Supersingular integral equations of the first kind and its approximate solutions

In this note, we consider a supersingular integral equations (SuperSIEs) 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, second, third and fourth kinds are used to find bounded, unbounded and semi-bounded solutions of SuperSIEs respectively. Exact calculations of singular integrals for Chebyshev polynomials allow us to obtain high accurate approximate solution. Gauss- Chebyshev quadrature formulas are used for high accurate computations of regular kernel integrals. Two examples are provided to verify the validity and accuracy of the proposed method. Comparisons with other methods are also given. Numerical examples reveal that approximate solutions are exact if solution of SuperSIEs is of the polynomial forms with corresponding weights. It is worth to note that proposed method works well for large value of node points and errors are drastically decreases.

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