Approximation of function belonging to generalized Hölder’s class by first and second kind Chebyshev wavelets and their applications in the solutions of Abel’s integral equations

In this paper, first and second kind Chebyshev wavelets are studied. New estimators E2k-1,0(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{2^{k-1},0}^{(1)}$$\end{document}, E2k-1,M(2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{2^{k-1},M}^{(2)}$$\end{document}, E2k-1,0(3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{2^{k-1},0}^{(3)}$$\end{document}, E2k-1,M(4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{2^{k-1},M}^{(4)}$$\end{document} for first kind Chebyshev wavelets and estimators E2k,0(5)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{2^{k},0}^{(5)}$$\end{document}, E2k,M(6)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{2^{k},M}^{(6)}$$\end{document}, E2k,0(7)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{2^{k},0}^{(7)}$$\end{document} and E2k,M(8)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{2^{k},M}^{(8)}$$\end{document} for second kind Chebyshev wavelets for a function f belonging to generalized Ho¨\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ddot{o}$$\end{document}lder’s class have been obtained. Also, a method based on first and second kind Chebyshev wavelet approximations has been presented for solving integral equations. Comparison of solutions obtained by both wavelets method has been studied. It is found that second kind Chebyshev wavelet method gives better and accurate solutions as compared to first kind Chebyshev wavelet method. This is a significant achievement of this research paper in wavelet analysis.


Introduction
During the past few decades, wavelets have found their ways in the fields of signal processing, time-frequency analysis, image processing, quantum mechanics, and data compression. Wavelets have been also used as basis functions to estimate the solutions of integral and differential equations.
The approximation of a function belonging to some class by wavelet method has been discussed by many researchers like Devore [7], Morlet [11,12], Meyer [10] and Debnath [6], etc. Sripathy et al. [14] discussed the Chebyshev wavelet based approximation for solving linear and non-linear differential equation. Adibi et al. [1] works on the numerical solution of Fredholm integral equation by first kind Chebyshev wavelet. As per our knowledge, no work seems for the approximation of the function belonging to generalized Hölder's class by first and second kind Chebyshev wavelet method. In this paper, first and second kind Chebyshev wavelet approximation of function f belonging to generalized Hölder's classes H (χ ) α [0, 1) and H (w) [0, 1) have been determined. Abel's integral equations are solved by first and second kind Chebyshev wavelet method. Several methods are known for approximating the solution of the integral equations and differential equations. Capobianco [4] discussed the method for solving first kind integral equation. Yousefi [16] presented the numerical solution of Abel integral equation by Legendre wavelet method. Chebyshev wavelets method for solving system of Volterra integral equations has been discussed by Iqbal et al. [8]. Abel integral equation has been studied by many researcher and some numerical methods were developed. In this paper, another wavelet, Chebyshev wavelet of first and second kind are applied for the solution of Abel's integral equation.
The solutions obtained by first and second kind Chebyshev wavelet method are compared with their exact solutions and the known Legendre wavelet method. It is observed that second kind Chebyshev wavelet method gives more accurate solutions of the integral equations in comparison to first kind Chebyshev wavelet method and the Legendre wavelet method. First and second kind Chebyshev wavelet approximations established in this paper are new, sharper and best possible in wavelet analysis.

First kind Chebyshev wavelet
Chebyshev wavelet of first kind, denoted by T n,m , is defined over the interval [0, 1) as

Second kind Chebyshev wavelet
Chebyshev wavelet of second kind, denoted by U n,m , is defined over the interval [0, 1) as where n = 0, 1, 2, . . . , 2 k − 1, m = 0, 1, 2, . . . , M and x is the normalized time. The polynomials U m (x) are second kind Chebyshev polynomials of degree m orthogonal with respect to the weight function 1], and satisfy the following recursive formula :

Generalized Hölder's class
Let χ be a positive, monotonic increasing function of t such that |t| α Remark It is important to note that if χ(t) = 1, then class H

Chebyshev wavelet function approximation
The function f ∈ L 2 [0, 1) is expressed in the form of Chebyshev wavelet series as where c n,m = f (x), ψ n,m (x) , in which ., . denotes the L 2 inner product. If infinite series in (2.1) is truncated then it is written as The Chebyshev wavelet approximation of its Chebyshev wavelet series is given by

Theorem
In this paper, we prove the following Theorems: .
Lastly, by Eq. (4.3), 1 0 (e n (x)) 2 w n (x)dx, due to disjointness of support of e n and e n . Hence .
Then (E |c n,m | 2 , by orthonormality property of T n,m (x) Therefore, E

Proof of Theorem 3.2
(i) Following the proof of the Theorem 3.1(i) and for f ∈ H (φ) [0, 1), we have Lastly, by Eq. (4.5) Therefore, E (ii) Following the proof of the Theorem 3.1(ii), |c n,m | 2 , by orthonormality property of T nm (x)

Proof of Theorem 3.3 (i) Error between f (x) and its second kind Chebyshev wavelet expansion in interval n
|s n (x)|dx, ζ n ∈ n 2 k , n + 1 2 k , by weighted mean value theorem.

Corollary
Following Corollaries can be deduced from Theorems 3.1 to 3.4: Proof of the Corollary 5.1 can be developed parallel to the proof of Theorem 3.1 by taking χ(t) = 1, ∀ t ∈ [0, 1).
where L is the operational matrix obtained from first or second kind Chebyshev wavelets. With the calculated L, the unknown coefficient vector Y can be calculated from using equation (6.4), (6.5) and the above equation: First Kind: Second Kind: , is the required approximate solution.

Illustrative examples
In this section, we applied Chebyshev wavelet method described in previous Section for solving integral equations and solve some examples.
Example 7.1 Consider the following Abel's integral equation of the first kind: having the exact solution y(x) = x 2 + x + 1.
Consider the approximate solution as y(x) = 9 m=0 c 0,m 0,m (x). The integral equation has been solved by applying the procedure described in Sect. 6 by first and second kind Chebyshev wavelets by taking M = 9 and k = 0.
The graphs of the obtained numerical solution through wavelet methods and the exact solution of example 7.1 are shown in the Fig. 1: It is observed from the table and figure that the numerical solution obtained by the second kind Chebyshev wavelet is approximately same as the exact solution.
having the exact solution y(x) = −6 Consider the approximate solution as y(x) = 9 m=0 c 0,m U 0,m (x). The integral equation has been solved by applying the procedure described in Sect. 6 for second kind Abel's integral equation by taking M = 9 and k = 0.
The graphs of the estimated numerical solution through wavelet methods and the exact solution of example 7.2 are shown in the Fig. 2: It is observed from the table and figure that the numerical solution obtained by the second kind Chebyshev wavelet is sufficiently close to the exact solution.