Numerical analysis of the influence of the initial parameters on the convergence rate of the approximate solution of the boundary value problem

In this article the methods for obtaining the approximate solution of usual and generalized Hilbert boundary value problems are proposed. The method of solution of usual Hilbert boundary value problem is based on the theorem about the representation of the kernel of the corresponding integral equation by τ = t and on the earlier proposed method for the computation of the Cauchy‐type integrals. The method for approximate solution of the generalized boundary value problem is based on the method for computation of singular integral of the formproposed by the author. All methods are implemented with the Mathcad and Maple.

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On a method for approximate solution of a mixed boundary value problem for an elliptic equation

Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva ◽

10.15507/2079-6900.23.202101.58-71 ◽

2021 ◽

Vol 23 (1) ◽

pp. 58-71

Author(s):

Mahmut E. Fairuzov ◽

Fedor V. Lubyshev

Keyword(s):

Boundary Condition ◽

Approximate Solution ◽

Elliptic Equation ◽

Boundary Value Problem ◽

Mixed Boundary ◽

Boundary Value ◽

Variable Coefficients ◽

Natural Boundary Condition ◽

Mixed Boundary Value Problem ◽

Integration Region

A mixed boundary value problem for an elliptic equation of divergent type with variable coefficients is considered. It is assumed that the integration region is a rectangle, and the boundary of the integration region is the union of two disjoint pieces. The Dirichlet boundary condition is set on the first piece, and the Neumann boundary condition is set on the other one. The given problem is a problem with a discontinuous boundary condition. Such problems with mixed conditions at the boundary are most often encountered in practice in process modeling, and the methods for solving them are of considerable interest. This work is related to the paper [1] and complements it. It is focused on the approbation of the results established in [1] on the convergence of approximations of the original mixed boundary value problem with the main boundary condition of the third boundary value problem already with the natural boundary condition. On the basis of the results obtained in this paper and in [1], computational experiments on the approximate solution of model mixed boundary value problems are carried out.

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Iterative method for solving the second boundary value problem (BVP) for Biharmonic type equation

Journal of Computer Science and Cybernetics ◽

10.15625/1813-9663/14/4/7975 ◽

2016 ◽

Vol 14 (4) ◽

pp. 66-72

Author(s):

Đặng Quang Á

Keyword(s):

Differential Equations ◽

Convergence Rate ◽

Iterative Method ◽

Boundary Value Problem ◽

Fourth Order ◽

Boundary Value ◽

Type Equation ◽

Second Order ◽

Second Order Equations

Solving BVPs for the fourth order differential equations by the reduction of them to BVPs for the second order equations with the aim to use the achievements for the latter ones attracts attention from many researchers. In this paper, using the technique developed by ourselves in recent works, we construct iterative method for the second BVP for biharmonic type equation. The convergence rate of the method is established.

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On an ill-posed boundary value problem for the Laplace equation in a circular cylinder

Russian Universities Reports. Mathematics ◽

10.20310/2686-9667-2021-26-133-35-43 ◽

2021 ◽

pp. 35-43

Author(s):

Evgeniy B. Laneev ◽

Dmitriy Yu. Bykov ◽

Anastasia V. Zubarenko ◽

Olga N. Kulikova ◽

Darya A. Morozova ◽

...

Keyword(s):

Integral Equation ◽

Approximate Solution ◽

Circular Cylinder ◽

Boundary Value Problem ◽

Laplace Equation ◽

Boundary Value ◽

Stable Solution ◽

Medical Diagnostics ◽

Cauchy Data ◽

Ill Posed

In this paper, we consider a mixed problem for the Laplace equation in a region in a circular cylinder. On the lateral surface of a cylidrical region, the homogeneous boundary conditions of the first kind are given. The cylindrical area is bounded on one side by an arbitrary surface on which the Cauchy conditions are set, i.e. a function and its normal derivative are given. The other border of the cylindrical area is free. This problem is ill-posed, and to construct its approximate solution in the case of Cauchy data known with some error it is necessary to use regularizing algorithms. In this paper, the problem is reduced to a Fredholm integral equation of the first kind. Based on the solution of the integral equation, an explicit representation of the exact solution of the problem is obtained in the form of a Fourier series with the eigenfunctions of the first boundary value problem for the Laplace equation in a circle. A stable solution of the integral equation is obtained by the Tikhonov regularization method. The extremal of the Tikhonov functional is considered as an approximate solution. Based on this solution, an approximate solution of the problem in the whole is constructed. The theorem on convergence of the approximate solution of the problem to the exact one as the error in the Cauchy data tends to zero and the regularization parameter is matched with the error in the data is given. The results can be used for mathematical processing of thermal imaging data in medical diagnostics.

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Boundary-Layer Similarity Near the Edge of a Rotating Disk

Journal of Applied Mechanics ◽

10.1115/1.3423646 ◽

1975 ◽

Vol 42 (3) ◽

pp. 584-590 ◽

Cited By ~ 5

Author(s):

R. J. Bodonyi ◽

K. Stewartson

Keyword(s):

Boundary Layer ◽

Approximate Solution ◽

Flow Field ◽

Boundary Value Problem ◽

Asymptotic Expansions ◽

Numerical Solutions ◽

Boundary Value ◽

Rotating Disk ◽

Angular Speed ◽

Rotating Fluid

Numerical solutions of the similarity equations governing the flow near the edge of a finite rotating disk are found to be possible only for −2.06626 ≤ α ≤ 1, where α is the ratio of the disk’s angular speed to that of the rigidly rotating fluid far from the disk. Furthermore, for α ≤ −1 the solutions of the boundary-value problem are not unique, and along one of the solution branches a singular structure of the flow field is approached as α → −1. Using the method of matched asymptotic expansions an approximate solution is found along the singular branch which explains some of the problems encountered in finding numerical solutions.

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