An Iterative Method to Recover the Heat Conductivity Function of a Nonlinear Heat Conduction Equation

The left-boundary data of temperature and heat flux are used to estimate an unknown heat conductivity function in a nonlinear heat conduction equation. A Lie-group adaptive method (LGAM) is developed to derive a Lie-group equation defined at two different times, which can be used to recover a spatial-dependence heat conductivity function through a few iterations. Also, the one-dimensional Calderón inverse problem is addressed by applying the present methodology to recover a steady-state heat conductivity coefficient in terms of space variable. The convergence speed and accuracy of the present LGAM are examined by linear and nonlinear numerical examples under noisy input data.