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.

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