Application of Faber polynomials to approximate solution of the Riemann problem
In the paper, Faber polynomials are used to derive an approximate solution of the Riemann problem on a Lyapunov curve. Moreover, an estimation of the error of the approximated solution is presented and proved.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507853192691712 |
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| author | Pylak, D. Sheshko, M. A. Wójcik, P. Пилак, Д. Шешко, М. А. Войчик, П. |
| author_facet | Pylak, D. Sheshko, M. A. Wójcik, P. Пилак, Д. Шешко, М. А. Войчик, П. |
| author_sort | Pylak, D. |
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| datestamp_date | 2019-12-05T09:32:42Z |
| description | In the paper, Faber polynomials are used to derive an approximate solution of the Riemann problem on a Lyapunov curve.
Moreover, an estimation of the error of the approximated solution is presented and proved. |
| first_indexed | 2026-03-24T02:15:54Z |
| format | Article |
| fulltext |
UDC 517.5
M. A. Sheshko, D. Pylak, P. Wójcik (John Paul II Catholic Univ. Lublin, Poland)
APPLICATION OF FABER POLYNOMIALS TO APPROXIMATE SOLUTION
OF THE RIEMANN PROBLEM
ЗАСТОСУВАННЯ ПОЛIНОМIВ ФАБЕРА ДО НАБЛИЖЕНОГО РОЗВ’ЯЗАННЯ
ПРОБЛЕМИ РIМАНА
In the paper, Faber polynomials are used to derive an approximate solution of the Riemann problem on a Lyapunov curve.
Moreover, an estimation of the error of the approximated solution is presented and proved.
Полiноми Фабера застосовано для отримання наближеного розв’язку проблеми Рiмана на кривiй Ляпунова. Наве-
дено i обґрунтовано оцiнку похибки цього наближеного розв’язку.
1. Introduction. Let L be a closed Lyapunov curve on the complex plane and G(t) \not = 0 and
g(t) be given functions of Hölder continuous class H(\mu ), 0 < \mu \leq 1, defined on L. The Riemann
boundary-value problem for analytic functions consists in finding a pair of functions F+(z), z \in D+,
and F - (z), z \in D - , analytic on the inside (D+) and outside (D - ) of the curve L, respectively,
such that the following condition is fulfilled
F+(t) = G(t)F - (t) + g(t), F - (\infty ) = 0, t \in L. (1)
Let us recall that a simple continuous curve is called Lyapunov curve if it satisfies the following
conditions:
(i) at every point of L there exists a well-defined tangent,
(ii) 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
| \theta (s2) - \theta (s1)| \leq k| s1 - s2| \alpha , 0 < \alpha \leq 1.
The Riemann problem (1) has numerous applications [8, 22, 6, 21]. The main arise in the theory
of singular integral equations. The homogeneous Riemann problem
\bigl(
g(t) \equiv 0
\bigr)
was first considered
by Hilbert [11], and the nonhomogeneous problem (1) by Privalov [23]. They reduced it to the
problem of solving integral equations. Next, Gakhov in the monograph [7] presented an effective
solution of (1) in terms of Cauchy type integrals.
We will recall this solution. Let \varkappa = \mathrm{I}\mathrm{n}\mathrm{d}G(t) \geq 0, then the solution has the following
form [8, 22]:
F\pm (z) = X\pm (z)
\bigl(
\Psi \pm (z) + P\varkappa - 1(z)
\bigr)
, (2)
where
X+(z) = \mathrm{e}\mathrm{x}\mathrm{p}\Gamma +(z), z \in D+, X - (z) = z - \varkappa \mathrm{e}\mathrm{x}\mathrm{p}\Gamma - (z), z \in D - , (3)
P\varkappa - 1(z) = \gamma 0 + \gamma 1z + . . .+ \gamma \varkappa - 1z
\varkappa - 1, and \gamma 0, \gamma 1, . . . , \gamma \varkappa - 1 are arbitrary constants. Here
c\bigcirc M. A. SHESHKO, D. PYLAK, P. WÓJCIK, 2016
1696 ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 12
APPLICATION OF FABER POLYNOMIALS TO APPROXIMATE SOLUTION OF THE RIEMANN PROBLEM 1697
\Gamma \pm (z) =
1
2\pi i
\int
L
\mathrm{l}\mathrm{n}
\bigl(
\tau - \varkappa G(\tau )
\bigr)
\tau - z
d\tau , z \in D+, (4)
and
\Psi \pm (z) =
1
2\pi i
\int
L
g(\tau )
X+(\tau )
d\tau
\tau - z
, z \in D+. (5)
If \varkappa < 0 then the solution F\pm (z), given by the formula (2) with P\varkappa - 1(z) \equiv 0, exists if and only
if the following conditions hold:\int
L
g(\tau )
X+(\tau )
\tau j - 1 d\tau = 0, j = 1, 2, . . . , | \varkappa | . (6)
Over the last decades the Riemann problem has been intensively investigated. Many generaliza-
tions and modifications can be found in the literature [6, 20]. One of the most famous problem is the
nonlinear conjugation problem of power type [3, 21]. Many research has been done under various as-
sumptions about the curve and coefficients [13, 14]. However, even in the classical Riemann problem
the Cauchy-type integrals occurring in (4), (5) have very complicated forms. Their exact values can
be calculated only in special cases. Therefore, to solve the problem (1), we apply the approximate
methods [2]. Well-known numerical solutions of the problem (1) have been constructed for the case
of zero index and the case when the contour is a circle or a segment of a real line [24, 25].
In this paper Faber polynomials are used to derive an approximate solution of the boundary
problem (1) on any Lyapunov curve in the case of an arbitrary index. Moreover, the convergence of
the approximate solution is proved and the rate of convergence is established.
Faber polynomials and their numerous modifications are a very useful tool in modern investigation
of analytic functions [18, 25] and approximation theory. One can find their applications to a numerical
solution of the Dirichlet problem in the plane [4], approximate solutions of singular integral equations
[17, 26] and many other numerical methods for analytic functions (see monograph by P. K. Suetin [25]
and papers [1, 5, 9, 10, 12, 15, 16, 19]).
2. Approximate solution. If \varkappa > 0 then the right-hand side of the formula (2) contains \varkappa
arbitrary constants. Therefore, we have to find the conditions for the uniqueness of the solution. In
our opinion, the most convenient conditions are the following:
- \mathrm{R}\mathrm{e}\mathrm{s}
z=\infty
\bigl(
F - (z)zj - 1
\bigr)
= Aj , j = 1, 2, . . . ,\varkappa , (7)
where Aj , j = 1, 2, . . . ,\varkappa , are given numbers.
By (2), taking into account the Laurent expansions of X - (z) and \Psi - (z) about the point z = \infty ,
i.e.,
X - (z) =
1
z\varkappa
\Bigl(
1 +
p1
z
+
p1
z2
+ . . .
\Bigr)
,
\Psi - (z) =
h1
z
+
h2
z2
+ . . . ,
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 12
1698 M. A. SHESHKO, D. PYLAK, P. WÓJCIK
from (7) we derive the system of linear equations
A1 = \gamma \varkappa - 1,
A2 = p1\gamma \varkappa - 1 + \gamma \varkappa - 2,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A\varkappa = p\varkappa - 1\gamma \varkappa - 1 + p1\gamma 1 + . . .+ \gamma 0.
It enables us to compute the unknown coefficients \gamma 0, \gamma 1, . . . , \gamma \varkappa - 1.
In what follows, it will be convenient to have an expansion in Faber series of the function F\pm (z)
defined by (2). For this purpose, we first find the expansion of the function \Gamma \pm (z) . By Theorem 6
from [25, p. 192] we obtain
\Gamma +(z) =
\infty \sum
k=0
ak\Phi k(z), z \in D+, (8)
where
ak =
1
2\pi i
\int
L
\mathrm{l}\mathrm{n}
\bigl(
\tau - \varkappa G(\tau )
\bigr)
\Phi k+1(\tau )
\Phi \prime (\tau ) d\tau =
1
2\pi i
\int
| t| =1
\mathrm{l}\mathrm{n}
\bigl[
(\varphi (t)) - \varkappa G(\varphi (t))
\bigr]
tk+1
d\tau ,
and
\Gamma - (z) =
\infty \sum
k=1
bk
\Phi k(z)
, z \in D - , (9)
where
bk =
1
2\pi i
\int
L
\mathrm{l}\mathrm{n}
\bigl(
\tau - \varkappa G(\tau )
\bigr)
\Phi k - 1(\tau )\Phi \prime (\tau ) d\tau =
1
2\pi i
\int
| t| =1
\mathrm{l}\mathrm{n}
\bigl[
\varphi - \varkappa (t)G(\varphi (t))
\bigr]
tk - 1 d\tau .
The function w = \Phi (z) is a Riemann mapping, i.e., a conformal and univalent mapping from D -
of \BbbC z onto the exterior of the unit circle | w| > 1 of \BbbC w, while \varphi (t) is a limit value of the inverse
function z = \varphi (w), and \Phi k(z), k = 0, 1, . . . , are the Faber polynomials on the area D+.
Similarly as in the previous case, we have the following expansions:
\Psi +(z) =
\infty \sum
k=0
ck\Phi k(z), z \in D+, (10)
\Psi - (z) =
\infty \sum
k=1
dk
\Phi k(z)
, z \in D - , (11)
where
ck =
1
2\pi i
\int
| t| =1
g
\bigl(
\varphi (t)
\bigr)
X+
\bigl(
\varphi (t)
\bigr) d\tau
tk+1
,
dk =
1
2\pi i
\int
| t| =1
g
\bigl(
\varphi (t)
\bigr)
X+
\bigl(
\varphi (t)
\bigr) tk - 1d\tau .
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 12
APPLICATION OF FABER POLYNOMIALS TO APPROXIMATE SOLUTION OF THE RIEMANN PROBLEM 1699
Thus, in the case of nonnegative index \varkappa \geq 0, the desired functions F\pm (z) can be expressed as
F+(z) = \mathrm{e}\mathrm{x}\mathrm{p}
\Biggl( \infty \sum
k=0
ak\Phi k(z)
\Biggr) \Biggl( \infty \sum
k=0
ck\Phi k(z) + P\varkappa - 1(z)
\Biggr)
, z \in D+,
F - (z) = z - \varkappa \mathrm{e}\mathrm{x}\mathrm{p}
\Biggl( \infty \sum
k=1
bk
\Phi k(z)
\Biggr) \Biggl( \infty \sum
k=1
dk
\Phi k(z)
+ P\varkappa - 1(z)
\Biggr)
, z \in D - .
As the approximate solution of the problem (1), (7), we take the function
F\pm
n (z) = X\pm
n (z)
\bigl(
\Psi \pm
n + P\varkappa - 1(z)
\bigr)
, z \in D\pm , (12)
where
X+
n (z) = \mathrm{e}\mathrm{x}\mathrm{p}\Gamma +
n (z), X -
n (z) = z - \varkappa \mathrm{e}\mathrm{x}\mathrm{p}\Gamma -
n (z),
\Gamma +
n (z) =
n\sum
k=0
ak\Phi k(z), \Gamma -
n (z) =
n\sum
k=1
bk
\Phi k(z)
,
(13)
and
\Psi +
n (z) =
n\sum
k=0
c\ast k\Phi k(z), \Psi -
n (z) =
n\sum
k=1
d\ast k
\Phi k(z)
,
c\ast k =
1
2\pi i
\int
| t| =1
g (\varphi (t))
X+
n (\varphi (t))
d\tau
tk+1
, d\ast k =
1
2\pi i
\int
| t| =1
g (\varphi (t))
X+
n (\varphi (t))
tk - 1d\tau .
If \varkappa < 0 then an approximate solution of the problem (1) can be found from the boundary condition
F+
n (t) = Gn(t)F
-
n (t) + g(t) +X+
n (t)
\Bigl( q1
t
+
q2
t2
+ . . .+
q| \varkappa |
t| \varkappa |
\Bigr)
, t \in L, (14)
where
Gn(t) = X+
n (t)
\bigl(
X -
n (t)
\bigr) - 1
,
and the functions X\pm
n (t) are defined by (13). The coefficients q1, q2, . . . , q| \varkappa | should be chosen in
such a way that the solvability conditions for Eq. (14) are satisfied. These conditions, by (6), have
the form
1
2\pi i
\int
L
\Bigl( q1
\tau
+
q2
\tau 2
+ . . .+
q| \varkappa |
\tau | \varkappa |
\Bigr)
\tau j - 1 d\tau = - 1
2\pi i
\int
L
g(\tau )
X+
n (\tau )
\tau j - 1 d\tau , j = 1, 2, . . . , | \varkappa | .
Hence, we obtain
qj = - 1
2\pi i
\int
L
g(\tau )
X+
n (\tau )
\tau j - 1 d\tau , j = 1, 2, . . . , | \varkappa | .
In accordance with (2), the solution of the problem (14) has the following structure:
F\pm
n (z) = X\pm
n (z)\Psi \pm
n (z), z \in D\pm , (15)
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 12
1700 M. A. SHESHKO, D. PYLAK, P. WÓJCIK
where X\pm
n (t) are defined by (13), and
\Psi +
n (z) =
n\sum
k=0
\alpha k\Phi k(z), \Psi -
n (z) =
n\sum
k=1
\beta k
\Phi k(z)
,
\alpha k =
1
2\pi i
\int
| t| =1
Hn(t)
dt
tk+1
, \beta k =
1
2\pi i
\int
| t| =1
Hn(t)t
k - 1 dt,
and
Hn(t) =
g
\bigl(
\varphi (t)
\bigr)
X+
n
\bigl(
\varphi (t)
\bigr) + q1
\varphi (t)
+
q2\bigl(
\varphi (t)
\bigr) 2 + . . .+
q| \varkappa | \bigl(
\varphi (t)
\bigr) | \varkappa | .
3. Estimation of errors. Now we will provide the error estimations of the approximate solutions
obtained above. Let the function G\ast (t) = \mathrm{l}\mathrm{n}(t - \varkappa G(t)), t \in L, be continuously differentiable up to
the order r and the rth derivative fulfill the Hölder inequality with the constant 0 < \mu < 1. Then we
say that the function G\ast (t) belongs to the class W rH\mu . According to [25, p. 262] we have\bigm| \bigm| \bigm| \bigm| \bigm| \Gamma +(z) -
n\sum
k=0
ak\Phi k(z)
\bigm| \bigm| \bigm| \bigm| \bigm| \leq K1
\mathrm{l}\mathrm{n}(n)
nr+\mu
, z \in D+, (16)
\bigm| \bigm| \bigm| \bigm| \bigm| \Gamma - (z) -
n\sum
k=1
bk
\Phi k(z)
\bigm| \bigm| \bigm| \bigm| \bigm| \leq K2
\mathrm{l}\mathrm{n}(n)
nr+\mu
, z \in D - , (17)
where Kj are constants independent of n. Taking into account the inequality
| 1 - ez| \leq (e - 1) | z| , | z| \leq 1,
and using the maximum modulus principle, we obtain
\bigm| \bigm| X\pm (z) - X\pm
n (z)
\bigm| \bigm| \leq \mathrm{m}\mathrm{a}\mathrm{x}
t\in L
\bigm| \bigm| X\pm (z) - X\pm
n (z)
\bigm| \bigm| \leq K3
\mathrm{l}\mathrm{n}n
nr+\mu
, z \in D\pm .
Similarly, we get the estimations for (16) and (17), i.e.,\bigm| \bigm| \bigm| \bigm| \bigm| \Psi +(z) -
n\sum
k=0
ck\Phi k(z)
\bigm| \bigm| \bigm| \bigm| \bigm| \leq K4
\mathrm{l}\mathrm{n}n
nr+\mu
, z \in D+,
\bigm| \bigm| \bigm| \bigm| \bigm| \Psi - (z) -
n\sum
k=1
dk
\Phi k(z)
\bigm| \bigm| \bigm| \bigm| \bigm| \leq K5
\mathrm{l}\mathrm{n}n
nr+\mu
, z \in D - ,
whenever g(t) \in W rH\mu . To estimate the modulus
\bigm| \bigm| \Psi \pm (z) - \Psi \pm
n (z)
\bigm| \bigm| , we first estimate the difference
n\sum
k=0
ck\Phi k(z) -
n\sum
k=0
c\ast k\Phi k(z).
By [25, p. 155], we have
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 12
APPLICATION OF FABER POLYNOMIALS TO APPROXIMATE SOLUTION OF THE RIEMANN PROBLEM 1701\bigm| \bigm| \bigm| \bigm| \bigm|
n\sum
k=0
ck\Phi k(z) -
n\sum
k=0
c\ast k\Phi k(z)
\bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq 1
2\pi
\int
L
\bigm| \bigm| \bigm| \bigm| \bigm| g(\tau )X+
n (\tau ) - X+(\tau )
X+(\tau )X+
n (\tau )
n\sum
k=0
\Phi k(z)
\Phi k+1(\tau )
\Phi \prime (\tau )
\bigm| \bigm| \bigm| \bigm| \bigm| | d\tau | \leq
\leq K6
\mathrm{l}\mathrm{n}n
nr+\mu
\int
L
\bigm| \bigm| \bigm| \bigm| \bigm|
n\sum
k=0
\Phi k(z)
\Phi k+1(\tau )
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \Phi \prime (\tau )
\bigm| \bigm| | d\tau | \leq K7
\mathrm{l}\mathrm{n}2 n
nr+\mu
, z \in L.
From the above, we obtain\bigm| \bigm| \bigm| \bigm| \bigm|
n\sum
k=1
dk
\Phi k(z)
-
n\sum
k=1
d\ast k
\Phi k(z)
\bigm| \bigm| \bigm| \bigm| \bigm| \leq K8
\mathrm{l}\mathrm{n}2 n
nr+\mu
, z \in L.
Hence, we get
\bigm| \bigm| \Psi +(z) - \Psi +
n (z)
\bigm| \bigm| = \bigm| \bigm| \bigm| \bigm| \bigm|
\infty \sum
k=0
ck\Phi k(z) -
n\sum
k=0
c\ast k\Phi k(z)
\bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
\bigm| \bigm| \bigm| \bigm| \bigm|
\infty \sum
k=0
ck\Phi k(z) -
n\sum
k=0
ck\Phi k(z)
\bigm| \bigm| \bigm| \bigm| \bigm| +
\bigm| \bigm| \bigm| \bigm| \bigm|
n\sum
k=0
ck\Phi k(z) -
n\sum
k=0
c\ast k\Phi k(z)
\bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq K9
\mathrm{l}\mathrm{n}2 n
nr+\mu
, z \in D+,
and
\bigm| \bigm| \Psi - (z) - \Psi -
n (z)
\bigm| \bigm| = \bigm| \bigm| \bigm| \bigm| \bigm|
\infty \sum
k=1
dk
\Phi k(z)
-
n\sum
k=1
d\ast k
\Phi k(z)
\bigm| \bigm| \bigm| \bigm| \bigm| \leq K10
\mathrm{l}\mathrm{n}2 n
nr+\mu
, z \in D - .
Now we can estimate the modulus
\bigm| \bigm| F\pm (z) - F\pm
n (z)
\bigm| \bigm| . We obtain\bigm| \bigm| F\pm (z) - F\pm
n (z)
\bigm| \bigm| =
=
\bigm| \bigm| X\pm (z)
\bigl(
\Psi \pm (z) + P\varkappa - 1(z)
\bigr)
- X\pm
n (z)
\bigl(
\Psi \pm
n (z) + P\varkappa - 1(z)
\bigr) \bigm| \bigm| =
=
\bigm| \bigm| \bigl( X\pm (z)\Psi \pm (z) - X\pm
n (z)\Psi \pm (z)
\bigr)
+
\bigl(
X\pm
n (z)\Psi \pm (z) - X\pm
n (z)\Psi \pm
n (z)
\bigr)
+
+
\bigl(
X\pm (z) - X\pm
n (z)
\bigr)
P\varkappa - 1(z)
\bigm| \bigm| \leq K11
\mathrm{l}\mathrm{n}2 n
nr+\mu
, z \in D\pm . (18)
Remark 1. If \varkappa < 0 then the estimation (18) holds as well.
We have thereby proved the following theorem.
Theorem 1. Let the functions G\ast (t) = \mathrm{l}\mathrm{n}(t - \varkappa G(t)) and g(t) appearing in (1) belong to the class
W rH\mu , r \geq 0, 0 < \mu < 1, and let F\pm (z), F\pm
n (z) denote the exact and approximate solutions (2)
and (12), respectively. Then the estimation (18) holds.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 12
1702 M. A. SHESHKO, D. PYLAK, P. WÓJCIK
4. Numerical experiment. Let L be an ellipse with foci \pm 1 and semiaxes a =
5
4
and b =
3
4
.
We find the exact and approximate solutions of the following Riemann problem:
F+(t) =
t
t2 - 1
F - (t) +
t3 - t2 + 1
t(t - 1)
, t \in L.
Here G(t) =
t
t2 - 1
, g(t) =
t3 - t2 + 1
t(t - 1)
. Moreover, the ellipse L can be expressed by the
equation t =
1
2
\biggl(
2ei\theta +
1
2
e - i\theta
\biggr)
, 0 \leq \theta \leq 2\pi .
Functions \Phi (z) and \varphi (w) have the forms
w = \Phi (z) =
1
2
\Bigl(
z +
\sqrt{}
z2 - 1
\Bigr)
,
z = \varphi (w) =
1
2
\biggl(
2w +
1
2w
\biggr)
.
Furthermore, in this case Faber polynomials have the form
\Phi n(z) =
2
Rn
Tn(z), n = 1, 2, . . . ,
where Tn(x), - 1 \leq x \leq 1 are the Chebyshev polynomials of the first kind.
It can be easily obtained that the index \varkappa = - 1 and the conditions of solvability (6) are satisfied.
Applying the formulae (3) – (5) we have
\Gamma +(z) = 0, z \in D+, \Gamma - (z) = - \mathrm{l}\mathrm{n}
z2
z2 - 1
, z \in D - ,
X+(z) = 1, z \in D+, X - (z) = z - 1
z
, z \in D - ,
\Psi +(z) = z, z \in D+, \Psi - (z) = - 1
z(z - 1)
, z \in D - .
Finally from (2), with P\varkappa - 1(z) \equiv 0, we obtain
F+(z) = z, z \in D+, F - (z) = - z + 1
z2
, z \in D - .
To find the approximate solution, we determine the functions \Gamma +
n (z), \Gamma
-
n (z) and \Psi +
n (z), \Psi
-
n (z) as
follows:
\Gamma +
n (z) = 0, z \in D+, \Gamma -
n (z) =
Ent
n+2
4\sum
k=1
- 4
(2k - 1)(z +
\surd
z2 - 1)4k - 2
, z \in D - ,
\Psi +(z) = z, z \in D+, \Psi - (z) =
n\sum
k=1
ik - 1(1 - ( - 1)k) - 2k
(z +
\surd
z2 - 1)k
, z \in D - .
The approximate solution F\pm
n (z) can be obtained from (15).
The exact and approximate values of the function F - (z) for chosen points in D - with n = 20,
are presented in Table 1.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 12
APPLICATION OF FABER POLYNOMIALS TO APPROXIMATE SOLUTION OF THE RIEMANN PROBLEM 1703
Table 1
t = 2.00 F - (t) = - 0.7500000000
F -
n (t) = - 0.7499999999
t = 1.62 + 0.71i F - (t) = - 0.7378050367 + 0.4615476936i
F -
n (t) = - 0.7378050374 + 0.4615476937i
t = 0.618 + 1.141i F - (t) = - 0.0424777057 + 1.1747032784i
F -
n (t) = - 0.0424777052 + 1.1747033045i
t = - 0.618 + 1.141i F - (t) = 0.6913297244 + 0.1803529499i
F -
n (t) = 0.6913297119 + 0.1803529346i
t = - 1.62 + 0.71i F - (t) = 0.3008804422 - 0.0087581894i
F -
n (t) = 0.3008804418 - 0.0087581893i
t = - 2.00 F - (t) = 0.2500000000
F -
n (t) = 0.2500000000
t = - 1.62 - 0.71i F - (t) = 0.3008804422 + 0.0087581894i
F -
n (t) = 0.3008804418 + 0.0087581893i
t = - 0.618 - 1.141i F - (t) = 0.6913297244 - 0.1803529499i
F -
n (t) = 0.6913297119 - 0.1803529346i
t = 0.618 - 1.141i F - (t) = - 0.0424777057 - 1.1747032784i
F -
n (t) = - 0.0424777052 - 1.1747033045i
t = 1.62 - 0.71i F - (t) = - 0.7378050367 - 0.4615476936i
F -
n (t) = - 0.7378050374 - 0.4615476937i
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Received 27.02.15
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 12
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| id | umjimathkievua-article-1954 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:15:54Z |
| publishDate | 2016 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/d1/d0b84d7a543381295b488759fab873d1.pdf |
| spelling | umjimathkievua-article-19542019-12-05T09:32:42Z Application of Faber polynomials to approximate solution of the Riemann problem Застосування полiномiв Фабера до наближеного розв’язання проблеми Рiмана Pylak, D. Sheshko, M. A. Wójcik, P. Пилак, Д. Шешко, М. А. Войчик, П. In the paper, Faber polynomials are used to derive an approximate solution of the Riemann problem on a Lyapunov curve. Moreover, an estimation of the error of the approximated solution is presented and proved. Полiноми Фабера застосовано для отримання наближеного розв’язку проблеми Рiмана на кривiй Ляпунова. Наведено i обґрунтовано оцiнку похибки цього наближеного розв’язку. Institute of Mathematics, NAS of Ukraine 2016-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1954 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 12 (2016); 1696-1704 Український математичний журнал; Том 68 № 12 (2016); 1696-1704 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1954/936 Copyright (c) 2016 Pylak D.; Sheshko M. A.; Wójcik P. |
| spellingShingle | Pylak, D. Sheshko, M. A. Wójcik, P. Пилак, Д. Шешко, М. А. Войчик, П. Application of Faber polynomials to approximate solution of the Riemann problem |
| title | Application of Faber polynomials to approximate
solution of the Riemann problem |
| title_alt | Застосування полiномiв Фабера до наближеного розв’язання проблеми Рiмана |
| title_full | Application of Faber polynomials to approximate
solution of the Riemann problem |
| title_fullStr | Application of Faber polynomials to approximate
solution of the Riemann problem |
| title_full_unstemmed | Application of Faber polynomials to approximate
solution of the Riemann problem |
| title_short | Application of Faber polynomials to approximate
solution of the Riemann problem |
| title_sort | application of faber polynomials to approximate
solution of the riemann problem |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1954 |
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