Approximate Conformal Mappings and Elasticity Theory

Ivanshin, Pyotr N.; Shirokova, Elena A.

doi:10.1155/2016/4367205

Cited by 12 publications

(17 citation statements)

References 11 publications

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“…After separating the imaginary part of Equation 3.1 we obtain the following Fredholm integral equation:

g_{0} (t) = - \frac{1}{normalπ} \int_{0}^{2 π} f_{0} false(normalτ false) {(\log [z false(normalτ false) - z false(t false)])}_{τ}^{'} d normalτ + \frac{1}{normalπ} \int_{0}^{2 π} g_{0} false(normalτ false) {(\arg [z false(normalτ false) - z false(t false)])}_{τ}^{'} d normalτ .

We consider the factor

(e^{i τ} - e^{i t})

in the expression

(z (τ) - z (t))

in order to separate the improper PV integral in the Fredholm equation of the second kind as follows :

\begin{matrix} left \\ z (τ) - z (t) = \sum_{k = - m}^{m} c_{k} true(e^{i k τ} - e^{i k t} true) \\ = true(e^{i τ} - e^{i t} true) true[\sum_{k = 1}^{m} c_{k} e^{i k t} \end{matrix}

…”

Section: Cauchy Integral Methods For the Solution Of The Dirichlet Promentioning

confidence: 99%

“…The solvability of Equation 3.5 was proved in with the help of the following Lemma which can be formulated in the following form:Lemma Let

Y (t)

2 π

periodic with the bounded second derivative (i.e .,

| Y^{''} (t) | < T

) and let

G (τ, t)

2 π

periodic with respect to both variables. Assume that there exist numbers

j, p > 1

and a constant

U > 0

such that

false| \frac{\partial^{j + p} G (τ, t)}{\partial t^{j} \partial {normalτ}^{p}} false| \leq U .

Then, the approximate solution of the uniquely resolvable Fredholm integral equation of the second kind

X (t) = \int_{0}^{2 π} G false(normalτ, t false) X false(normalτ false) d normalτ + Y false(t false),

can be reduced to solution of finite linear system with error estimated by

O true(1 / N^{2} true)

where

N

is the finite linear system rank .…”

Section: Cauchy Integral Methods For the Solution Of The Dirichlet Promentioning

confidence: 99%

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A Cauchy integral method of the solution of the 2D Dirichlet problem for simply or doubly connected domains

Shirokova

El-Shenawy

2018

Numerical Methods Partial

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“…After separating the imaginary part of Equation 3.1 we obtain the following Fredholm integral equation:

g_{0} (t) = - \frac{1}{normalπ} \int_{0}^{2 π} f_{0} false(normalτ false) {(\log [z false(normalτ false) - z false(t false)])}_{τ}^{'} d normalτ + \frac{1}{normalπ} \int_{0}^{2 π} g_{0} false(normalτ false) {(\arg [z false(normalτ false) - z false(t false)])}_{τ}^{'} d normalτ .

We consider the factor

(e^{i τ} - e^{i t})

in the expression

(z (τ) - z (t))

in order to separate the improper PV integral in the Fredholm equation of the second kind as follows :

\begin{matrix} left \\ z (τ) - z (t) = \sum_{k = - m}^{m} c_{k} true(e^{i k τ} - e^{i k t} true) \\ = true(e^{i τ} - e^{i t} true) true[\sum_{k = 1}^{m} c_{k} e^{i k t} \end{matrix}

…”

Section: Cauchy Integral Methods For the Solution Of The Dirichlet Promentioning

confidence: 99%

“…The solvability of Equation 3.5 was proved in with the help of the following Lemma which can be formulated in the following form:Lemma Let

Y (t)

2 π

periodic with the bounded second derivative (i.e .,

| Y^{''} (t) | < T

) and let

G (τ, t)

2 π

periodic with respect to both variables. Assume that there exist numbers

j, p > 1

and a constant

U > 0

such that

false| \frac{\partial^{j + p} G (τ, t)}{\partial t^{j} \partial {normalτ}^{p}} false| \leq U .

Then, the approximate solution of the uniquely resolvable Fredholm integral equation of the second kind

X (t) = \int_{0}^{2 π} G false(normalτ, t false) X false(normalτ false) d normalτ + Y false(t false),

can be reduced to solution of finite linear system with error estimated by

O true(1 / N^{2} true)

where

N

is the finite linear system rank .…”

Section: Cauchy Integral Methods For the Solution Of The Dirichlet Promentioning

confidence: 99%

A Cauchy integral method of the solution of the 2D Dirichlet problem for simply or doubly connected domains

Shirokova

El-Shenawy

2018

Numerical Methods Partial

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“…We consider the factor ( − ) in the expression ( ( ) − ( )), = 0, 1 in order to separate the improper PV integral in the Fredholm equation of the second kind as follows [8,9]:…”

Section: Introductionmentioning

confidence: 99%

“…The solvability of (11) is proved in [8] where the Fourier series solution form of the functions s ( ), s = 0, 1, leads us to an infinite linear system of equations which can be reduced to a finite one according to the following lemma.…”

Section: Introductionmentioning

confidence: 99%

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The Approximate Solution of 2D Dirichlet Problem in Doubly Connected Domains

El-Shenawy

Shirokova

2018

Advances in Mathematical Physics

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We propose a new method for constructing an approximate solution of the two-dimensional Laplace equation in an arbitrary doubly connected domain with smooth boundaries for Dirichlet boundary conditions. Using the fact that the solution of the Dirichlet problem in a doubly connected domain is represented as the sum of a solution of the Schwarz problem and a logarithmic function, we reduce the solution of the Schwartz problem to the Fredholm integral equation with respect to the boundary value of the conjugate harmonic function. The solution of the integral equation in its turn is reduced to solving a linear system with respect to the Fourier coefficients of the truncated expansion of the boundary value of the conjugate harmonic function. The unknown coefficient of the logarithmic component of the solution of the Dirichlet problem is determined from the following fact. The Cauchy integral over the boundary of the domain with a density that is the boundary value of the analytical in this domain function vanishes at points outside the domain. The resulting solution of the Dirichlet problem is the sum of the real part of the Cauchy integral in the given domain and the logarithmic function. In order to avoid singularities of the Cauchy integral at points near the boundary, the solution at these points is replaced by a linear function. The resulting numerical solution is continuous in the domain up to the boundaries. Three examples of the solution of the Dirichlet problem are given: one example demonstrates the solution with constant boundary conditions in the domain with a complicated boundary; the other examples provide a comparison of the approximate solution with the known exact solution in a noncircular domain.

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Solution the Dirichlet Problem for Multiply Connected Domain Using Numerical Conformal Mapping

Abzalilov

Ivanshin

Shirokova

2019

Complex Anal. Oper. Theory

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Approximate Conformal Mappings and Elasticity Theory

Abstract: Here, we present the new method of approximate conformal mapping of the unit disk to a one-connected domain with smooth boundary without auxiliary constructions and iterations. The mapping function is a Taylor polynomial. The method is applicable to elasticity problems solution.

Cited by 12 publications

References 11 publications

A Cauchy integral method of the solution of the 2D Dirichlet problem for simply or doubly connected domains

A Cauchy integral method of the solution of the 2D Dirichlet problem for simply or doubly connected domains

The Approximate Solution of 2D Dirichlet Problem in Doubly Connected Domains

Solution the Dirichlet Problem for Multiply Connected Domain Using Numerical Conformal Mapping

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