Determination of a time-dependent coefficient in the bioheat equation

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

“…To overcome this instability, we employ the second-order Tikhonov regularization method which gives r λ = ( where λ > 0 is a regularization parameter to be prescribed and R 2 is a second-order differential regularization matrix, given by [15,29],…”

Inverse time-dependent source problems for the heat equation with nonlocal boundary conditions

“…internal pointwise, boundary, integral or boundary integral, [16]. In comparison with previous related studies, [4,6,7,9,10,12], in the current investigation, the birth rate in the population age model [15] is an arbitrary constant and the boundary conditions and the additional observation are both nonlocal and integral, respectively. One interesting point to remark is that in the case of arbitrary birth rate, the algebraic multiplicity of the auxiliary spectral problem is, in general, equal to 2 and this leads to a different treatment of generalized Fourier series analysis.…”

“…u(x, t) dx = E(t), t ∈ [0, T ], (2.5) which represents the density of population between the ages from 0 to 1 at the moment of time t. In other diffusion applications, condition (2.5) refers to the specification of mass [2]. Related inverse problems with b = −1 and (2.4) replaced by the convection boundary condition u x (0, t) + αu(0, t) = 0, t ∈ [0, T ], (2.6) where α is a Robin convection coefficient, have been considered elsewhere, [4,7,9,12]. When the function F is given, the problem of finding u(x, t) from the initial boundary value problem (2.1)-(2.4) is called the direct problem.…”

“…The timedependent coefficient can be a free term, as in a linear inverse source problem, [4], or multiplying the dependent variable, as in a quasi-linear perfusion coefficient identification problem, [18]. The boundary conditions are usually of Dirichlet, Neumann or Robin type but, more recently, nonlocal boundary conditions arising in population age models have also been considered, [5,6,7]. Also, additional measurements which are sufficient to render a unique solution have been of various types, e.g.…”

Identification of a memory kernel in a nonlinear integrodifferential parabolic problem

“…Crucial in the analysis is that the term depending on the solution u in the right-hand side of (1) is in integral form. Note that other parameter (time-dependent) identification problems can be found in [16,17,18,19,20,21,22,23,24,25,26,27,28].…”