Numerical solutions of hypersingular integrals and integral equations of the first kind

In this thesis, two problems are considered: i) An automatic quadrature scheme is presented for the evaluation of hypersingular integral of the form Qi(f; x) =Z 1¡1 wi(t)f(t)(t ¡ x)2 dt; x 2 [¡1; 1]; i = 0; 1; 2; (1) where w0(x) = 1; w1(x) = p1 ¡ x2; w2(x) = 1 p1¡x2 are the weights, and the function f imperatives to have certain smoothness or continuity properties. ii)We also described the approximate solutions of hypersingular integral equations of the form Z 1¡1 Q(t)h K(t; x)(t ¡ x)2 + L(t; x)idt = f(x); x 2 (¡1; 1); (2)where K(t; x) and L(t; x) are regular square-integrable functions of t and x, andK(x; x) 6= 0. The density function Q(t) satis¯es the HÄolder-continuous ¯rst deriva-tive, means that Q(t) 2 C1;®[¡1; 1]. The real function f is approximated by the orthogonal Chebyshev polynomials of the ¯rst and second kinds Tn(x) and Un(x) respectively. For the ¯rst problem in (1), an automatic quadrature scheme (AQS) for hypersin-gular integrals is derived. The numerical results show that the Chebyshev polyno- mials give a very good approximation by choosing the appropriate weight function. Particular attention is paid to the error estimate of the numerical solutions of Eq. (1). The error rate is calculated by Chebyshev norm for the class of functions CN+2;®[¡1; 1], which is de¯ned as k eN kc = max ¡1·a·t·b·1 jf(t) ¡ PN(t)j: (3)For the second problem in (2), we ¯rst consider the characteristic hypersingular integral equation of the form1 ¼ = Z 1 ¡1 Á(t)dt (t ¡ x)2 = f(x); jxj < 1; (4) where K(t; x) = 1 and L(t; x) = 0. By applying the Galerkin method, Eq. (4) can be reduced to a system of linear algebraic equations. The exactness of the numerical solutions of Eq. (4), when the density function Á(t) is a polynomial of degree 3, is proved. While for the case of K(t; x) = 1 and L(t; x) 6= 0, an e±cient expansion method for approximating the solution of Eq. (2) is presented. MATLAB codes are developed to obtain the numerical results for all proposed problems. The numerical examples assert the theoretical results.

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