### CAS wavelet approximation of functions of Holder class and solutions of Fredholm integral equation

#### Abstract

In this paper, cosine and sine wavelet is considered. Two new CAS wavelet estimators E(1) 2k;2M+1(f) and E(2) 2k;2M+1(f) for the approximation of a function f whose first derivative f' and second derivative f '' belong to Hölder's class Hα [0; 1) of order 0 <α≤1, have been obtained. These estimators are sharper and best in wavelet analysis.Using CAS wavelet a computational method has been developed to solve Fredholm integral equation of second kind. In this process, Fredholm integral equations are reduced into a system of linear equations. Approximation of function by CAS wavelet method is applied in obtaining the solution of Fredholm integral equation of second kind. CAS wavelet coefficient matrices are prepared using the properties of CAS wavelets.Two examples are illustrated to show the validity and efficiency of the technique discussed in this paper.

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C.K.Chui, Wavelets: A mathematical tool for signal analysis, SIAM, Philadelphia PA,(1997).

Alpert, B.K. : A class of bases in L2 for the sparse representation of integral operators. SIAM J. Math. Anal., 24: 246-262(1993).

Dehghan, M. and A. Saadatmandi :Chebyshev nite dierence method for Fredholm integro-dierential equation. Int.J. Comput. Math.,85:123-129(2008).

S. Saha Ray and P.K. Sahu :Numerical Methods for Solving Fredholm Integral Equations of Second Kind. Hindawi Publishing Corporation, 426916(2013).

M.M. Shamooshaky, P. Assari and H. Adibi : CAS Wavelet Method for the Numerical Solution of Boundary Integral Equations with Logarithmic Singular Kernels. International Jornal of mathematical modeling and computations. 377-387(2014).

Xiaoyang Zheng and Zhengyuan Wei : Estimates of Approximation Error by Legendre Wavelet. Applied Mathematics. 694-700(2016).

Lal and Priya :Approximation of a function f of Generalized Lipschitz Class by its Extended Legendre Wavelet Series. Int.J.Comput.Math. 4:147(2018).

Zygmund A.: Trigonometric Series, vol.I. Cambridge University Press,Cambridge (1959).

Arbabi,Nazari,Darvishi : A two dimensional Haar wavelets method for solving systems of PDEs. Applied Mathematics and Computation 292 (2017) 33-77.

Rehman and Khan : The Legendre wavelet method for solving fractional dierential equations. Commun Nonlinear Sci Numer Simulat 16(2011) 4163-4173

DOI: http://dx.doi.org/10.23755/rm.v39i0.549

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