Citation
Mahiub, Mohammad Abdulkawi
(2010)
Numerical Solutions Of Cauchy Type Singular Integral Equations Of The First Kind Using Polynomial Approximations.
PhD thesis, Universiti Putra Malaysia.
Abstract
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 endpoints x = ±1,
Case II. y(x) is bounded at both endpoints 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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