel.
Substituting (13) in (10), we get Hence, the approximate solution for generalized Abel integral equation ( 1) is obtained by putting the value of from ( 14) in (9).
and compute the corresponding errors and level them as are computed for the four chosen values of  as mentioned above.In all the figures some of the error terms   j E t are multiplied by 10 or some power of 10 for suitable scaling.Also we tabulated the approximate and exact solutions through Tables 1-4 for the four examples given below for various values of .t

Illustrative Examples
The following examples are solved with and without noise terms to illustrate the efficiency and stability of our method.Note that in all the examples to follow, the series ( 5) is truncated at level 6 m  and hence the almost operational matrix in ( 13) is of order .
where 2 1 F  is the regularized hypergeometric function.
This has the exact solution   , where , 1, , , 1 The following examples are solved with and without noise to illustrate the efficiency and stability of our method by choosing two different values of the noises This has the exact solution .Example 5. Next we consider, the following generalized Abel integral equation with and

 
where is incomplete beta function, defined by  , Figure 13 shows two approximate solutions obtained by applying the operational matrix of integration of order (dotted blue) and the operational matrix of integration of order (solid red).Both the approximate solutions obtained by the two different matrices have similar and almost overlapping evolutions except at the boundary points 0 and 1.So, we may conclude that the exact solution will have similar evolution.

Conclusions
We have introduced an almost Bernstein operational matrices of integration to propose a new and stable algorithm for numerical solution of generalized Abel integral equation.It is found that the method is accurate and stable as shown by the numerical examples.Moreover, the algorithm is easy to use since this is a direct method and the solution is obtained by applying the operational matrix of integration directly to the algorithm.
whereas the forcing term   f x and the unknown function   x  belong to those class of functions which admit representations of the form

7 7  1 .
Example Consider the generalized Abel integral equation with     1 a x b x   and The accuracy of the proposed algorithm is demonstrated by calculating the parameters of absolute error and average deviation also known as root mean square error (RMS).They are calculated using the following equations:

Figure 1 1 
Figure 1 illustrates the effect of the absolute errors without noise for different values of  , whereas Figures 2 and 3 show the absolute errors with noise term 1  added to the forcing term   f x for = 1000, 500 respectively.Table 1 compares the approximate and exact values of Example 1.

2 Example 2 .
Note that the calculation of N  is performed by taking = 1000, 500 in Equation (15).In all the examples, the exact and noisy profiles are denoted by N Consider the generalized Abel integral equation with     1 a x b x   and

Figure 4
Figure 4 illustrates the effect of the absolute errors

Figure 1 . 1 
Figure 1.Comparison of absolute errors, Example 1. without noise for different values of  , whereas Figures 5 and 6 show the absolute errors with noise term 1  added to the forcing term   f x for =1000, 500 respectively.Table 2 compares the approximate and exact values of Example 2.

Table 1
compares the approximate and exact values of Example 1.

Table 2
compares the approximate and exact values of Example 2.

Figure 10. Comparison of absolute errors, Example 4.
Table 3 compares the approximate and exact values of Example 3.