Singular Integral Equations with Cauchy Kernels (original) (raw)
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.
One-dimensional Singular Integral Equations
Complex Variables and Elliptic Equations, 2003
A class of singular integral equations having solution in closed form is considered. This class contains the characteristic equation and other equations associated with it. Equations are solved by their reduction to some Riemann boundary value problems in generalized Ho¨lder spaces H ' ðÀÞ, where À is a closed rectifiable Jordan curve for which the Plemelj-Privalov theorem holds.
On the Solution of a Class of Cauchy Integral Equations
Journal of Mathematical Analysis and Applications, 2000
In a previous paper we have presented a new method for solving a class of Cauchy integral equations. In this work we discuss in detail how to manage this method numerically, when only a finite and noisy data set is available: particular attention is focused on the question of the numerical stability.
Solvability of Quadratic Integral Equations with Singular Kernel
Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences), 2022
In this paper, we discussed the existence and uniqueness of solution of the singular Quadratic integral equation (SQIE). The Fredholm integral term is assumed in position with singular kernel. Under certain conditions and new discussions, the singular kernel will tend to a logarithmic kernel. Then, using Chebyshev polynomial, a main of spectral relationships are stated and used to obtain the solution of the singular Quadratic integral equation with the logarithmic kernel and a smooth kernel. Finally, the Fredholm integral equation of the second kind is established and its solution is discussed, also numerical results are obtained and the error, in each case, is computed.
On the approximation of singular integral equations by equations with smooth kernels
Integral Equations and Operator Theory, 1995
Singular integral equations with Cauchy k ernel and piecewise{continuous matrix coe cients on open and closed smooth curves are replaced by i n tegral equations with smooth kernels of the form (t ;) (t ;) 2 ; n 2 (t)" 2 ] ;1 , " ! 0, where n(t), t 2 ;, is a continuous eld of unit vectors non{tangential to ;. We give necessary and su cient conditions under which the approximating equations have unique solutions and these solutions converge to the solution of the original equation. For the scalar case and the space L 2 (;) these conditions coincide with the strong ellipticity of the given equation.
Analysis of Abel-type nonlinear integral equations with weakly singular kernels
Boundary Value Problems, 2014
In this paper, we investigate Abel-type nonlinear integral equations with weakly singular kernels. Existence and uniqueness of nontrivial solution are presented in an order interval of a cone by using fixed point methods. As a byproduct of our method, we improve a gap in the proof of Theorem 5 in Buckwar (Nonlinear Anal. TMA 63:88-96, 2005). As an extension, solutions in closed form of some Erdélyi-Kober-type fractional integral equations are given. Finally theoretical results with three illustrative examples are presented. MSC: Primary 26A33; secondary 45E10; 45G05