Projection Method for Bounded and Unbounded Solution of Hypersingular Integral Equations of the First Kind

In this note, we consider 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 D = [1, – 1] × [–1, 1]. Projection method together with Chebyshev polynomials of the first and second kinds are used to find bounded and unbounded solutions of HSIEs respectively. Exact calculations of hypersingular and singular integrals for Chebyshev polynomials allows us to obtain high accurate approximate solution. Gauss-Chebyshev quadrature with Gauss-Lobotto nodes are presented as the high accurate computation of regular kernel integrals. Six examples are provided to verify the validity and accuracy of the proposed method. Comparisons with other methods are also given. Numerical examples reveals that approximate solutions are exact if solution of HSIEs is of the polynomial forms with corresponding weights. It is worth to note that proposed method works well for large value of node points n and errors are drastically decreases. Comparisons of SPU times are also shown to demonstrate effectiveness of the method and less complexity computations.