Abstract
In this paper, first and second kind Chebyshev wavelets are studied. New estimators E_{2^{k-1},0}^{(1)}, E_{2^{k-1},M}^{(2)}, E_{2^{k-1},0}^{(3)}, E_{2^{k-1},M}^{(4)} for first kind Chebyshev wavelets and estimators E_{2^{k},0}^{(5)}, E_{2^{k},M}^{(6)}, E_{2^{k},0}^{(7)} and E_{2^{k},M}^{(8)} for second kind Chebyshev wavelets for a function f belonging to generalized Hddot{o}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.
Highlights
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
Adibi et al [1] works on the numerical solution of Fredholm integral equation by first kind Chebyshev wavelet
Several methods are known for approximating the solution of the integral equations and differential equations
Summary
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. No work seems for the approximation of the function belonging to generalized Hölder’s class by first and second kind Chebyshev wavelet method. First and second kind Chebyshev wavelet approximation of function f belonging to generalized Hölder’s classes Hα(χ)[0, 1) and H (w)[0, 1) have been determined. Abel’s integral equations are solved by first and second kind Chebyshev wavelet method. 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. First and second kind Chebyshev wavelet approximations established in this paper are new, sharper and best possible in wavelet analysis
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