A potential—theoretic formulation of the problem of conformally mapping a simply connected region (or its complement) onto the unit disk leads to a Fredholm integral equation of the first kind, known as Symm’s integral equation, which has a kernel with a logarithmic singularity. Unlike Fredholm integral equations of the second kind, e.g., Theodorsen’s equation, in which the singularity of the kernel at points near but not on the boundary creates computational difficulties, Symm’s integral equation is found easily solvable by numerical methods, such as the orthonormal polynomials method or its modified form, Lagrange’s interpolation method, and spline approximations which are discussed in this chapter. Numerical evaluation of Green’s functions, as developed in Chapter 6, is another viable alternative to obtain the approximate mapping function.
KeywordsIntegral Equation Fredholm Integral Equation Spline Approximation Interior Angle Exterior Region
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