Approximate solution of a dominant singular integral equation with conjugation
UDC 517.5In the present paper, the method of successive approximations and Faber polynomials are used to derive an approximate solution of a dominant singular integral equation with Holder continuous coefficients and conjugation on the Lyapunov curve. Moreover, conditions of convergence in the $L_2$...
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| author | Pylak, D. Wójcik, P. Pylak, D. Wójcik, P. P. |
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| description | UDC 517.5In the present paper, the method of successive approximations and Faber polynomials are used to derive an approximate solution of a dominant singular integral equation with Holder continuous coefficients and conjugation on the Lyapunov curve. Moreover, conditions of convergence in the $L_2$ and $H(α)$ spaces are presented.
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DOI: 10.37863/umzh.v73i9.758
UDC 517.5
D. Pylak, P. Wójcik (Lublin Univ. Technology, Poland)
APPROXIMATE SOLUTION OF A DOMINANT SINGULAR
INTEGRAL EQUATION WITH CONJUGATION
НАБЛИЖЕНИЙ РОЗВ’ЯЗОК ДОМIНАНТНОГО СИНГУЛЯРНОГО
IНТЕГРАЛЬНОГО РIВНЯННЯ ЗI СПРЯЖЕННЯМ
In the present paper, the method of successive approximations and Faber polynomials are used to derive an approximate
solution of a dominant singular integral equation with Hölder continuous coefficients and conjugation on the Lyapunov
curve. Moreover, conditions of convergence in the L2 and H(\alpha ) spaces are presented.
У цiй роботi за допомогою методу послiдовних наближень та полiномiв Фабера отримано наближений розв’язок
домiнантного сингулярного iнтегрального рiвняння з неперервними за Гельдером коефiцiєнтами та спряженням на
кривiй Ляпунова. Крiм того, запропоновано умови збiжностi у просторах L2 та H(\alpha ).
1. Introduction. Let us consider a singular equation with Cauchy kernel of the form
a1(t)\varphi (t) + b1(t)
1
\pi i
\int
L
\varphi (\tau )
\tau - t
d\tau + a2(t)\varphi (t) + b2(t)
1
\pi i
\int
L
\varphi (\tau )
\tau - t
d\tau = f(t), t \in L, (1.1)
where a1(t), b1(t), a2(t), b2(t), f(t) are given complex-valued Hölder continuous functions on a
Lyapunov curve L and \varphi (t) is an unknown function. We seek the solution in the class of Hölder
continuous functions.
Equation (1.1) has numerous applications [3, 4, 8, 11, 12, 17]. It is a generalization of the well-
known from the literature dominant singular integral equation [7, 11]. An explicit solution of the
dominant equation was found by I. N. Vekula [16]. He reduced it to the Riemann problem which
had been solved previously by F. D. Gakhov [6]. Equation (1.1) contains a conjugate of unknown
function. As the exact solution of such a equation can be found only in some rare particular cases
[10, 18] approximate methods are developed. In [14] the computational schemes of collocation and
mechanical quadrature methods for solving (1.1) are proposed when L is a smooth simple closed
contour containing inside it the point z = 0. In [2] approximation methods for singular integral
operators with continuous coefficients and conjugation on curves with corners are investigated with
respect to their stability.
In the present paper solution of equation (1.1) is based on the solution of some boundary problem
[13] being a generalization of Riemann problem [7, 11]. Next the method of successive approximation
and Faber polynomials are used to construct an approximate solution of equation (1.1) on a closed
Lyapunov curve. The Faber polynomials and their modifications play an important role in modern
methods of approximation of complex functions [1, 5, 9, 15, 19, 20].
Let us recall that a simple continuous curve is called Lyapunov curve if it satisfies the following
conditions:
at every point of L there exists a well-defined tangent,
c\bigcirc D. PYLAK, P. WÓJCIK, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9 1289
1290 D. PYLAK, P. WÓJCIK
the angle \theta (s) between OX axis and the tangent to L at the point M whose distance from a
fixed point, measured along the curve L, is equal to s, satisfies Hölder condition\bigm| \bigm| \theta (s\prime ) - \theta (s\prime \prime )
\bigm| \bigm| \leq K| s\prime - s\prime \prime | \mu ,
where K > 0, 0 < \mu \leq 1 are constants independent of the position of the points s\prime , s\prime \prime .
It is known [15, p. 90] that in such case every Cauchy-type integral
f(z) =
1
2\pi i
\int
L
\varphi (\zeta )
\zeta - z
d\zeta , z \in G,
is expandable in uniformly convergent Faber series.
2. Approximate solution. Let us consider the Cauchy-type integral
\phi (z) =
1
2\pi i
\int
L
\varphi (\tau )
\tau - z
d\tau , z \in D\pm ,
where D+ and D - are interior and exterior domains with the boundary L, respectively, and Sokhots-
ki – Plemelj formulae [7]
\varphi (t) = \phi +(t) - \phi - (t), t \in L, (2.1)
1
\pi i
\int
L
\varphi (\tau )
\tau - t
d\tau = \phi +(t) + \phi - (t), t \in L. (2.2)
Substituting the left-hand sides of equations (2.1) and (2.2) into equation (1.1) we get
\alpha 1(t)\phi
+(t) + \beta 1(t)\phi +(t) = \alpha 2(t)\phi
- (t) + \beta 2(t)\phi - (t) + f(t), (2.3)
\phi +(\infty ) = 0, (2.4)
where
\alpha 1(t) = a1(t) + b1(t), \beta 1(t) = a2(t) + b2(t),
\alpha 2(t) = a1(t) - b1(t), \beta 2(t) = a2(t) - b2(t).
Thus, equation (1.1) has been reduced to the boundary problem of finding two analytic functions
\phi +(z), z \in D+, and \phi - (z), z \in D - , which limit values fulfil (2.3) on the curve L and, moreover,
function \phi +(z) vanishes at infinity. Taking into account the system consisting of equation (2.3) and
its conjugation, and then eliminating the function \phi - (t), we obtain
\phi +(t) = A(t)\phi - (t) +B(t)\phi +(t) + C(t), t \in \Gamma , (2.5)
where
A(t) =
| \alpha 2(t)| 2 - | \beta 2(t)| 2
\alpha 1(t)\alpha 2(t) - \beta 1(t)\beta 2(t)
, B(t) =
\alpha 1(t)\beta 2(t) - \alpha 2(t)\beta 1(t)
\alpha 1(t)\alpha 2(t) - \beta 1(t)\beta 2(t)
,
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
APPROXIMATE SOLUTION OF A DOMINANT SINGULAR INTEGRAL EQUATION WITH CONJUGATION 1291
C(t) =
\alpha 2(t)f(t) - \beta 2(t)f(t)
\alpha 1(t)\alpha 2(t) - \beta 1(t)\beta 2(t)
.
Let A(t) \not = 0 \forall t \in \Gamma and the origin of a coordinate system \BbbC z belong to the area D+. Moreover,
let \mathrm{I}\mathrm{n}\mathrm{d}A(t) = \kappa \geq 0. Then by [7] the canonical functions X+(z), z \in D+, and X - (z), z \in D - ,
of the linear conjugation problem
X+(t) = A(t)X - (t), t \in \Gamma , (2.6)
have the forms
X+(z) = \mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
\Gamma +(z)
\bigr)
, z \in D+,
and
X - (z) = z - \kappa \mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
\Gamma - (z)
\bigr)
, z \in D - ,
respectively. Here
\Gamma \pm (z) =
1
2\pi i
\int
\Gamma
\mathrm{l}\mathrm{n} (\tau - \kappa A(\tau ))
\tau - z
d\tau , z \in D\pm .
Taking into account (2.6), the boundary problem (2.5) can be rewritten in the form
\phi +(t)
X+(t)
- \phi - (t)
X - (t)
= G(t)
\phi +(t)
X+(t)
+ g(t), t \in \Gamma ,
where
G(t) = B(t)
X+(t)
X+(t)
, g(t) =
C(t)
X+(t)
.
Furthermore, by setting
F\pm (z) =
\phi \pm (z)
X\pm (z)
, z \in D\pm , (2.7)
we obtain
F+(t) - F - (t) = G(t)F+(t) + g(t), t \in \Gamma . (2.8)
The solution of the problem (2.8) we will seek in the following form:
F (z) =
1
2\pi i
\int
\Gamma
\mu (\tau )
\tau - z
d\tau + P\kappa - 1(z), z \in D\pm , (2.9)
where \mu (\tau ) is an unknown function satisfying the Hölder condition and P\kappa - 1(z) = \gamma 0 + \gamma 1z + . . .
. . .+ \gamma \kappa - 1z
\kappa - 1 is a polynomial with arbitrary complex-valued coefficients \gamma 0, \gamma 1, . . . , \gamma \kappa - 1.
Applying the Sokhotski – Plemelj formulae to the function F (z), the problem (2.8) can be trans-
formed to the following singular integral equation:
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
1292 D. PYLAK, P. WÓJCIK
\mu (t) = G(t)
\left( 1
2
\mu (t) +
1
2\pi i
\int
\Gamma
\mu (\tau )
\tau - t
d\tau
\right) +G(t)P\kappa - 1(t) + g(t), t \in \Gamma , (2.10)
which can be solved by the method of successive approximations.
First, we establish the coefficients \gamma 0, \gamma 1, . . . , \gamma \kappa - 1 of the polynomial P\kappa - 1(z). They will be
uniquely determined if we supplement (2.3) with new conditions
- \mathrm{R}\mathrm{e}\mathrm{s}
z=\infty
\bigl(
zj - 1\phi - (z)
\bigr)
= Aj , j = 1, 2, . . . , \kappa , (2.11)
where Aj are arbitrary constants (see [11]). In view of (2.7) and (2.9) we have
\phi - (z) = X - (z)
\left( 1
2\pi i
\int
\Gamma
\mu (\tau )
\tau - z
d\tau + P\kappa - 1(z)
\right) .
Thus, using the expansion
X - (z) =
1
z\kappa
\Bigl(
1 +
p1
z
+
p2
z2
+ . . .
\Bigr)
of a canonical function X - (z) in a neighborhood of z = \infty , from (2.11) we derive the system of
equations
\gamma \kappa - 1 = A1,
\gamma \kappa - 2 + p1\gamma \kappa - 1 = A2,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
\gamma 0 + p1\gamma 1 + . . .+ p\kappa - 1\gamma \kappa - 1 = A\kappa .
The solution of this system can be given by a recurrent formulae of the form
\gamma \kappa - i = Ai -
\kappa - 1\sum
j=\kappa - i+1
pj\gamma j , i = 1, . . . , \kappa . (2.12)
Now let us solve equation (2.10) by successive approximation method taking the function
\mu 0(t) = g\ast (t) = G(t)P\kappa - 1(t) + g(t), t \in \Gamma ,
as an initial guess. The next approximations are given by the formula
\mu n+1(t) = G(t)
\left( 1
2
\mu n(t) +
1
2\pi i
\int
\Gamma
\mu n(\tau )
\tau - t
d\tau
\right) + g\ast (t), n = 0, 1, 2, . . . . (2.13)
Then the successive approximations for the problem (2.5), (2.11) have the form
\phi \pm
n (z) = X\pm (z)
\left( 1
2\pi i
\int
\Gamma
\mu n(\tau )
\tau - z
d\tau + P\kappa - 1(z)
\right) , z \in D\pm , n = 0, 1, 2, . . . . (2.14)
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
APPROXIMATE SOLUTION OF A DOMINANT SINGULAR INTEGRAL EQUATION WITH CONJUGATION 1293
Hence, applying Sokhotski – Plemelj formulae to the above functions, by (2.1) it finally follows that
approximate solution of the problem (1.1), (2.11) is given by the formula
\varphi n(t) = X - (t)
\left( 1
2
(A(t) + 1)\mu n(t)+
+
1
2\pi i
(A(t) - 1)
\int
\Gamma
\mu n(\tau )
\tau - t
d\tau + (A(t) - 1)P\kappa - 1(t)
\right) , t \in \Gamma , n = 0, 1, 2, . . . ,
where coefficients of the polynomial P\kappa - 1 satisfy relation (2.12).
Let us consider the case \mathrm{I}\mathrm{n}\mathrm{d}A(t) = \kappa < 0. As we have to assume P\kappa - 1(z) \equiv 0 in (2.9), we get
\phi \pm (z) = X\pm (z)
1
2\pi i
\int
\Gamma
\mu (\tau )
\tau - z
d\tau , z \in D\pm .
Since in this case the order of the pool of the function X - (z) in a neighborhood of z = \infty is | \kappa | ,
then the conditions
1
2\pi i
\int
\Gamma
\mu (\tau )\tau j - 1 d\tau = 0, j = 1, 2, . . . , | \kappa | ,
are necessary and sufficient to preserve the analyticity of the function \phi - (z), \phi - (\infty ) = 0, in a
neighborhood of the point z = \infty . The above equalities follow from the expansion
1
2\pi i
\int
\Gamma
\mu (\tau )
\tau - z
d\tau =
\infty \sum
k=1
ck
zk
, z \in D - ,
where
ck = - 1
2\pi i
\int
\Gamma
\mu (\tau )\tau k - 1 d\tau , k = 1, 2, . . . .
The approximation \phi \pm
n (z) with \kappa < 0 can be determined from the formula
\phi \pm
n (z) = X\pm (z)
\left( 1
2\pi i
\int
\Gamma
\mu n(\tau )
\tau - z
d\tau -
| \kappa | \sum
k=1
c
(n)
k
zk
\right) ,
where
c
(n)
k = - 1
2\pi i
\int
\Gamma
\mu n(\tau )\tau
k - 1 d\tau , k = 1, 2, . . . , | \kappa | .
Similarly like in the case of nonnegative index we apply Sokhotski – Plemelj formulae to the functions
\phi \pm
n (z) and by (2.1) we get the solution of the problem (1.1)
\varphi n(t) = X - (t)
\left( 1
2
(A(t) + 1)\mu n(t)+
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
1294 D. PYLAK, P. WÓJCIK
+
1
2\pi i
(A(t) - 1)
\int
\Gamma
\mu n(\tau )
\tau - t
d\tau + (A(t) - 1)
| \kappa | \sum
k=1
c
(n)
k
zk
\right) , t \in \Gamma , n = 0, 1, 2, . . . .
In the case of \kappa \geq 0 we use Faber polynomial to find an approximate value of the expression
1
2
\mu n(t) +
1
2\pi i
\int
\Gamma
\mu n(\tau )
\tau - t
d\tau , n = 1, 2, . . . , \mu 0(t) = g\ast (t),
occurring in (2.13). For this purpose we introduce the integral of Cauchy-type
Fn(z) =
1
2\pi i
\int
\Gamma
\mu n(\tau )
\tau - z
d\tau , z \in D+, n = 0, 1, . . . . (2.15)
By [15, p. 215] we have
Fn(z) =
\infty \sum
k=0
a
(n)
k \Phi k(z), z \in D+, (2.16)
where \Phi k(z), k = 0, 1, 2, . . . , are Faber polynomials of degree k and
a
(n)
k =
1
2\pi i
\int
\Gamma
\mu n(\tau )
\Phi k+1(\tau )
\Phi \prime (\tau ) d\tau =
1
2\pi i
\int
| t| =1
\mu n(\Psi (t))
tk+1
dt. (2.17)
The function \Phi (z) transforms conforemely and univalently the area D - in the coordinate system
\BbbC z to the exterior of the circle | w| > 1 in the coordinate system \BbbC w and satisfies the conditions
\Phi (\infty ) = 0, \Phi \prime (\infty ) > 0, where z = \Psi (w) is the inversion of the function w = \Phi (z).
Using the Sokhotski – Plemelj formulae, from (2.15), (2.16) we obtain
1
2
\mu n(t) +
1
2\pi i
\int
\Gamma
\mu n(\tau )
\tau - t
d\tau =
\infty \sum
k=0
a
(n)
k \Phi k(t), t \in \Gamma .
Then by (2.13) we get
\mu n+1(t) = G(t)
\infty \sum
k=0
a
(n)
k \Phi k(t) + g\ast (t), t \in \Gamma , n = 0, 1, . . . .
Finally, we have to find \phi \pm
n (z), n = 0, 1, . . . . If z \in D+, then by (2.14) and expansion (2.16) we
obtain
\phi +
n (z) = X+(z)
\infty \sum
k=0
a
(n)
k \Phi k(z) +X+(z)P\kappa - 1(z), z \in D+,
where a
(n)
k are given by the equalities (2.17).
If z \in D - we find the corresponding formulae in a similar way. In this case we have
\phi -
n (z) = X - (z)
\infty \sum
k=1
b
(n)
k
\Phi k(z)
+X - (z)P\kappa - 1(z), z \in D - , n = 0, 1, . . . ,
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
APPROXIMATE SOLUTION OF A DOMINANT SINGULAR INTEGRAL EQUATION WITH CONJUGATION 1295
where coefficients b
(n)
k are defined using the expansion
1
2\pi i
\int
\Gamma
\mu n(\tau )
\tau - z
d\tau =
\infty \sum
k=1
b
(n)
k
\Phi k(z)
, z \in D - .
3. Approximation criterion. Now we will give the conditions for the convergence of a sequence
of functions \{ \mu n(t)\} , n = 0, 1, . . . , determined by (2.13) in the spaces L2(\Gamma ). For this purpose we
rewrite equation (2.10) in the form
\mu - T\mu = g\ast ,
where
T\mu \equiv G(t)
\left( 1
2
\mu (t) +
1
2\pi i
\int
\Gamma
\mu (\tau )
\tau - t
d\tau
\right)
and
g\ast (t) = G(t)P\kappa - 1(t) + g(t).
As it was proved in [13], the following theorem is true.
Theorem 1. Let a1(t), b1(t), a2(t), and b2(t) be given continuous functions defined on the
closed Lyapunov curve \Gamma satisfying the condition
a1(t)a2(t) - b1(t)b2(t) \not = 0 \forall t \in \Gamma . (3.1)
Moreover, let
\mathrm{I}\mathrm{n}\mathrm{d}A(t) = \mathrm{I}\mathrm{n}\mathrm{d}
\Bigl(
b1(t)b2(t) - a1(t)a2(t)
\Bigr)
= \kappa \geq 0.
If
q =
1
2
(1 + S2)\mathrm{m}\mathrm{a}\mathrm{x}
t\in \Gamma
\bigm| \bigm| \bigm| \bigm| \bigm| a1(t)b2(t) - b1(t)a2(t)
a1(t)a2(t) - b1(t)b2(t)
\bigm| \bigm| \bigm| \bigm| \bigm| < 1, (3.2)
where S2 is a norm of the singular integral operator
S\mu =
1
\pi i
\int
\Gamma
\mu (\tau )
\tau - t
d\tau
in the space L2, then the boundary problem (2.5), (2.11) has a unique solution for an arbitrary
function C(t) \in L2(\Gamma ). This solution can be found by the successive approximations method.
In the case \mathrm{I}\mathrm{n}\mathrm{d}A(t) = \kappa < 0 if the condition (3.2) is fulfilled and the necessary and sufficient
conditions of the form
1
2\pi i
\int
\Gamma
(I - T ) - 1g(\tau )\tau j - 1 d\tau = 0, j = 1, 2, . . . , | \kappa | , (3.3)
hold, where
g(t) =
a2(t)c(t) - b2(t)c(t)
X+(t)
,
then the boundary-value problem (2.5) has a unique solution.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
1296 D. PYLAK, P. WÓJCIK
Similarly, in the Banach space H\alpha , 0 < \alpha \leq 1, consisting of functions determined on the curve
\Gamma and belonging to the function class H(\alpha ), with the norm given by the formula
\| f\| H\alpha = \mathrm{m}\mathrm{a}\mathrm{x}
t\in \Gamma
| f(t)| + \mathrm{s}\mathrm{u}\mathrm{p}
t1,t2\in \Gamma
| f(t2) - f(t1)|
| t2 - t1| \alpha
,
we have the following theorem [13].
Theorem 2. Let the functions a1(t), b1(t), a2(t), and b2(t) belong to the function class H(\alpha ),
0 < \alpha < 1, and let the condition (3.1) be satisfied. If \mathrm{I}\mathrm{n}\mathrm{d}A(t) = \kappa \geq 0 and the condition\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| a1(t)b2(t) - b1(t)a2(t)
a1(t)a2(t) - b1(t)b2(t)
X+(t)
X+(t)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
H\alpha
M\alpha < 1 (3.4)
is fulfilled for some constant M\alpha then the boundary-value problem (2.5), (2.11) is uniquely solvable.
If \mathrm{I}\mathrm{n}\mathrm{d}A(t) < 0 and if the convergence condition (3.4) and the necessary and sufficient condi-
tions (3.3) are satisfied, then the boundary-value problem (2.5) has a unique solution.
References
1. I. Caraus, The numerical solution for system of singular integro-differential equations by Faber – Laurent polynomials,
Numer. Anal. and Appl., 3401, ser. Lect. Notes Comput. Sci., 219 – 223 (2005).
2. V. D. Didenko, B. Silbermann, S. Roch, Some peculiarities of approximation methods for singular integral equations
with conjugation, Methods and Appl. Anal., 7, № 4, 663 – 686 (2000).
3. R. Duduchava, On general singular integral operators of the plane theory of elasticity, Rend. Sem. Mat. Univ.
Politec. Torino, 42, № 3, 15 – 41 (1984).
4. R. Duduchava, General singular integral equations and basic problems of planar elasticity theory, Trudy Tbiliss.
Mat. Inst. AN GSSR, 82, 45 – 89 (1986).
5. S. W. Ellacott, A survey of Faber methods in numerical approximation, Comput. and Math. Appl., 12, № 5, 1103 – 1107
(1986).
6. F. D. Gakhov, On the Riemann boundary value problem, Mat. Sb., 44, № 4, 673 – 683 (1937).
7. F. D. Gakhov, Boundary value problems, Dover Publ., Mineloa (1990).
8. A. I. Kalandiya, Mathematical methods of two-dimensional elasticity, Mir, Moscow (1975).
9. E. Ladopoulos, G. Tsamasphyros, Approximations of singular integral equations on Lyapunov contours in Banach
spaces, Comput. and Math. Appl., 50, № 3-4, 567 – 573 (2005).
10. G. S. Litvinchiuk, The boundary problems and singular equations with displacement, Nauka, Moscow (1997).
11. N. I. Muskhelishvili, Singular integral equations: boundary problems of function theory and their application to
mathematical physics, Dover Publ., Mineloa (2008).
12. N. I. Muskhelishvili, Some basic problems of the mathematical theory of elasticity, Springer Sci. & Business Media
(2013).
13. M. A. Sheshko, P. Karczmarek, D. Pylak, P. Wójcik, Application of Faber polynomials to the approximate solution
of a generalized boundary value problem of linear conjugation in the theory of analytic functions, Comput. Math.
Appl., 67, № 8, 1474 – 1481 (2014).
14. I. Spinei, V. Zolotarevschi, Direct methods for solving singular integral equations with complex conjugation, Comput.
Sci., 6, № 1, 83 – 91 (1998).
15. P. K. Suetin, Series of Faber polynomials, Gordon and Breach Sci. Publ., New York (1998).
16. I. N. Vekua, On singular linear integral equations, Dokl. Akad. Nauk SSSR, 26, № 8, 335 – 338 (1940).
17. I. N. Vekua, Generalized analytic functions, Pergamon Press, Oxford (1962).
18. N. Vekua, Systems of singular integral equations and certain boundary value problems, Nauka, Moscow (1970).
19. P. Wójcik, M. A. Sheshko, S. M. Sheshko, Application of Faber polynomials to the approximate solution of singular
integral equations with the Cauchy kernel, Different. Equat., 49, № 2, 198 – 209 (2013).
20. V. Zolotarevskii, Z. Li, I. Caraus, Approximate solution of singular integro-differential equations by reduction over
Faber – Laurent polynomials, Different. Equat., 40, № 12, 1764 – 1769 (2004).
Received 27.02.19
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
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| id | umjimathkievua-article-758 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:03:55Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a5/6f8820d9b12ada004d3438d62a4a58a5.pdf |
| spelling | umjimathkievua-article-7582025-03-31T08:46:40Z Approximate solution of a dominant singular integral equation with conjugation Approximate solution of a dominant singular integral equation with conjugation Pylak, D. Wójcik, P. Pylak, D. Wójcik, P. P. singular integral equations Cauchy kernel successive approximation singular integral equations Cauchy kernel successive approximation UDC 517.5In the present paper, the method of successive approximations and Faber polynomials are used to derive an approximate solution of a dominant singular integral equation with Holder continuous coefficients and conjugation on the Lyapunov curve. Moreover, conditions of convergence in the $L_2$ and $H(α)$ spaces are presented. &nbsp; УДК 517.5 Наближений розв’язок домiнантного сингулярного iнтегрального рiвняння зi спряженням У цій роботі за допомогою методу послідовних наближень та поліномів Фабера отримано наближений розв'язок домінантного сингулярного інтегрального рівняння з неперервними за Гельдером коефіцієнтами та спряженням на кривій Ляпунова. Крім того, запропоновано умови збіжності у просторах $L_2$ та $H(α).$ Institute of Mathematics, NAS of Ukraine 2021-09-16 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/758 10.37863/umzh.v73i9.758 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 9 (2021); 1289 - 1296 Український математичний журнал; Том 73 № 9 (2021); 1289 - 1296 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/758/9112 Copyright (c) 2021 Paweł Wójcik, d Pylak |
| spellingShingle | Pylak, D. Wójcik, P. Pylak, D. Wójcik, P. P. Approximate solution of a dominant singular integral equation with conjugation |
| title | Approximate solution of a dominant singular integral equation with conjugation |
| title_alt | Approximate solution of a dominant singular integral equation with conjugation |
| title_full | Approximate solution of a dominant singular integral equation with conjugation |
| title_fullStr | Approximate solution of a dominant singular integral equation with conjugation |
| title_full_unstemmed | Approximate solution of a dominant singular integral equation with conjugation |
| title_short | Approximate solution of a dominant singular integral equation with conjugation |
| title_sort | approximate solution of a dominant singular integral equation with conjugation |
| topic_facet | singular integral equations Cauchy kernel successive approximation singular integral equations Cauchy kernel successive approximation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/758 |
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