A Fourier method for initial value problems with mixed boundary conditions
The classical heat equation, written in polar coordinates (r,\(\theta)\) is considered in the unit circle. At the arc \(r=1\), \(\theta\in (-c,c)\) heat flux is prescribed whereas at the complementary arc temperature does not change with time. Initial temperature is also given. The authors present in detail the analysis of obtaining the solution in form of a series containing trigonometric and Bessel functions in the case of time- independent heat flux. Actually, they derive such a series for the difference of the stationary and the transient solution. After discussing a few concrete examples they describe how the case of time-dependent flux can be treated by aid of Duhamel's principle.
series containing trigonometric and Bessel functions
scientific article; zbMATH DE number 4035072