The free parameter occurring in will then be determined by the boundary conditions for . In the following, this kind of expansion with suitable basis functions will be applied to the solution of a problem of nonlinear heat transfer, and its convergence will be treated as far as relevant for this special case. For all the cases investigated in this article, we apply the recursive approach (), since the structure of the chosen basis functions is relatively simple. For more sophisticated basis functions, the combinatorial formulas () and () have to be implemented in order to reduce CPU time. This may be done in future investigations.
As a canonical example, we study the partial differential equation of transient nonlinear heat transfer with temperature-dependent thermal conductivity . We demonstrate the application of Bürmann series to the practical problem of heat transfer in ZnO ceramics The half space is filled by this material, which has initial constant temperature , and the temperature as . At the surface temperature at is instantaneously raised to a constant temperature . Using the results of measurements of the thermal conductivity in ZnO , we can formulate the problem by writing
Nonlinear problems arise in many heat transfer applications, and several analytical and numerical methods for solving these problems are described in the literature. Here, the method of variation of parameters is shown to be a relatively simple method for obtaining solutions to four specific heat transfer problems: 1. a radiating annular fin, 2. conduction-radiation in a plane-parallel medium, 3. convective and radiative exchange between the surface of a continuously moving strip and its surroundings, and 4. convection from a fin with temperature-dependent thermal conductivity and variable cross-sectional area. The results for each of these examples are compared to those obtained using other analytical and numerical methods. The accuracy of the method is limited only by the accuracy with which the numerical integration is performed. The method of variation of parameters is less complex and relatively easy to implement compared to other analytical methods and some numerical methods. It is slightly more computationally expensive than traditional numerical approaches. The method presented may be used to verify numerical solutions to nonlinear heat transfer problems.