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

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

Determination of the time-dependent thermal grooving coefficient

Changes in morphology of a polycrystalline material may occur through interface motion under the action of a driving force. An important special case that is considered in this paper is the thermal grooving that occurs when a grain boundary intersects the flat surface of a recently solidified metal slab giving rise to the formation of a thin symmetric groove. In case the transient surface diffusion is the main forming mechanism this yields a fourth-order time-dependent partial differential equation with unknown time-dependent surface diffusivity. In order to determine it, the profile of the free grooving surface at a fixed location is recorded in time. The grooving boundaries are supported by self-adjoint boundary conditions. We provide sufficient conditions on the input data for which the resulting coefficient identification problem is proved to be well-posed. Furthermore, we develop a predictor–corrector finitedifference spline method for obtaining an accurate and stable numerical solution to the nonlinear coefficient identification problem. Numerical results illustrate the performance of the inversion of both exact and noisy data

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

In this paper, we consider inverse problems of finding the time-dependent source function for the population model with population density nonlocal boundary conditions and an integral over-determination measurement. These problems arise in mathematical biology and have never been investigated in the literature in the forms proposed, although related studies do exist. The unique solvability of the inverse problems are rigorously proved using generalized Fourier series and the theory of Volterra integral equations. Continuous dependence on smooth input data also holds but, as in reality noisy errors are random and non-smooth, the inverse problems are still practically ill-posed. The degree of ill-posedness is characterised by the numerical differentiation of a noisy function. In the numerical process, the boundary element method together with either a smoothing spline regularization or the first-order Tikhonov regularization are employed with various choices of regularization parameter. One is based on the discrepancy principle and another one is the generalized cross-validation criterion. Numerical results for some benchmark test examples are presented and discussed in order to illustrate the accuracy and stability of the numerical inversion

The influence of the rheological parameters of a hydro-viscoelastic system consisting of a viscoelastic plate, viscous fluid and rigid wall on the frequency response of this system

In this paper the forced vibration of a hydro-viscoelastic system consisting of a viscoelastic plate, compressible viscous fluid and rigid wall is considered. The focus is on the investigation of the influence of the rheological parameters of the plate material and the viscosity of the fluid on the frequency response of this system. The constitutive relations for the plate material are given through the fractional-exponential operators, and the exact equations of the visco-elastodynamics in the plane-strain state are employed for describing the plate motion. The fluid motion is described through the linearized Navier–Stokes equations and it is assumed that the velocity and force vectors are continuous across the interface plane between the fluid and the plate. Numerical results on the frequency response of the normal stress acting on the interface plane and of the normal velocity of the points of this plane are presented for various values of the rheological parameters of the plate material. These results are also distinguished with respect to inviscid and viscous fluid cases. As a result of the analyses of these results, corresponding conclusions are made on the influence of the rheological parameters of the plate material on the aforementioned frequency responses. </jats:p