ABSTRACT
This chapter presents methods to solve problems with one independent variable involving first-order differential equations and third-kind boundary conditions. Mathematically, this class of cases can be summarized as a third-kind boundary condition. The chapter considers the heating of a solid with controlled heat transfer rate and presents the solution through the methodology of separation of variables. For the case of a temperature-controlled batch reactor, the solution is obtained by separable equation approach. However, the problem can be solved through Laplace transform as well. A realistic equation for the energy source because of the reaction is described by the Arrhenius formulation. Among the possible methods to achieve a solution in this case is the weighted residual method. The chapter also illustrates choosing the form of approximations including the first and second approximation and the collocation method.
