On the Approximate Solution of the First Boundary Value Problem for $\nabla ^4 u = f$

1973 ◽  
Vol 10 (6) ◽  
pp. 967-982 ◽  
Author(s):  
Julius Smith
Keyword(s):  
Approximate Solution ◽  
Boundary Value Problem ◽  
Boundary Value

scholarly journals ABOUT THE APPROXIMATE SOLUTION OF THE USUAL AND GENERALIZED HILBERT BOUNDARY VALUE PROBLEMS FOR ANALYTICAL FUNCTIONS

2000 ◽  
Vol 5 (1) ◽  
pp. 119-126
Author(s):  
V. R. Kristalinskii
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Cauchy Type ◽  
Analytical Functions ◽  
Method Of Solution ◽  
Cauchy Type Integrals ◽  
Hilbert 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.

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

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.

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

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.

Boundary-Layer Similarity Near the Edge of a Rotating Disk

1975 ◽  
Vol 42 (3) ◽  
pp. 584-590 ◽  
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.

scholarly journals APPROXIMATE SOLUTION OF AN INITIAL-BOUNDARY VALUE PROBLEM FOR AN NONHOMOGENEOUS SECOND-ORDER DIFFERENTIAL EQUATION OF MIXED TYPE WITH TWO DEGENERATE LINES

2020 ◽  
pp. 81-98
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Boundary Value Problem ◽  
Mixed Type ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Approximate Solutions ◽  
Second Order ◽  
Initial Boundary Value

This work is devoted to the study of an approximate solution of the initial-boundary value problem for the second order mixed type nonhomogeneous differential equation with two degenerate lines. Similar equations have many different applications, for example, boundary value problems for mixed type equations are applicable in various fields of the natural sciences: in problems of laser physics, in magneto hydrodynamics, in the theory of infinitesimal bindings of surfaces, in the theory of shells, in predicting the groundwater level, in plasma modeling, and in mathematical biology. In this paper, based on the idea of A.N. Tikhonov, the conditional correctness of the problem, namely, uniqueness and conditional stability theorems are proved, as well as approximate solutions that are stable on the set of correctness are constructed. In obtaining an apriori estimate of the solution of the equation, we used the logarithmic convexity method and the results of the spectral problem considered by S.G. Pyatkov. The results of the numerical solutions and the approximate solutions of the original problem were presented in the form of tables. The regularization parameter is determined from the minimum estimate of the norm of the difference between exact and approximate solutions.

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

2021 ◽  
pp. 394-403
Author(s):  
Evgeniy B. Laneev ◽  
Viktor A. Anisimov ◽  
Polina A. Lesik ◽  
Viktoriya I. Remezova ◽  
Andrey A. Romanov ◽  
...  
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Circular Cylinder ◽  
Boundary Value Problem ◽  
Regularization Parameter ◽  
Boundary Value ◽  
Stable Solution ◽  
Cauchy Data ◽  
Cylindrical Domain ◽  
Imaging Data

In this paper, we consider a mixed problem for a metaharmonic equation in a domain in a circular cylinder. The cylindrical area is bounded on one side by an arbitrary surface on which the Cauchy conditions are set, i. e. the function and its normal derivative are set. The other border of the cylindrical area is free. On the lateral surface of the cylindrical domain, homogeneous boundary conditions of the first kind are given. The problem is illposed and its approximate solution, stable to errors in the Cauchy data, is constructed using regularization methods. The problem is reduced to a first kind Fredholm integral equation. Based on the solution of the integral equation obtained in the form of a Fourier series by the eigenfunctions of the first boundary value problem for the Laplace equation in a circle, an explicit representation of the exact solution of the problem is constructed. A stable solution of the integral equation is obtained by the method of Tikhonov regularization. The extremal of the Tikhonov functional is considered as an approximate solution. Based on this solution, an approximate solution of the problem as a whole is constructed. A 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 early diagnostics in medicine.

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