Browsing by Author "Harish Chandra Yadav"

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An error estimation of absolutely continuous signals and solution of Abel's integral equation using the first kind pseudo-Chebyshev wavelet technique
(Elsevier B.V., 2025) Susheel Kumar; A. K. AWASTHI; Sudhir Kumar Mishra; Harish Chandra Yadav; n. Abhilasha; Shyam N. Lal
This paper introduces a novel computational strategy for addressing challenges in approximation theory. It focuses on the use of first-kind pseudo-Chebyshev wavelet approximations and the methodology and evaluation of the error for a specific function are outlined, along with practical instances to showcase the method's effectiveness and efficiency. This approach is motivated by the need for highly efficient and precise methods for function representation and the error reduction in this domain. The paper also establishes the error of a function associated with the class of absolutely continuous functions using first-kind pseudo-Chebyshev wavelets via orthogonal projection operators, highlighting their precision and theoretical optimality within the domain of wavelet analysis. Additionally, the use of wavelet approximation to solve Abel's integral equation is demonstrated by computing the approximate solution using first kind pseudo-Chebyshev wavelet. © 2024 The Authors
APPROXIMATION OF FUNCTIONS BELONGING TO CM,α[0, 1) CLASS AND SOLUTION OF CHANDRASEKHAR’S WHITE DWARFS AND PANTOGRAPH DIFFERENTIAL EQUATION BY GENOCCHI WAVELETS
(Poincare Publishers, 2023) Shyam Lal; Harish Chandra Yadav; Abhilasha
In this paper, the approximation of the solution function f for Chandrasekhar’s white dwarfs and the Pantograph differential equation of class CM,α[0, 1) by the (2k−1, M)th partial sums of their Genocchi wavelet expansion in the interval [0, 1) has been estimated. The Genocchi wavelet technique has been employed to determine the solution of Chandrasekhar’s white dwarfs and the Pantograph differential equation. The solution obtained through the Genocchi wavelet method approaches their exact solution and is compared to the Chebyshev wavelet method, Legendre wavelet method, and ODE-45 method. This represents an achievement of wavelet analysis in this research article. © Poincare Publishers.
Approximation of functions belonging to Hölder’s class and solution of Lane-Emden differential equation using Gegenbauer wavelets
(University of Nis, 2023) Shyam Lal; Harish Chandra Yadav
In this paper, a very new technique based on the Gegenbauer wavelet series is introduced to solve the Lane-Emden differential equation. The Gegenbauer wavelets are derived by dilation and translation of an orthogonal Gegenbauer polynomial. The orthonormality of Gegenbauer wavelets is verified by the orthogonality of classical Gegenbauer polynomials. The convergence analysis of Gegenbauer wavelet series is studied in Hölder’s class. Hölder’s class Hα[0, 1) and Hϕ[0, 1) of functions are considered, Hϕ[0,1) class consides with classical Hölder’s class Hα[0, 1) if ϕ(t) = tα, 0 < α ≤ 1. The Gegenbauer wavelet approximations of solution functions of the Lane-Emden differential equation in these classes are determined by partial sums of their wavelet series. In briefly, four approximations E(1) 2k−1,0, E(1) 2k−1,M, E(2) 2k−1,0, E(2) 2k−1,M of solution functions of classes Hα[0, 1), Hϕ[0, 1) by (2k−1, 0)th and (2k−1, M)th partial sums of their Gegenbauer wavelet expansions have been estimated. The solution of the Lane-Emden differential equation obtained by the Gegenbauer wavelets is compared to its solution derived by using Legendre wavelets and Chebyshev wavelets. It is observed that the solutions obtained by Gegenbauer wavelets are better than those obtained by using Legendre wavelets and Chebyshev wavelets, and they coincide almost exactly with their exact solutions. This is an accomplishment of this research paper in wavelet analysis. © Faculty of Sciences and Mathematics, University of Niš, Serbia.
APPROXIMATIONS IN HÖLDER’S CLASS AND SOLUTION OF BESSEL’S DIFFERENTIAL EQUATIONS BY EXTENDED HAAR WAVELET
(Poincare Publishers, 2023) Shyam Lal; Harish Chandra Yadav
In this paper, extended Haar wavelet has been introduced in the interval [0, λ), λ > 0. It reduces to classical Haar wavelet for λ = 1. The orthonormality of extended Haar wavelets has been discussed. The convergence analysis of an extended Haar wavelet series of a function f belonging to Hölder’s classes Hα [0, λ) & H2α [0, λ) have been studied. Consequently, the approximations of function f belonging to the generalised Hölder’s class have been estimated. The solutions of Bessel’s differential equation of order zero have been obtained by the extended Haar operational matrix method for λ = 1 & 2. These solutions for λ = 1 & 2 are compared with their exact solutions. It is observed that the extended Haar wavelet solutions and their exact solutions are almost the same. This validates the adopted procedure for solutions of Bessel’s differential equation by an extended Haar operational matrix. This is a significant achievement in wavelet analysis. © Poincare Publishers.
EXTENDED CHEBYSHEV WAVELET OF FIRST KIND AND ITS APPLICATIONS IN APPROXIMATION OF A FUNCTION BELONGING TO HÖLDER’S CLASS AND SOLUTION OF FREDHOLM INTEGRAL EQUATION OF SECOND KIND
(The Indian Mathematical Society, 2024) Shyam Lal; Harish Chandra Yadav
In this paper, six approximations of solution functions of the Fredholm integral equation in Hölder’s class by first kind extended Chebyshev wavelet expansion in the interval [0, 1) have been estimated. The solutions of the Fredholm integral equation of the second kind by extended Chebyshev wavelets of the first kind have been obtained. The solutions obtained by an extended Chebyshev wavelet of the first kind are approximately the same as their exact solutions. This is a significant achievement of this research paper in wavelet analysis. © Indian Mathematical Society, 2024.