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.

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