Nonlinear Boundary-Value Problem for One-Dimensional Heating (Cooling) of a Sphere, a Cylinder and an Infinite Plate. The Inverse Problem of the Temperature Field.

A method of approximate solution of an unsteady one-dimensional heat-conduction equation with a time-variable or nonlinear boundary condition is presented. By considering finite differences of the time variable and using the equations for modified Bessel functions, an approximate solution is obtained in the form of a sum of linearly independent functions, the coefficients of which are determined from the boundary conditions by solving a first-order recurrence equation (if the Biot number depends on the time and the radiation is rejected) or a fourth-order recurrence equation (if the Biot number depends on the time and the radiation is taken into account). The equations obtained are used for solving the inverse temperature problem. The considerations outlined are illustrated by numerical examples.