Simultaneous identification of the right-hand side and time-dependent coefficients in a two-dimensional parabolic equation

This paper investigates the simultaneous identification of time-dependent lowest and source terms in a two-dimensional (2D) parabolic equation from the additional measurements. To investigate the solvability of the inverse problem, we first examine an auxiliary inverse boundary value problem and prove its equivalence to the original problem in a certain sense. Then, applying the contraction mappings principle existence and uniqueness of the solution of an equivalent problem is proved. Furthermore, using the equivalency, the existence and uniqueness theorem for the classical solution of the original problem is obtained and some discussions on the numerical solutions for this inverse problem are presented including numerical examples

The nonlocal inverse problem of the identification of the lowest coefficient and the right-hand side in a second-order parabolic equation with integral conditions

Abstract In the present paper a nonlocal inverse boundary-value problem for a second-order parabolic equation is considered. This investigation introduces the identification of the lowest unknown coefficient and time-depend right-hand side in a second-order parabolic equation on overdetermination at the internal points. Sufficient conditions for the existence and uniqueness of the classical solution to inverse problem of a second-order parabolic equation are obtained for small time

Construction of exact control for a one-dimensional heat equation with delay

We prove an exact controllability result for a one-dimensional heat equation with delay in both lower and highest order terms and nonhomogeneous Dirichlet boundary conditions. Moreover, we give an explicit representation of the control function steering the system into a given final state. Under certain decay properties for corresponding Fourier coefficients which can be interpreted as a sufficiently high Sobolev regularity of the data, both control function and the solution are proved to be regular in the classical sense both with respect to time and space variables

On classical solvability for a linear 1D heat equation with constant delay

In this paper, we consider a linear heat equation with constant coefficients and a single constant delay. Such equations are commonly used to model and study various problems arising in ecology and population biology when describing the temporal evolution of human or animal populations accounting for migration, interaction with the environment and certain aftereffects caused by diseases or enviromental polution, etc. Whereas dynamical systems with lumped parameters have been addressed in numerous investigations, there are still a lot of open questions for the case of systems with distributed parameters, especially when the delay effects are incorporated. In the present paper, we consider a general non-homogeneous one-dimensional heat equation with delay in both higher and lower order terms subject to non-homogeneous initial and boundary conditions. For this, we prove a unique existence of a classical solution as well as its continuous dependence on the data