Semi Bounded Solution of Hypersingular Integral Equations of the First Kind on the Rectangle

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.

Trigonometric Ratios Using Algebraic Methods

“…Results from number theory and theory of equations were presented wherever required. Trigonometric functions also arise in the study of the hypersingular integral equations [61] and the superstability solutions of The pexiderized trigonometric functional equations [62] Some of the numerical results can also be derived using the Microsoft Excel [63]- [68]. The other alternatives are the symbolic packages, such as the Mathematica [69,70].…”

On projection method for numerical solution of hypersingular integral equations of the first kind combined with quadrature methods

“…To study the existence of a unique solution for the hypersingular integral equation (1), following the assertions in [21], we suppose K(x, x) ≠ 0, then the kernel k(x, s) can be expressed as follows…”

“…Later, the authors in [20] used the projection method together with Chebyshev polynomials with the assumption that the kernel of the hypersingular integral equation k(x, s), is constant on the diagonal of the domain. They applied truncated series of Chebyshev polynomials of the third and fourth kinds to find semi bounded solutions for (1) with constant K(x, s) on the diagonal [21]. They expressed the kernel as…”

A Novel Collocation Method for Numerical Solution of Hypersingular Integral Equation with Singular Right-Hand Function

“…In certain cases, there exist some analytical methods for solving HSIEs, but in general, there is no known analytical method. Thus, different numerical methods are introduced and developed for solving HSIEs [8,9,[12][13][14][15][16][17][18][19][20][21][22][23][24][25][26].…”