Identification general boundary conditions of the diffusion operator
The problems of identification of both general and separated boundary conditions of the diffusion operator by its eigenvalues are investigated. For general boundary conditions it is shown that if at least one of the coefficients in the differential equation is asymmetric function, then the problem of identifying boundary conditions by all eigenvalues has two solutions, and if both of the coefficients are symmetric functions, then the problem of identifying the boundary conditions over all there are infinitely many solutions to the eigenvalues. It is also proved that if at least one of the coefficients in the differential equation is an asymmetric function, then two solutions can be obtained by four eigenvalues, if the rank of some matrix is four. For separated boundary conditions it is shown that if at least one of the coefficients in the differential equation is an asymmetric function, then the problem of identifying boundary conditions by all eigenvalues has a unique solution, and if both of the coefficients are symmetric functions, then the problem of identifying boundary conditions has two solutions. It is also shown that under certain conditions a unique solution can be obtained from three eigenvalues, and two solutions by two eigenvalues. Appropriate examples are considered.