ABSTRACT
Figure 9.3. Comparison of exact (line 1) and proposed (line 2) methods for the calculation of diffracted waves for a bagel obstacle
Figure 9.4. Comparison of exact (line 1) and proposed (line 2) methods for the calculation of diffracted waves for a three›leaf rose
This section follows in the main the classical results of Belotserkovsky and Lifanov (1993). Let us start to expound the methods suitable for numerical treatment of singular integrals and integral equations from a quadrature formula for the Cauchy›type singular integrals. Let us introduce the so›called canonical partition of the interval (a, b) with its subdivision to n+1 equal subintervals of the length h = (b – a)/(n + 1). This implies two sets of nodes: {tj} with a = t0, t1, t2, . . . , tn, tn+1 = b, tj = a + jh, j = 0, 1, . . . ,n + 1, and {xi} as the
central points of each subinterval (ti, ti+1), i = 0, 1, . . . ,n, so that xi = a + ( i + 1/2
) h,
i = 0, . . . ,n. It is proved that if g(x) ∈ C1(a, b), then∫ b a
g(t) dt xi – t
= h
g(tj ) xi – tj
+O
[ lnn
n(xi – a)(b – xi) ]
, n→∞ ∼ h→ 0. (9.48)
Now let us now study a singular integral equation with a characteristic kernel∫ b a
g (t) dt x – t
= f (x), x ∈ (a, b) , f (x) ∈ C1 (a, b) , (9.49)
and we recall some classical results related to this equation (see, for example, Gakhov, 1966; Muskhelishvili, 1965). For equation (9.49) there exist three different classes of its solution: 1) Solution unbounded at both ends of the interval (a, b). Such a solution is not unique, and as in the theory of ordinary differential equations of the rst order this is dened to within an arbitrary constantC . 2) Solution bounded at any of the two ends of the integration interval. Such a solution is unique. 3) With some additional condition for the right›hand side (i.e., not for every function f (x)) there can exist a solution bounded at both ends of the interval. We will be interested here only in case 1), where the exact analytical solution to equation (9.49) is given explicitly as
g(t) = 1 pi2 √(t – a) (b – t)
[ piC –
√(x – a) (b – x) t – x
f (x) dx ]
. (9.50)
Note that arbitrary constant C is related with the integral of the unknown function g(x):
C =
g(x) dx. (9.51)
Our further strategy is to construct a direct numerical collocation technique to solve characteristic Eq. (9.49) for arbitrary right›hand side, so that the constructed solution is a correct approximation for exact analytical solution (9.50). Then the method developed for the characteristic case will allow us to spread these results to a full singular integral equation where the analytical solution is not known.
