Singular Integral Equations with Cauchy Kernels (original) (raw)
Singular integral equations with cauchy kernel on the half-line
International Journal of Engineering Science, 1987
(') N. I. Mushelishvili, Application of integrals of Cauchy type to a class of singular integral equations, Travaux de l'Institut mathématique de Tbilissi (Trudy Tbilisskogo matematicheskogo instituta) vol. 10 (1941) pp. 1-44; also see pp. 161-162. (2) I. Vecoua, Integral equations with a singular kernel of Cauchy type, Travaux de l'Institut mathématique de Tbilissi (Trudy Tbilisskogo matematicheskogo instituta) vol. 10 (1941) pp.
Singular integral equations with a Cauchy kernel
Journal of Computational and Applied Mathematics, 1986
A method is proposed to solve various linear and nonlinear integral equations with a Cauchy kernel. It yields closed-form expressions for the solutions. These involve the evaluation of infinite Hilbert integrals. Some examples of solutions of linear and non-linear equations are then presented.
Singular Integral Equations with Multiplicative Cauchy-type Kernels
Fasciculi Mathematici, 2013
In this paper we consider singular integral equations of the first kind with multiplicative Cauchy-type kernels defined on n-dimensional domains. We give their general solutions in the class of Hölder continuous functions and propose the statements of uniqueness problem.
Approximate solution of singular integral equations of the first kind with Cauchy kernel
Applied Mathematics Letters, 2009
In this work a study of efficient approximate methods for solving the Cauchy type singular integral equations (CSIEs) of the first kind, over a finite interval, is presented. In the solution, Chebyshev polynomials of the first kind, T n (x), second kind, U n (x), third kind, V n (x), and fourth kind, W n (x), corresponding to respective weight functions W (1) (x) = (1 − x 2) − 1 2 , W (2) (x) = (1 − x 2) 1 2 , W (3) (x) = (1 + x) 1 2 (1 − x) − 1 2 and W (4) (x) = (1 + x) − 1 2 (1 − x) 1 2 , have been used to obtain systems of linear algebraic equations. These systems are solved numerically. It is shown that for a linear force function the method of approximate solution gives an exact solution, and it cannot be generalized to any polynomial of degree n. Numerical results for other force functions are given to illustrate the efficiency and accuracy of the method.
On the Analytical Solutions of Two Singular Integral Equations with Hilbert Kernels
Journal of Integral Equations and Applications, 2005
The analytical solution of two singular integral equations with Hilbert kernel of the first and second kind, respectively, is derived from the known solution of the corresponding singular integral equations with Cauchy kernel of the first and second kind by introducing a proper change of variables.
Semi-Bounded Solutions of Singular Integral Equations of Cauchy Type
2009
In this work, semi-bounded numerical solutions of the Cauchy type singular integral equations of the first kind over the interval [−1, 1] is presented. In the integral equation, the truncated Chebyshev series of the third kind V j and fourth kind W j with the weight functions ω 1 (x) = (1 + x) 1/2 (1 − x) −1/2 and ω 2 (x) = (1 + x) −1/2 (1 − x) 1/2 , respectively, are used to approximate the unknown function. The exactness of the method is shown for characteristic singular integral equation when the forcing function is linear. The numerical results are given to show the efficiency and accuracy of the method presented.
2018
In present paper, a numerical approach for solving Cauchy type singular integral equations is discussed. Lagrange interpolation with Gauss Legendre quadrature nodes and Taylor series expansion are utilized to reduce the computation of integral equations into some algebraic equations. Finally, five examples with exact solution are given to show efficiency and applicability of the method. Also, we give the maximum of computed absolute errors for some examples.