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.

KEYWORDS:

• Inverse heat source problem
• inverse heat conduction problem
• nonlinear source
• conjugate gradient method
• eigenfunction series expansion

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35K05
• 35K55
• 35K57

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

L. Zhuo would like to acknowledge the China Scholarship Council (CSC) for supporting his visiting to the University of Leeds. S. Meng would like to acknowledge the support from the Major State Basic Research Development Program of China (973 Program) (Grant No. 2015CB655200).