A direct problem and an inverse problem for the Laplace’s equation was solved in this paper. Solution to the direct problem in a rectangle was sought in a form of finite linear combinations of Chebyshev polynomials. Calculations were made for a grid consisting of Chebyshev nodes, what allows us to use orthogonal properties of Chebyshev polynomials. Temperature distributions on the boundary for the inverse problem were determined using minimization of the functional being the measure of the difference between the measured and calculated values of temperature (boundary inverse problem). For the quasi-Cauchy problem, the distance between set values of temperature and heat flux on the boundary was minimized using the least square method. Influence of the value of random disturbance to the temperature measurement, of measurement points (distance from the boundary, where the temperature is not known) arrangement as well as of the thermocouple installation error on the stability of the inverse problem was analyzed.
This article presents a new efficient method of determining values of gas flow parameters (e.g. axial dispersion coefficient, DL and Pèclet number, Pe). A simple and very fast technique based on the pulse tracer response is proposed. It is a method which combines the benefits of a transfer function, numerical inversion of the Laplace transform and optimization allows estimation of missing coefficients. The study focuses on the simplicity and flexibility of the method. Calculations were performed with the use of the CAS-type program (Maple®). The correctness of the results obtained is confirmed by good agreement between the theory and experimental data for different pressures and temperature. The CAS-type program is very helpful both for mathematical manipulations as a symbolic computing environment (mathematical formulas of Laplace-domain model are rather sophisticated) and for numerical calculations. The method of investigations of gas flow motion is original. The method is competitive with earlier methods.