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.
Highlights
IntroductionEncounters in several physical problems such as aerodynamics, hydrodynamics, and elasticity theory (see [1]-[7])
Hypersingular integral equations (HSIEs) of the first kind of the form K(x, t) = φ(t) π −1(t − x)2 + L1(x, t) dt = f (x), −1 < x < 1, (1)encounters in several physical problems such as aerodynamics, hydrodynamics, and elasticity theory
Uniform convergence and the rate of convergence of projection method are obtained in subspace of Hilbert space for HSIEs (1)
Summary
Encounters in several physical problems such as aerodynamics, hydrodynamics, and elasticity theory (see [1]-[7]). For singular integral equations many efficient methods are derived and proved convergence of the method as well as showed illustrative examples In 2011, Abdulkawi et al [16], considered the finite part integral equation (1) with K (x, t) = 1 and used Chebyshev polynomials of 1st and 2nd kind to find bounded solution of Eq (1). Eshkuvatov and Narzullaev ([19], accepted in 2018) have solved Eq (1) using projection method together with Chebyshev polynomials of the first and second kinds to find bounded and unbounded solutions of HSIEs (1) respectively. Existence of inverse of hypersingular operator and exact calculations of hypersingular integral for Chebyshev polynomials allows us to obtain high accurate approximate solution for the case of bounded and unbounded solutions, where the kernel K(x, t) is a constant on the diagonal of the domain D = [−1, 1] × [−1, 1].
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