Mathematics and Statistics 8(2): 106-120, 2020 DOI: 10.13189/ms.2020.080206 http://www.hrpub.org Semi Bounded Solution of Hypersingular Integral Equations of the First Kind on the Rectangle Zainidin Eshkuvatov 1,∗ , Massamdi Kommuji 1 , Rakhmatullo Aloev 2 , Nik Mohd Asri Nik Long 3 , Mirzoali Khudoyberganov 2 1 Faculty of Science and Technology, Universiti Sains Islam Malaysia, Malaysia 2 Faculty of Mathematics, National University of Uzbekistan, Uzbekistan 3 Department of Mathematics, Faculty of Science, Universiti Putra, Malaysia Received June 29, 2019; Revised September 17, 2019; Accepted September 23, 2019 Copyright c 2020 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Abstract A hypersingular integral equations (HSIEs) of the first kind on the interval [−1, 1] with the assumption that kernel of the hypersingular integral is constant on the diagonal of the domain is considered. Truncated series of Chebyshev polynomials of the third and fourth kinds are used to find semi bounded (unbounded on the left and bounded on the right and vice versa) solutions of HSIEs of first kind. Exact calculations of singular and hypersingular integrals with respect to Chebyshev polynomials of third and forth kind with corresponding weights allows us to obtain high accurate approximate solution. Gauss-Chebyshev quadrature formula is extended for regular kernel integrals. Three examples are provided to verify the validity and accuracy of the proposed method. Numerical examples reveal that approximate solutions are exact if solution of HSIEs is of the polynomial forms with corresponding weights. Keywords Approximation, Chebyshev polynomials, Convergence, Hypersingular integral equations. 1 Introduction Hypersingular integral equations (HSIEs) of the first kind of the form 1 π =  1 −1 ϕ(t)  K(x,t) (t − x) 2 + L 1 (x,t)  dt = f (x) , −1 <x< 1, (1) encounters in several physical problems such as aerodynamics, hydrodynamics, and elasticity theory (see [1]-[7]). In 1985, Golberg [1] consider Eq. (1) with the kernel K(x,t)=1 and proposed projection method with the truncated series of Chebyshev polynomials of the second kind together with Galerkin and collacation methods. Uniform convergence and the rate of convergence of projection method are obtained in subspace of Hilbert space for HSIEs (1). In 1992, Martin [2] obtained the analytic solution to the simplest one-dimensional hypersingular integral equation i.e. the case of K(x,t)=1 and L 1 (x,t)=0 in Eq. (1). In 2006, Mandal and Bera have proposed a simple approximate method (Polynomial approximation) for solving a general hypersingular integral equation of the first kind (1) with K(x,x) =0. The method is mostly concentrated with the bounded solution and illustrated proposed method by considering some simple examples. Mandal and Bhattacharya ([4], 2007) proposed approximate numerical solutions of some classes of singular integral equations including HSIEs (1) with K(x,t)=1 by using Bernstein polynomials as basis. The method was explained with illustrative examples. Convergence of the method is referred to book of Golberg and Chan [5]. In 2009-2010, Boykov et al. ([6]-[7]) proposed spline-collocation method and its justification for the solution of one-dimensional hypersingular integral equations, poly-hypersingular integral equations, and multi-dimensional hypersingular integral equations. Proved convergence of the method and illustrative examples demonstrated to show the accuracy and efficiency of the developed method. Gulsu and Uzturk ([8], 2014) have purposed approximation method for hypersingular integro-differential equations in the most general form under the mixed conditions in terms of the second kind