Numerical Solutions Of Cauchy Type Singular Integral Equations Of The First Kind Using Polynomial Approximations

In this thesis, the exact solutions of the characteristic singular integral equation of Cauchy type 1−1'(t)t − x dt = f(x), −1 < x < 1, (0.1) are described, where f(x) is a given real valued function belonging to the H¨older class and '(t) is to be determined. We also described the exact solutions of Cauchy type singular integral equations of the form /1−1'(t)t − xdt +/ 1−1 K(x, t) '(t) dt = f(x), −1 < x < 1, (0.2) where K(x, t) and f(x) are given real valued functions, belonging to the H¨older class, by applying the exact solutions of characteristic integral equation (0.1) and the theory of Fredholm integral equations. This thesis considers the characteristic singular integral equation (0.1) and Cauchy type singular integral equation (0.2) for the following four cases:Case I. '(x) is unbounded at both end-points x = ±1, Case II. y(x) is bounded at both end-points x = ±1, Case III. y(x) is bounded at x = −1 and unbounded at x = 1, Case IV. y(x) is bounded at x = 1 and unbounded at x = −1. The complete numerical solutions of (0.1) and (0.2) are obtained using polynomial approximations with Chebyshev polynomials of the first kind Tn(x), second kind Un(x), third kind Vn(x) and fourth kind Wn(x) corresponding to the weight functions w1(x) = (1 − x2)−1/2 , w2(x) = (1 − x2)1/2 , w3(x) = (1 + x)1/2 (1 − x)−1/2 andw4(x) = (1 + x)−1/2 (1 − x)1/2 , respectively

Numerical Evaluation of Cauchy Type Singular Integrals Using Modification of Discrete Vortex Method

In this thesis, characteristic singular integral equations of Cauchy type ()(),LtdtfxxLxtϕ=∈−∫ (1) where L is open or closed contour, are examined. The analytical solutions for equation (1) are described. Some examples of solution for certain functions f (x) are given. A quadrature formula for evaluation of Cauchy type singular integral (SI) of the form 11(),11tdtxxtϕ−−<<−∫ (2) is constructed with equal partitions of the interval [−1,1] using modification discrete vortex method (MMDV), where the singular point x is considered in the middle of one of the intervals [tj, tj+1], j=1,…, n. It is known that the bounded solution of equation (1) when L=[−1,1] is 12211,111ftxxdtttxϕπ−−=−−∫ (3) A quadrature formula is constructed to approximate the SI in (3) using MMDV and linear spline interpolation functions, where the singular point x is assumed to be at any point in the one of the intervals [tj,tj+1], j=1,…, n. The estimation of errors of constructed quadrature formula are obtained in the classes of functions C1[−1,1] and Hα(A,[−1,1]) for SI (2) and Hα(A,[−1,1]) for (3). For SI (2), the rate of convergence is improved in the class C1[−1,1], whereas in the class Hα(A,[−1,1]), the rate of convergence of quadrature formula is the same of that of discrete vortex method (MDV). FORTRAN code is developed to obtain numerical results and they are presented and compared with MDV for different functions f(t). Numerical experiments assert the theoretical results

Unbounded solution of characteristic singular integral equation using differential transform method

In this paper, The differential transform method is extended to solve the Cauchy type singular integral equation of the first kind. Unbounded solution of the Cauchy type singular&nbsp; Integral equation is discussed. Numerical results are shown to illustrate the efficiency and accuracy of the present solution

A note on the numerical solution for Fredholm integral equation of the second kind with Cauchy kernel.

In this study, numerical solution for the Fredholm integral equation of the second kind with Cauchy singular kernel is presented. The Chebyshev polynomials of the second kind are used to approximate the unknown function. Numerical results are given to show the accuracy of the present numerical solution. The present numerical solution to the Fredholm integral equation of the second kind with Cauchy kernel is accurate

Half-bounded numerical solution of singular integral equations with Cauchy kernel.

In this study, a numerical solution for singular integral equations of the first kind with Cauchy kernel over the finite segment [-1,1] is presented. The numerical solution is bounded at x =1 and unbounded at x = -1. The numerical solution is derived by approximating the unknown density function using the weighted Chebyshev polynomials of the fourth kind. The force function is approximated by using the Chebyshev polynomials of the third kind. The exactness of the numerical solution is shown for characteristic equation when the force function is a cubic

On the semi-bounded solution of cauchy type singular integral equations of the first kind.

This paper presents an efficient approximate method to obtain a numerical solution, which is bounded at the end point x = −1, for Cauchy type singular integral equations of the first kind on the interval [−1,1]. The solution is derived by approximating the unknown density function using the weighted Chebyshev polynomials of the third kind, and then computing the Cauchy singular integral which is obtained analytically. The known force function is interpolated using the Chebyshev polynomials of the fourth kind. The exactness of this approximate method is shown for characteristic equation when the force function is a cubic. Particular result is also given to show the exactness of this method

Numerical evaluation for Cauchy type singular integrals on the interval.

The singular integral (SI) with the Cauchy kernel is considered. New quadrature formulas (QFs) based on the modification of discrete vortex method to approximate SI are constructed. Convergence of QFs and error bounds are shown in the classes of functions Hα([−1,1])Hα([−1,1]) and C1([−1,1])C1([−1,1]). Numerical examples are shown to validate the QFs constructed

A note on the numerical solution of singular integral equations of Cauchy type

This manuscript presents a method for the numerical solution of the Cauchy type singular integral equations of the first kind, over a finite segment which is bounded at the end points of the finite segment. The Chebyshev polynomials of the second kind with the corresponding weight function have been used to approximate the density function. The force function is approximated by using the Chebyshev polynomials of the first kind. It is shown that the numerical solution of characteristic singular integral equation is identical with the exact solution, when the force function is a cubic function. Moreover, it also shown that this numerical method gives exact solution for other singular integral equations with degenerate kernels

Quadrature formula for approximating the singular integral of Cauchy type with unbounded weight function on the edges.

New quadrature formulas (QFs) for evaluating the singular integral (SI) of Cauchy type with unbounded weight function on the edges is constructed. The construction of the QFs is based on the modification of discrete vortices method (MMDV) and linear spline interpolation over the finite interval [−1,1]. It is proved that the constructed QFs converge for any singular point x not coinciding with the end points of the interval [−1,1]. Numerical results are given to validate the accuracy of the QFs. The error bounds are found to be of order O(hα|lnh|) and O(h|lnh|) in the classes of functions Hα([−1,1]) and C1([−1,1]), respectively