Solution of a class of the first kind singular integral equation with multiplicative Cauchy kernel

where D = (−1, 1)× (−1, 1) , f(x, y) is a given Hölder continuous function in D, and φ(x, y) is an unknown function. The equation (1) has applications in the theory of aeroelasticity [1]. Note that the equation without logarithmic singularities was many times considered in different classes of functions. In the literature the solutions of the equation (1) in bounded domains [2, 5, 6, 9] as well as unbounded [3, 4, 7, 8, 10], are known for both single and multiple integrals. Let us introduce the function classes that will be used here.


Introduction. Let us consider a singular integral equation of the form
(1) where D = (−1, 1) × (−1, 1) , f (x, y) is a given Hölder continuous function in D, and ϕ(x, y) is an unknown function.The equation (1) has applications in the theory of aeroelasticity [1].Note that the equation without logarithmic singularities was many times considered in different classes of functions.In the literature the solutions of the equation (1) in bounded domains [2,5,6,9] as well as unbounded [3,4,7,8,10], are known for both single and multiple integrals.
Let us introduce the function classes that will be used here.

Exact solution in the class h
. Then the general solution of the equation (1) in the class h 0 × h 0 has the form where and γ 1 (x), γ 2 (y) are arbitrary functions from the class h 0 .
If the solution ϕ(x, y) satisfies the conditions where g(x) and h(x) are given functions of the class h 0 such that then the equation (1) has the unique solution given by the following formula: one can express the equation (1) in the form (13) Solving the equation (13) in the class h 0 , we obtain [9] (15) where c 1 (y) is an arbitrary function from h 0 .Next, solving (11), we have where R (f ; x, y) is given by the formula ( 6), and c 2 (x), γ 1 (x), γ 2 (y) are arbitrary functions from h 0 .We get the same result solving the equations ( 14) and, consequently, (12).In order to determine the functions γ 1 (x), γ 2 (y), we substitute the general solution given by ( 5) to the conditions ( 7)- (9).Then using the Poincaré-Bertrandt formula, we prove that the unique solution of the equation ( 1) in the class h 0 × h 0 is given by the formula (10).

1). Then the unique solution of the equation (1) in the class h(−1, 1) × h(−1, 1) exists if and only if the following conditions:
are fulfilled and it is given by the following formula: Proof.Similarly to the previous proof we introduce denotations (11), ( 12) and express the equation (1) in the form (13) or ( 14), respectively.Solving the equation (13) in the class of bounded functions, having the condition (18) fulfilled, we get [9] (20) Next, from (14) and condition (17) we get It is easy to verify that the conditions (17), ( 18) are also sufficient for solvability of the equations ( 11) and ( 12), respectively in the class of bounded functions, and the solution of the equation ( 1) is given by the formula (19).u(x, y)

Approximate solution in the class
Then it is easy to show that the problem (1), ( 7), ( 8), ( 9) takes the form Now we find an approximate solution of the problem (23), (24), (25).For this purpose we approximate the function f (x, y) by the interpolating polynomial of the form (26) and define the approximate solution u mn (x, y) as a solution of the following problem: (27) To find the form of an approximate solution, we use the relation ( 22) and the exact solution in the class h 0 × h 0 given by ( 5) with the kernel (6).
Next, substituting the function f m,n (x, y) defined in (26) in place of the given function f (x, y) and using the formula where P k (x), Q j (y) are the polynomials of degree k and j, respectively, defined as the principal part of the Laurent expansion of the following functions: in a neighborhood of the infinity, c kj are the unknown coefficients.
To determine the coefficients c kj we substitute the right-hand side of the formula (31) to the equation ( 27), getting and taking into account the following formulae: we compare the corresponding coefficients getting formulae for coefficients c kj , k, j = 1, . . ., n.Then substituting the right-hand side of the formula (31) to the conditions (28), (29), we get and respectively.From the above equations, using formula (40) , n = 0, 2, 4, . . ., we get the coefficients c 00 , c 0j , c k0 , j, k = 1, . . ., n.
Finally, (41) k , q (s) j are the coefficients of the polynomials P r (x), Q s (y) given in (32) and (33).

Approximate solution in the class
. As previously, we introduce a new unknown function u(x, y) defined by Then the equation ( 1) takes the form (43) where the given function f (x, y) satisfies the conditions (17), (18).Next, we approximate the function f (x, y) by the polynomial (26) and define the approximate solution u mn (x, y) as a solution of the following equation: Note that the function f mn (x, y) does not have to satisfy the conditions (17), (18).Therefore the sum Q * 1 (x) + Q * 2 (y) is added.Substituting the right-hand side of the equation (44) to the conditions (17), (18), we get the relations (45) ln Dividing (45) by √ 1 − x 2 , integrating respect to x, and adding both sides of the equations ( 45) and (46), we get Similarly to the class h 0 ×h 0 , it can be proved that the approximate solution u mn (x, y) has the form where P k−2 (x), Q j−2 (y) are polynomials of degree k − 2 i j − 2, respectively, defined as the leading part of the Laurent expansion of the following functions: h 0 ×h 0 .To find an approximate solution of the equation (1) in the class h 0 ×h 0 we introduce a new unknown function u(x, y) by the relation (22) ) in a neighborhood of the infinity.Here c kj are the unknown coefficients.Substituting the right-hand side of the formula (48) to equation (44)−4.9440984× 10 −18 −0.0015 0.9986 1.32194482 × 10 −16 −0.5523 0.6686 −5.5595619 × 10 −18 −0.9853 −0.0006 6.6776657 × 10 −18 −0.3247 −0.8954 −3.0507894 × 10 −17 −0.0247 −0.2354 −5.260122 × 10 −18 0.4247 −0.7554 −1.0698026 × 10 −17 0.9487 −0.1554 4.2669215 × 10 −18

Table 1 .
Comparison of the exact and approximate solutions in the class h 0 × h 0 .