On a criterion for the solvability of one ill-posed problem for the biharmonic equation
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Tynysbek S. Kal’menov
and
Ulzada A. Iskakova
Abstract
A local boundary value problem for an inhomogeneous biharmonic equation in a rectangular domain is considered. Boundary conditions are given on the whole boundary of the domain. It is shown that the problem turns out to be self-adjoint. And herewith the problem is ill-posed. An example is constructed demonstrating that the solution stability to the problem is violated. Necessary and sufficient conditions of the existence of a strong solution to the investigated problem are found. The idea of the method is that the solution to the problem is constructed in the form of an expansion on eigenfunctions of this self-adjoint problem. This problem has an isolated point of continuous spectrum in zero. It is shown that there exists a series (a sequence) of eigenvalues converging to zero. Asymptotics of these eigenvalues is found. Namely this asymptotics defines a reason for the ill-posedness of the investigated problem. A space of well-posedness for the investigated problem is constructed.
Funding source: Ministry of Education and Science of the Republic of Kazakhstan
Award Identifier / Grant number: 0820/GF4
Funding statement: Research supported by the grant 0820/GF4 of the Ministry of Education and Science of the Republic of Kazakhstan.
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- Numerical solution of an ill-posed Cauchy problem for a quasilinear parabolic equation using a Carleman weight function
- On a criterion for the solvability of one ill-posed problem for the biharmonic equation
Articles in the same Issue
- Frontmatter
- Inverse problem about two-spectra for finite Jacobi matrices with zero diagonal
- Shape and parameter reconstruction for the Robin transmission inverse problem
- Inverse source problem based on two dimensionless dispersion-current functions in 2D evolution transport equations
- On the null space of a class of Fredholm integral equations of the first kind
- Numerical solution of an elliptic 3-dimensional Cauchy problem by the alternating method and boundary integral equations
- Reconstruction of local volatility for the binary option model
- Determination of finite difference coefficients for the acoustic wave equation using regularized least-squares inversion
- Numerical solution of an ill-posed Cauchy problem for a quasilinear parabolic equation using a Carleman weight function
- On a criterion for the solvability of one ill-posed problem for the biharmonic equation
