The paper deals with numerical methods for eigenvalue problem for the second order ordinary differential operator with variable coefficient subject to nonlocal integral condition. FD-method (functional-discrete method) is derived and analyzed for calculating of eigenvalues, particulary complex eigenvalues. The convergence of FD-method is proved. Finally numerical procedures are suggested and computational results are schown.
AbstractUniqueness of parabolic Cauchy problems is nowadays a classical problem and since Hadamard [Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Dover, New York, 1953], these kind of problems are known to be ill-posed and even severely ill-posed.
Until now, there are only few partial results concerning the quantification of the stability of parabolic Cauchy problems.
We bring in the present work an answer to this issue for smooth solutions under the minimal condition that the domain is Lipschitz.
Abstract
This paper is concerned with the positivity of solutions to the Cauchy problem for linear and nonlinear parabolic equations with the biharmonic operator as fourth order elliptic principal part. Generally, Cauchy problems for parabolic equations of fourth order have no positivity preserving property due to the change of sign of the fundamental solution. One has eventual local positivity for positive initial data, but on short time scales, one will in general have also regions of negativity.
The first goal of this paper is to find sufficient conditions on initial data which ensure the existence of solutions to the Cauchy problem for the linear biharmonic heat equation which are positive for all times and in the whole space.
The second goal is to apply these results to show existence of globally positive solutions to the Cauchy problem for a semilinear biharmonic parabolic equation.
Stability Analysis and Error Estimation of Variable Coefficient Convection-Diffusion Equation: Generalized Numerical Fluxes