Approximate solution of singular integral equations

K e y w o r d s I n t e g r a l equations, Cauchy type, Singular kernels. 1. I N T R O D U C T I O N Singular integral equations of the first kind, with a Cauchy type singular kernel, over a finite interval can be represented by the general equation f f _ l f(t)[ko(t,x)+k(t,x)]dt = g(x), 1 < x < 1, (1.1) 1 where ko(t, x) x) t) 0) (1.2) t x ' and k are regular square in tegrab le funct ions of t he two var iables t and x, and the kernel k0 c lear ly involves t he s ingula r i ty of t he Cauchy type. In tegra l equa t ions of form (1.1) and o ther different forms occur in var ie t ies of mixed b o u n d a r y va lue p rob lems of m a t h e m a t i c a l physics A. C. thanks the authorities of Ghent University for supporting a short visit during which the present work was completed. *Presently on leave, at the Department of Mathematical Sciences, NJIT, University Heights, Newark, NJ 07102, U.S.A. 0893-9659/04/$ see front matter Q 2004 Elsevier Ltd. All rights reserved. Typeset by JtA/IS-2~X doi:10.1016/j.aml.2003.04.007 554 A. CHAKRABARTI AND G. VANDEN BERGHE which include problems of two-dimensional deformations of isotropic elastic bodies involving cracks (see [1-3]) and scattering of two-dimensional surface water waves by vertical barriers (see [4-8]) and other related problems. The simplest integral equation of the form (1.1) is ~ f( t) dt = g(x), (1.3) 1 $ x for which k(t,x) = 1 and k(t,x) = 0 (see [1,3]), and there are four basically important and interesting cases of equation (1.1), even under such simplifying assumptions on the nature of the kernel (i.e., when k = 1 and k = 0), as given by the following. CASE (I). f(x) is unbounded at both the endpoints x = +1. CASE ( I I ) . f(x) is unbounded at the end x = -1 , but bounded at the end x = +1. CASE (III). f(x) is bounded at the end x = -1 , but unbounded at the end x = +1. CASE (IV). f(x) is bounded at both the endpoints x = +1. It is well known (see [1,3]) that the complete analytical solutions of the singular integral equation (1.3) in the above four cases can be determined by using the following formulae: Ao 1 f l (1 t 2) 1/2 g(t) Case (I): f(x) = (1 X2) 1/2 7r 2 (1 x2) 1/2 ~-1 (t x) dr, (1.4) where A0 is an arbitrary constant, Case (II): f(x) = -Tr'l dr, (1.5) Case (III): f(x) = -Tr zl dt, (1.6) Case (IV): (1 x 2 ) g(t) dr, (i.7) 1 (1 t2) 1/2 (t x) the solution existing in Case (IV), if and only if _1 g(t) (1.8) 1 (1 -~-)1/2 dt=O. Guided by the analytical results available, as given by expressions (1.4)-(1.7), for the solution of the simple singular equation (1.3), as well as by utilizing the idea (see [9]) of replacing the integrand by an appropriate approximate function, we explain, in the next section, a numerical scheme that can be developed and implemented, for obtaining the approximate solutions of the general singular integral equation (1.1). The particular case of equation (1.3) follows quite easily and the known analytical solutions are recovered in the cases of simple forms of the forcing function g(x), being polynomials of low degree. 2. T H E A P P R O X I M A T E S C H E M E We shall represent the unknown function f(x) in the form f(x) = h,(x)A,(x) ( 1 x2) 1/2' (r = 1, 2, 3, 4), (2.1) where hr (x) is a well-behaved function of x in the interval 1 < x < 1, and A1 (x) = 1, in Case (I), A2(x) = 1 x, in Case (II), A3(x) -1 -tx, in Case (III), and A4(x) -1 x 2, in Case (IV). Approximate Solution 555 Then we approximate the unknown function hr(x) by means of a polynomial of degree n, given by hr(x) ~ c r)xJ , (r = 1,2,3,4), (2.2) in the four cases as mentioned above, by using a "Chebyshev approximation", using the zeros xj (j = O, 1, 2 , . . . , n) of the Chebyshev polynomial Tn+l(x) = cos[(n + 1)arccos(x)] (see [10]) in [-1,1]. Using the approximate form (2.2) of the function hr(x) along with the representation (2.1), in the original integral equation (1.1), we obtain j__~0 cj~ ) /~(t)k(t, x)tJ 1 Xr(t)k(t, x)t j (1 t . ) 112 (t x) . 1 ( 7 ~ = 9(x), (2.3) (r = 1,2,3,4), ( l < x < 1). In the above equation (2.3), we next use the following "Chebyshev approximations" to the kernels k(t, x) and k(t, x), given by (for fixed x) ~(t,x) ~ ~ ~p(~)t., ,=0 (2.4) s k(t, z) ~ ~ kq(x)e, q=0 with known expressions for kp(x) and kq(x), obtainable in terms of the points tp, tq, where 1 < to < tl < " " < t,~ < 1 and 1 < to < tl < . . . < ts < 1, to,t l , . . . , tm being the zeros of T,~+l(t) in [-1, 1]. We thus obtain the following functional relation to be solved for the unknown constants cj (j = 0, 1 , . . . , n): i; . ,,_ (2.~) j=0 q=0 1 ( -1 < x < 1). Now, using the notations i-~ " t p + j ~r(t) dt-~ u~)j(x) (2.6) and i ; tq+jX'(t) dr=(2.7) (~) 1 (1 t2) 1/2 "Yq+J' (,-) where the up+j (x) can be determined to be certain polynomials by using standard contour integration and where (') is a constant, obtainable in terms of the J-functions, we obtain, from [q+j equation (2.5), EcJ r) kp(X)U j(x)-tkq(x)'y~j =g(x ) , ( r = 1 , 2 , 3 , 4 ) , l < x < l . (2.8) j=O q=0 Setting x = xt, l = 0, 1, 2 , . . . , n in relation (2.8), we obtain the following system of ( n + l ) × ( n + l ) linear equations for the determination of the unknown constants c~ ~), (j = 0, 1, 2 , . . . , n): n v'~ cj.(r)c~jj(r) = gt, (l = O, 1, 2. . . , n), (r = 1, 2, 3, 4), (2.9) j=O 556 A. CHAKRABARTI AND G. VANDEN BERGHE