Wyniki wyszukiwania - BazTech

1

Conditional stability estimate for an ill-posed elliptic equation by using nonlocal conditions

Benrabah Abderafik

Journal of Applied Mathematics and Computational Mechanics

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2022

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Vol. 21, nr 3

5--17

EN

We consider an ill-posed linear homogeneous fourth-order elliptic equation. We show that the problem is ill-posed in the sense of Hadamard, i.e., the solution does not depend continuously on the given data. We propose a regularization method via nonlocal conditions and under some a priori bound assumptions different estimates for the regularized solution are obtained. Numerical examples for a rectangle domain show the effectiveness of the new method in providing highly accurate numerical solutions as the noise level tends to zero.

2

Linear barycentric rational collocation method for solving biharmonic equation

Li Jin

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

587--603

EN

Two-dimensional biharmonic boundary-value problems are considered by the linear barycentric rational collocation method, and the unknown function is approximated by the barycentric rational polynomial. With the help of matrix form, the linear equations of the discrete biharmonic equation are changed into a matrix equation. From the convergence rate of barycentric rational polynomial, we present the convergence rate of linear barycentric rational collocation method for biharmonic equation. Finally, several numerical examples are provided to validate the theoretical analysis.

3

Determination of stress state in coal seam based on inverse problem solution using acoustic sounding data

Kudaibergenov Mels K., Iskakov Kazizat T., Kudaibergenova Bakytzhan S.

Mining Science

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2019

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Vol. 26

173--187

EN

The paper presents an algorithm for calculating the horizontal, vertical and tangential stresses in a horizontal coal seam lying between galleries. These stresses are expressed in terms of the Airy function, which satisfies a homogeneous biharmonic equation. For its numerical solution it is necessary to set boundary conditions. Practical limitations do not allow us to determine the tangential stresses on horizontal boundaries of the coal seam. Calculations of stresses are proposed to be carried out in two steps. The first step is to solve the inverse problem for the biharmonic equation to find unknown tangential stresses on horizontal boundaries of the coal seam. The inverse problem is solved by minimizing the residual functional. Its strong convexity is proved, which implies the existence and uniqueness of the solution of the inverse problem. The second step is to solve the boundary value problem for the biharmonic equation to calculate stresses in the coal seam. The result of a numerical experiment is presented.

4

Positivity preserving results for a biharmonic equation under Dirichlet boundary conditions

Omrane H. B., Khenissy S.

Opuscula Mathematica

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2014

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Vol. 34, no. 3

601--608

EN

We prove a dichotomy result giving the positivity preserving property for a biharmonic equation with Dirichlet boundary conditions arising in MEMS models. We adapt some ideas in [H.-Ch. Grunau, G. Sweers, Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions, Math. Ann. 307 (1997), 589–626].

5

On Plane, Steady, Creeping Flow, Generated Around an Arbitrary Cylinder by a Translating Flat Plate

Klonowska-Prosnak M.E., Prosnak W.J., Cześnik P.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2003

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Vol. 51, nr 2

113-144

EN

Plane, steady, creeping flow around an arbitrary cylinder situated in vicinity of a flat plate is considered, the flow being generated by translation of this plate along itself with constant speed, perpendicular to generatrices. The stream function of the flow satisfies the biharmonic equation, so that all properties of the flow - such as velocity and pressure fields as well as fields of other stress tensor components - are expressed in terms of the Goursat functions. Hence, the problem of determination of the flow reduces to determination of these functions. The two-step approach is applied to the solution of this problem. The first step consists in conformal mapping of the original domain of solution onto an annulus - by means of a suitable set of mapping functions. The second step consists in development of the two Goursat functions in Laurent series extended by two logarithmic terms each. Unknown coefficients of the series have to satisfy a system of linear algebraic equations, following from boundary conditions. The system is arrived at by means of the pseudospectral method. The so obtained velocity field is applied to generation of streamline patterns. Such a pattern is presented in the paper, and compared with an analogous one, corresponding to potential flow.

6

The Dirichlet problem for biharmonic equation in case the half plane

Lupu M.

Demonstratio Mathematica

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1999

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Vol. 32, nr 1

41-53

EN

Using the representations of the solution of Dirichlet problem for the half plane, the basic biharmonic problem (BP) is solved. Applying the half plane theorem on Almansi type representation of the solution is given by direct or analytical methods. New formulas are proved and for special cases some applications are presented.

7

Stress intensity factors computations using the singularity substraction technique incorporated with the Tau Method

Pan C. K, Liu K. M.

Computer Assisted Mechanics and Engineering Sciences

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1998

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Vol. 5, nr 1

55-64

EN

In this paper we discuss the use of the singularity subtraction technique incorporated with the Tau Method for the numerical solution of singular partial differential equations which are relevant to the linear elastic fracture mechanics. To treat the singularity, we apply the singularity subtraction technique to the singular boundary value problems. The problems arising in this application are not in the standard form required by the Tau software. By introducing the pseudo-differential equations l k=0, k=1(1)m, to detrmine the stress intensity and higher order factors lk results in the standard boundary value problems. We consider two model crack problems including Motz ' anti-plane crack problem and plane strain problem defined by the biharmonic equation. We obtain results of considerable accuracy which compare favorably with those published in the recent literature.