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

Abstract

We investigate an inverse problem of finding a time-dependent heat source in a parabolic equation with nonlocal boundary and integral overdetermination conditions. The boundary conditions of this problem are regular but not strengthened regular. The principal difference of this problem is: the system of eigenfunctions is not complete. But the system of eigen-and associated functions forming a basis. Under some natural regularity and consistency conditions on the input data the existence, uniqueness and continuously dependence upon the data of the solution are shown by using the generalized Fourier method.

Nonlocal spectral problem for a second-order differential equation with an involution

“…Spectral problems arising in connection with differential operators with involution were considered in [14][15][16][17][18] for first-order operators and in [19,20] for second-order operators. Spectral problems for ordinary differential operators with non-strongly regular boundary conditions and their applications for parabolic problems were investigated in [21][22][23][24][25].…”

Properties in L p of root functions for a nonlocal problem with involution

“…Spectral topics for first-and second-order operations that have involution in their main terms are discussed in [13-15, 32, 33]. Applications of the spectral approach for PDEs with involution and/or nonlocal boundary conditions are discussed in [1,2,25,27,[29][30][31]. For the spectral properties of conventional differential operators in non-Hilbert spaces, one could refer to [4,6,7,19,20,34].…”

Stability of the Time-Dependent Identification Problem for the Delay Hyperbolic Equation

“…There is permanently a major interest for the theory of source identification problems for partial differential equations since they have widespread applications in modern physics and technology. For this effort, the stability of various source identification problems for partial differential and difference equations has also been studied extensively by many researchers (see, e.g., [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25] and the references given therein).…”