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

Benrabah Abderafik

Journal of Applied Mathematics and Computational Mechanics

2022

Vol. 21, nr 3

5--17

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.

Linear barycentric rational collocation method for solving biharmonic equation

Li Jin

Demonstratio Mathematica

2022

Vol. 55, nr 1

587--603

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.

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

1998

Vol. 5, nr 1

55-64

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.