We consider the undulatory delta hydrofoil. The equations of the leading edge are
(64)
In order to ensure the uniqueness of the solution of the integral equation, some analytical results from [2] suggest to presume that there exists a continuous finite function g such that . We have therefore
(65)
where FP stands for the finite part of the hypersingular integral as it is introduced by Ch. Fox in [21]. Since the inner integral vanishes for , we assume that
(66)
where is finite for . We consider on D a net consisting of the nodes (grid points, control points) , , . For the hypersingular integral occurring in (65) we may use the quadrature formula for equidistant control points given by Dumitrescu [22],
(67)
(68)
(69)
We shall give a quadrature formula for calculating . Denoting
(70)
(71)
(72)
we have
(73)
whence it follows that
(74)
(75)
(76)
Finally we deduce
(77)
with
(78)
For we get
(79)
Assuming that , we have
(80)
(81)
(82)
For calculating we employ the quadrature formula
(83)
At last we find
(84)
(85)
The singularities of the kernels , , are weaker than . We replace these kernels with , , for obtaining approximation formulas similar to the formulas for . We get for :
(86)
(87)
The kernels , , , , , , , , and are continuous and we utilize the approximation formulas
(88)
(89)
For calculating we use the series expansions of the Bessel and Strouve functions and we take into account that
(90)
where , . and are integrals which are evaluated numerically with the trapezoidal rule. For calculating the Bessel (MacDonald) functions and we may utilize the series expansions. We may also utilize the libraries offered by MATLAB. For calculating the kernels and we use the integral representations
(91)
(92)
The integrals are evaluated numerically with the trapezoidal rule. The approximation formulas for the kernels , , , , , , , , , , and were also given in [5] and [6]. The new approximation formulas for the kernels , , are given for the first time herein. Denoting , we obtain, discretizing the two-dimensional integral equation (32):
(93)