Partial Differential Equations: Finite Difference Approaches

The two previous chapters have considered differential equations which involve only one independent variable — typically a spatial variable or a time variable. The present chapter expands the range of equations to one or more differential equation in two or more independent variables which may be spatial variables or spatial variables and a time variable. The simplest partial differential equations involve only two variables while some physically interesting engineering problems involve as many as four independent variables (three spatial variables and a time variable). Even more general problems with more independent variables can of course be considered.

The discussion in this chapter will concentrate primarily on partial differential equations in only two independent variables. This is due in part to the exponential growth in needed solution points as the number of dimensions increases. Also for a large number of engineering problems the symmetry of the problem makes it possible to reduce the number of primary variables to two. Finally, much of the insight into real engineering problems can be obtained from a study of partial differential equations in only two dimensions. The discussion in this chapter also concentrates on finite difference approaches to solving partial differential equations. This approach is most appropriate to problems where time is one variable and a spatial variable is the second variable. For many engineering problems involving two or more spatial variables, the finite element approach discussed in the next chapter is a more appropriate technique, because the discretization of spatial points can more easily be make to conform to appropriate spatial boundaries of a given problem.