Re: Muskhelishvili Singular Integral Equations Djvu
Robert Bhushan
A system of integral equations is obtained for the tangential components of the velocities on the upper and lower surfaces of a profile which does not have a parametric singularity connected with the thickness of the airfoil. An example of the solution of these equations by the discrete vortex method is given. This shows that they are highly effective for solving problems of flow over an airfoil of any, including arbitrarily small, thickness.
During the last three decades, the singular integral equation methods with applications to several basic fields of engineering mechanics, like elasticity, plasticity, aerodynamics and fracture mechanics have been studied and improved by several scientists (see [1] -[6] ). Hence, it is interest to solve numerically this type of integral equations (see [7] [8] ). Chebyshev polynomials are of great importance in many areas of mathematics particularly approximation theory (see [9] [10] ).
In this paper we analyze the numerical solution of singular integral equations by using Chebyshev polynomials of first, second, third and fourth kind to obtain systems of linear algebraic equations, these systems are solved numerically. The methodology of the present work expected to be useful for solving singular integral equations of the first kind, involving partly singular and partly regular kernels. The singularity is assumed to be of the Cauchy type. The method is illustrated by considering some examples.
where and are given real-valued continuous functions belonging to the class Holder of continues functions and. In Equation (1.1) the singular kernel is interpreted as Cauchy principle value. Integral equation of form (1.1) and other different forms have many applications (see [1] [2] [6] [11] [12] ). The theory of this equation is well known and it is presented in [13] [14] . An approximate method for solving (1.1) using a polynomial approximation of degree has been proposed in [7] .
3) The solution is bounded at end but unbounded at end4) The solution is unbounded at end but bounded at endare given by [15] . In this paper the used approximate method for solving Equation (1.1) stems from recent work [10] wherein an approximate method has been developed to solve the simple Equation (1.2). The approximate method developed below appears to be quite appropriate for solving the most general type Equation (1.1). Some examples are presented to illustrate the method.
where are unknown coefficients, to be determined, and in case (I):, in case (II):, in case (III): and in case (VI):, where, , and are The Chebyshev polynomials of the first, second, third and fourth kinds respectively can be defined by the recurrence relations [9] [16] .
Numerical results (Tables 1-3) show that the errors of approximate solutions of Examples 1-3 in different Cases with small value of n are very small. These show that the methods developed are very accurate and in fact for a linear function give the exact solution.
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