Determination of the time-dependent reaction coefficient and the heat flux in a nonlinear inverse heat conduction problem

Abstract

Diffusion processes with reaction generated by a nonlinear source are commonly encountered in practical applications related to ignition, pyrolysis and polymerization. In such processes, determining the intensity of reaction in time is of crucial importance for control and monitoring purposes. Therefore, this paper is devoted to such an identification problem of determining the time-dependent coefficient of a nonlinear heat source together with the unknown heat flux at an inaccessible boundary of a one-dimensional slab from temperature measurements at two sensor locations in the context of nonlinear transient heat conduction. Local existence and uniqueness results for the inverse coefficient problem are proved when the first three derivatives of the nonlinear source term are Lipschitz continuous functions. Furthermore, the conjugate gradient method (CGM) for separately reconstructing the reaction coefficient and the heat flux is developed. The ill-posedness is overcome by using the discrepancy principle to stop the iteration procedure of CGM when the input data is contaminated with noise. Numerical results show that the inverse solutions are accurate and stable.

Implementation of DRBEM for the determination of the heat flux in an inverse problem

“…These results mentioned above are good agreement with the results for the inverse natural convection problem and inverse heat conduction problem in [19] and [17], respectively.…”

“…In addition, advection-di¤usion and reaction-di¤usion equations for an inverse problem case have been considered in [15] and [16], respectively. Later on, one-dimensional inverse heat conduction problem has solved in [17].…”

Simultaneous identification of the solely time‐dependent potential and source control terms from known moisture moments

“…IPs related to the classical diffusion equation ( $$$ \eta =0 $$$) have been extensively examined through theoretical and numerical investigations. For instance, IPs of estimating the time/space‐varying heat source can be found in [20–23], and IPs of reconstructing the time‐dependent reaction coefficient have been investigated in the literature [24–27]. Also, it is important to note that in Hasanov [28], a variational approach to a nonlinear nonlocal identification problem related to the nonlinear ion transport equation is studied.…”

Identification of the Solely Time-Dependent Zero-Order Coefficient in a Linear Bi-Flux Diffusion Equation from an Integral Measurement

“…The IPs for the classical diffusion equation (β = 1) are satisfactorily studied theoretically and numerically. For instance, IPs of determining the time-dependent heat source are studied in [13,14,15,16], and IPs of reconstructing the time-dependent reaction coefficient are investigated in [17,18,19,20,21] with different boundary conditions (local, non-local or non-classical conditions). On contrary to the IPs for the classical diffusion equation, our aim is to study the Bi-flux diffusion equation to identify the time-dependent zero-order coefficient k(τ) along with the particle concentration Z(χ, τ) theoretically, for the first time, in the rectangular domain, using the IC (1.3), homogeneous BCs (1.4) and the AC (1.5).…”