Combined method of separation of variables. 3. Nonstationary heat conduction in solids with the first-kind boundary condition

The general idea of the combined method of separation of variables, as applied to the solution of problems on the nonstationary heat conduction in solid bodies canonical in shape (plate, cylinder, sphere) with the fi rst-kind boundary condition, is elucidated. Four effi cient schemes of calculating the eigenvalues of the Sturm–Liouville boundary-value problem are presented. An original method is proposed for determining initial amplitudes with a relatively high accuracy. Using the example of approximate solution of the heat-conduction problem for a solid cylinder with the fi rst-kind boundary condition, the signifi cant simplicity and, at the same time, the high effi ciency of the combined method of separation of variables is graphically demonstrated.
The general idea of the combined method of separation of variables, as applied to the solution of problems on the nonstationary heat conduction in solid bodies canonical in shape (plate, cylinder, sphere) with the fi rst-kind boundary condition, is elucidated. Four effi cient schemes of calculating the eigenvalues of the Sturm–Liouville boundary-value problem are presented. An original method is proposed for determining initial amplitudes with a relatively high accuracy. Using the example of approximate solution of the heat-conduction problem for a solid cylinder with the fi rst-kind boundary condition, the signifi cant simplicity and, at the same time, the high effi ciency of the combined method of separation of variables is graphically demonstrated.
Author:  V. A. Kot
Keywords:  heat conduction equation, boundary-value problem, method of separation of variables, eigenvalues, eigenfunctions, approximate solution, integral relations, differential equation
Page:  311-344