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.

JO - Archives of Thermodynamics L1 - http://journals.pan.pl/Content/105637/PDF/05_paper.pdf L2 - http://journals.pan.pl/Content/105637 PY - 2016 IS - No 4 EP - 88 KW - Laplace’s equation KW - boundary inverse problem KW - quasi-Cauchy problem KW - stability of the inverse problem A1 - Joachimiak, Magda A1 - Ciałkowski, Michał A1 - Frąckowiak, Andrzej PB - The Committee on Thermodynamics and Combustion of the Polish Academy of Sciences JF - Archives of Thermodynamics DA - 2016 T1 - Solution of inverse heat conduction equation with the use of Chebyshev polynomials SP - 73 UR - http://journals.pan.pl/dlibra/docmetadata?id=105637 DOI - 10.1515/aoter-2016-0028 ER -