scholarly journals Algorithm for using the boundary element method on the example of a mixed boundary value problem for the Poisson equation

2018 ◽  
Author(s):  
L. T. Boyko
Keyword(s):  
Integral Equation ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Element ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Normal Derivative ◽  
Algebraic Equations ◽  
Boundary Contour ◽  
Linear Algebraic Equations

The possibilities of the algorithm for applying the boundary element method to solving boundary value problems are discussed on the example of the two-dimensional Poisson differential equation. The algorithm does not change significantly when the type of boundary conditions changes: the Dirichlet problem, the Neumann problem, or a mixed boundary value problem. The idea of the algorithm is taken from the work of John T. Katsikadelis [1]. The algorithm is described in detail in the next sequence of actions. 1) The boundary- value problem for a two-dimensional finite domain is formulated. The desired function in the domain, its values, and its normal derivative on the boundary contour are connected by means of the second Green formula. 2) We pass from the boundary value problem for the Poisson equation to the boundary value problem for the Laplace equation. This simplifies the process of constructing an integral equation. We obtain the integral equation on the boundary contour using the boundary conditions. 3) In the integral equation, we divide the boundary contour into a finite number of boundary elements. The desired function and its normal derivative are considered constant values on each boundary element. We compose a system of linear algebraic equations considering these values. 4) We modify the system of linear algebraic equations taking into account the boundary conditions. After that, we solve it using the Gauss method. The computer program has been developed according to the developed algorithm. We used it in the learning process. The software implementation of the algorithm takes into account the capabilities of modern computer technology and modern needs of the educational process. The work of the program is shown in the test case. Further modification of the described algorithm is possible

scholarly journals NUMERICAL SOLUTION OF THE BOUNDARY PROBLEM FOR PARABOLIC EQUATION WITH NON-SELF-ADJOINT OPERATOR AND RELATED BOUNDARY CONDITIONS

2020 ◽  
pp. 7-11
Author(s):  
B. Dovgiy ◽  
L. Vakal ◽  
E. Vakal
Keyword(s):  
Parabolic Equation ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Diffusion Processes ◽  
Boundary Value ◽  
Adjoint Operator ◽  
Second Order ◽  
Algebraic Equations ◽  
Sweep Method ◽  
Linear Algebraic Equations

A boundary value problem for a second-order parabolic equation with a non-self-adjoint operator is considered. Such problems are mathematicalmodels for a number of problems, describing convective-diffusion processes of matter transfer, breakdown mechanisms of laser activity in plasma, etc. While studying the physics of breakdown, one should take into account the avalanche-like increase in the number of free electrons due to multiphoton ionization processes under the influence of optical pulses. This requires the inclusion of related boundary conditions in the problem formulation. An important circumstance that must be taken into account when developing a method for solving the problem is fulfillment of a certain conservation law for its solution. To solve the boundary value problem an approach based on the finite difference method is proposed. The approximation of the equation and boundary conditions is constructed so that the difference scheme is completely conservative. It approximates the original problem with the second order in the spatial variable and in time, and it has the second order of convergence. To effectively solve a system of linear algebraic equations at each time layer, the sweep method for complex systems in combination with the non-monotonic sweep method for systems with a tridiagonal matrix is used. Software based on computer mathematics MATLAB is developed to perform numerical calculations. It is obtained an approximate solution of an applied problem for different instants of time, as well as values of an absorption coefficient, the change in sign of which determines the transition of the plasma in a laser-active state.

scholarly journals The nonlocal boundary value problem with perturbations of mixed boundary conditions for an elliptic equation with constant coefficients. II

2020 ◽  
Vol 12 (1) ◽  
pp. 173-188
Author(s):  
Ya.O. Baranetskij ◽  
P.I. Kalenyuk ◽  
M.I. Kopach ◽  
A.V. Solomko
Keyword(s):  
Elliptic Equation ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Nonlocal Conditions ◽  
Mixed Boundary Conditions ◽  
Multipoint Problem ◽  
Uniqueness Of The Solution

In this paper we continue to investigate the properties of the problem with nonlocal conditions, which are multipoint perturbations of mixed boundary conditions, started in the first part. In particular, we construct a generalized transform operator, which maps the solutions of the self-adjoint boundary-value problem with mixed boundary conditions to the solutions of the investigated multipoint problem. The system of root functions $V(L)$ of operator $L$ for multipoint problem is constructed. The conditions under which the system $V(L)$ is complete and minimal, and the conditions under which it is the Riesz basis are determined. In the case of an elliptic equation the conditions of existence and uniqueness of the solution for the problem are established.

scholarly journals Approximate method of solving one periodic optimal regulated boundary value problem

2019 ◽  
pp. 66-69
Author(s):  
I. Askerov
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Small Parameter ◽  
General Problem ◽  
Asymptotic Method ◽  
Nonlinear Differential Equations ◽  
Boundary Value ◽  
Algebraic Equations ◽  
Lagrange Equations ◽  
Nonlinear Algebraic Equations

In the present work we considered the solution of one periodic optimal regulated boundary value problem by the asymptotic method. For the solution of the problem with extended functional writing, boundary conditions and Euler-Lagrange equations were found. The approach to the solution of the problem depending on a small parameter by seeking a system of nonlinear differential equations and solving Euler-Lagrange equations, the solution of the general problem in the first approach comes down to solving two nonlinear algebraic equations.

scholarly journals Electrostatic potential calculation by application of walk-on-spheres method

2018 ◽  
pp. 152-160
Author(s):  
Liudmila Vladimirova ◽  
Irina Rubtsova ◽  
Nikolai Edamenko
Keyword(s):  
Integral Equation ◽  
Problem Solving ◽  
Boundary Value Problem ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Mixed Boundary Value Problem ◽  
Equation Solving ◽  
Electrostatic Potential Calculation ◽  
Walk On Spheres Algorithm ◽  
Boundary Form

The paper is devoted to mixed boundary-value problem solving for Laplace equation with the use of walk-on-spheres algorithm. The problem under study is reduced to finding a solution of integral equation with the kernel nonzero only at some sphere in the domain considered. Ulam-Neumann scheme is applied for integral equation solving; the appropriate Markov chain is introduced. The required solution value at a certain point of the domain is approximated by the expected value of special statistics defined on Markov paths. The algorithm presented guarantees the average Markov trajectory length to be finite and allows one to take into account boundary conditions on required solution derivative and to avoid Markov paths ending in the neighborhood of the boundaries where solution values are not given. The method is applied for calculation of electric potential in the injector of linear accelerator. The purpose of the work is to verify the applicability and effectiveness of walk-on-spheres method for mixed boundary-value problem solving with complicated boundary form and thus to demonstrate the suitability of Monte Carlo methods for electromagnetic fields simulation in beam forming systems. The numerical experiments performed confirm the simplicity and convenience of this method application for the problem considered.

scholarly journals A PROBLEM WITH PARAMETER FOR THE INTEGRO-DIFFERENTIAL EQUATIONS

2021 ◽  
Vol 26 (1) ◽  
pp. 34-54
Author(s):  
Elmira A. Bakirova ◽  
Anar T. Assanova ◽  
Zhazira M. Kadirbayeva
Keyword(s):  
Differential Equation ◽  
General Solution ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Cauchy Problems ◽  
Algebraic Equations ◽  
Integro Differential Equation ◽  
Loaded Differential Equation ◽  
Linear Algebraic Equations

The article proposes a numerically approximate method for solving a boundary value problem for an integro-differential equation with a parameter and considers its convergence, stability, and accuracy. The integro-differential equation with a parameter is approximated by a loaded differential equation with a parameter. A new general solution to the loaded differential equation with a parameter is introduced and its properties are described. The solvability of the boundary value problem for the loaded differential equation with a parameter is reduced to the solvability of a system of linear algebraic equations with respect to arbitrary vectors of the introduced general solution. The coefficients and the right-hand sides of the system are compiled through solutions of the Cauchy problems for ordinary differential equations. Algorithms are proposed for solving the boundary value problem for the loaded differential equation with a parameter. The relationship between the qualitative properties of the initial and approximate problems is established, and estimates of the differences between their solutions are given.

scholarly journals Vector Riemann–Hilbert problem with almost periodic and meromorphic coefficients and applications

2015 ◽  
Vol 471 (2180) ◽  
pp. 20150262 ◽  
Author(s):  
Y. A. Antipov
Keyword(s):  
Boundary Value Problem ◽  
Finite Number ◽  
Hilbert Problem ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Almost Periodic ◽  
First Case ◽  
Linear Algebraic Equations ◽  
Poles And Zeros ◽  
Riemann Hilbert Problem

The vector Riemann–Hilbert problem is analysed when the entries of its matrix coefficient are meromorphic and almost periodic functions. Three cases for the meromorphic functions, when they have (i) a finite number of poles and zeros (rational functions), (ii) periodic poles and zeros, and (iii) an infinite number of non-periodic zeros and poles, are considered. The first case is illustrated by the heat equation for a composite rod with a finite number of discontinuities and a system of convolution equations; both problems are solved explicitly. In the second case, a Wiener–Hopf factorization is found in terms of the hypergeometric functions, and the exact solution of a mixed boundary value problem for the Laplace equation in a wedge is derived. In the last case, the Riemann–Hilbert problem reduces to an infinite system of linear algebraic equations with the exponential rate of convergence. As an example, the Neumann boundary value problem for the Helmholtz equation in a strip with a slit is analysed.

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