Boundary Value Problems as Ordinary Differential Equations

To solve ordinary differential equations, it is necessary to supply boundary conditions or initial conditions. The general solution to a second order equation, for example, has two independent solutions and the boundary conditions are used to determine how much of each of them are in the final solution. The characteristic that makes the ordinary differential equation a boundary value problem is that the conditions are provided at different points (in space) or different values of the independent variable. Boundary values frequently arise in transport problems, and here they are solved analytically, using the finite difference method, the orthogonal collocation method, the finite element method, the method of orthogonal collocation on finite elements, and initial value methods. In addition, there are some more advanced techniques described.