Direct and inverse problems for the heat equation with a dynamic-type boundary condition

Abstract

This paper considers the initial-boundary value problem for the heat equation with a dynamic type boundary condition. Under some regularity, consistency and orthogonality conditions, the existence, uniqueness and continuous dependence upon the data of the classical solution are shown by using the generalized Fourier method. This paper also investigates the inverse problem of finding a time-dependent coefficient of the heat equation from the data of integral overdetermination condition.

An inverse time-dependent source problem for the heat equation with a non-classical boundary condition

“…When the function r(t) is given, the problem of finding u(x, t) from the heat equation (1.1), initial condition (1.2), and boundary conditions (1.3) and (1.4) is termed as the direct (or forward) problem. The well-posedness of this direct problem has been established elsewhere, [21]. This model can be used in heat transfer and diffusion processes with a source parameter present in (1.1).…”

“…The following Bessel-type inequalities are true for the systems {y n (x)} and {u n (x)} (n = 0, 1, 2, ...; n ̸ = n 0 ), see [21].…”

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

“…Problems of the solvability of inverse problems for equations of parabolic type were considered in the papers by Ivanchov [14,15], Kozhanov [16], Vasin [17], Pyatkov [18], Kabanikhin [19], Ismailov [20][21][22][23], and many others. Regarding recent results on the theory of partial differential equation with discontinuous coefficients, we refer the reader to [24][25][26].…”

Determination of a time-dependent heat source under not strengthened regular boundary and integral overdetermination conditions

“…In [3] a system of nonlinear impulsive differential equations with two-point and integral boundary conditions is investigated. In [16] the initial-boundary value problem for the heat equation with a dynamictype boundary condition is considered. In [23] one family of problems simulating the determination of target components and density of sources from given values of the initial and final states is considered.…”