Numerical solution of one-dimensional parabolic inverse problem

Abstract

This paper is concerned with numerical techniques for the solution of a one-dimensional parabolic inverse problem with a control parameter. The discrete approximation of the problem is based on the finite difference schemes. These techniques are presented for identifying the control parameter which produces at each time a desired temperature at a given point in a spatial domain. The main idea behind the finite difference methods for obtaining the solution of a given partial differential equation is to approximate the derivatives appearing in the equation by a set of values of the function at a selected number of points. The most usual way to generate these approximations is through the use of Taylor series. The methods developed here are based on the modified equivalent partial differential equation as described by Warming and Hyett [J. Comput. Phys. 14 (2) (1974) 159]. This approach allows the simple determination of the theoretical order of accuracy, thus allowing methods to be compared with one another. Also from the truncation error of the modified equivalent equation, it is possible to eliminate the dominant error terms associated with the finite-difference equations that contain free parameters (weights), thus leading to more accurate methods. The results of a numerical experiment are presented, and accuracy and the Central Processor (CPU) time needed for the parabolic inverse problem are discussed.