An inverse coefficient problem for a parabolic equation in the case of nonlocal boundary and overdetermination conditions

Abstract

Communicated by A. KirschIn this paper, the inverse problem of finding the time-dependent coefficient of heat capacity together with the solution of heat equation with nonlocal boundary and overdetermination conditions is considered. The existence, uniqueness and continuous dependence upon the data are studied. Some considerations on the numerical solution for this inverse problem are presented with the examples.

An inverse problem of finding the time‐dependent diffusion coefficient from an integral condition

“…The analysis is similar to that of [4] for the identification of the time-dependent blood perfusion coefficient in the bio-heat equation. Consider the spectral problem…”

“…In this case, φ(x) is the distribution of admixture in the chip for x ∈ (0, 1) at the initial time t = 0, while u(x, t) is its distribution at time t. Condition (3) means that the admixtures in the left and right boundaries of the chip are the same. The adiabatic condition (4) means that the right boundary x = 1 of the chip is perfectly insulated. Condition (5) means that part of the substance is concentrated (segregated) on the left side x = 0 of the chip, [9,10].…”

Determination of a diffusion coefficient in a quasilinear parabolic equation

“…They are also Riesz bases in L 2 OE0; 1 (see [7]). We obtain the following representation for the solution of (1)-(3) for arbitrary a.t / by using the Fourier method:…”

An inverse problem for a 2D parabolic equation with nonlocal overdetermination condition

“…The problem (1)- (5) is equivalent to the equation (22) in the following sense: if (a, u) is a solution to problem (1)- (5), then a is a solution of (22) and, on the other hand, if a ∈ C([0, T]) is a solution of (22), then (a, u) is a solution to the problem (1)- (5), where u is determined by the equations (11).…”

“…The other approaches to this problem addressing the question of existence and uniqueness are the Fourier method utilized by Ismailov M.I., Kanca F. [11], Oussaeif T.-E., Bouziani A. [16] and the theory of reproducing kernels used by Mohammadi M., Mokhtari R. and Isfahani F.T.…”