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.
Here we introduce the Cauchy integral method for the solution of the Dirichlet problem in doubly connected domains. The proposed method gives an analytical approximate solution to this problem. This analytical solution is differentiable at the interior points and can be calculated at any point in the domain of solution and this is the main advantage of the Cauchy integral method. The method can be applied to arbitrary simply and multiply connected domains with smooth boundaries. Furthermore, it is applicable for the domains with piecewise smooth boundary curves which can be approximated by Fourier polynomial. This technic can be extended to apply the method to Poisson equation, biharmonic equation, and some other types of problems with different types of boundary conditions (Neumann and mixed). It can be applied for solving Riemann-Hilbert problem for analytical functions. The base of these problems is the Dirichlet problem. Here we introduce the solution of 2D Laplace equation with Dirichlet conditions in order to declare the basic idea of the method. In Section
The mathematical theory regarding Laplace’s equation is often referred to as the potential theory, given the significance the equation holds for describing physical phenomena such as gravitational and electrical potentials. Laplace equation with Dirichlet boundary conditions arises in different areas such as electrostatics (where it describes the electrostatic potential in a charge-free region), gravitation (where it describes the gravitational potential in free space), steady state flow of inviscid fluids, and steady state heat conduction. Many authors (e.g., [
The numerical solution of two-dimensional Laplace equation with Dirichlet boundary conditions in doubly connected domain has been introduced by many authors; for example, the complex variable boundary element methods has been presented in [
The proposed method constructs the approximate Cauchy integral solution of 2D Dirichlet problem in doubly connected domains. The method is based on the reduction of the problem to the Fredholm integral equation of the second kind for the boundary values of the conjugate harmonic function. The singularity of the obtained integral equation is overcome by using the Hilbert formula. The solution of the resulting integral equation is reduced to the solution of a truncated linear system by using the truncated Fourier series. Finally, the solution of Dirichlet problem has the form of the real part of the Cauchy integral. The solution at the points near the boundaries is approximated by linear functions.
Let
Consider a doubly connected domain
Assume that
The solvability of (
Let the numbers
According to the previous lemma and [
The solution of the linear system of (
We can satisfy (
Here we introduce a new technic to improve the approximate solution at the points near boundaries. The technic depends on the approximation of the harmonic function solution of Dirichlet problem at these points by linear functions.
Let us define the reference curves:
Define
Firstly, for every point
Secondly, the approximate solution of 2D Dirichlet problem is calculated at this point by substituting in the following linear equation:
The Cauchy integral method was applied to the 2D Dirichlet problem and highly accurate results for regular and irregular doubly connected domains with smooth boundaries are obtained. Numerical examples are presented to verify the accuracy of the proposed method in the earlier sections.
Let us define the doubly connected domain with nonstarlike boundaries as in [
The method was applied and we use a constant boundary values which equal
The contour plot of the approximate solution with constant boundary conditions in Example
Consider the doubly connected domain defined in [
where
The domain in Example
The boundary conditions are derived from the closed form of the exact solution
The absolute error along circle with radius
Consider the doubly connected domain shown in Figure
where
Domain in Example
Figure
The absolute error along a circle with radius 0.6 in Example
The Cauchy integral method gives highly accurate results for the solution of 2D Dirichlet problem for irregular doubly connected domains. The proposed algorithm improves the approximate solution at the points near the boundaries. The method is applicable for domains bounded by any smooth curve approximated by Fourier polynomial. Numerical experiments are given to verify the efficiency of the method.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
This work is performed according to the Russian Government Program of Competitive Growth of Kazan Federal University.