1. Problem Formulation

Formulating the boundary-value problems for partial differential equations (PDE) means finding the unknown functions of not one (as in the case of ordinary differential equations), but several variables, for example f(x, y, z) or f(x, t). Such functions involve derivatives with respect to different variables (partial derivatives); for example, derivatives with respect to spatial coordinates x, y, and z, or with respect to time t. Consequently, in all cases, a proper mathematical formulation of such problems additionally requires that a certain number of boundary conditions (and, possibly, initial conditions if one of the variables is time) should be specified.

The necessary number and format of the initial and boundary conditions are specified and explained in the corresponding divisions of mathematical physics. In general, partial differential equations are used to describe various physical phenomena. They can also be used to successfully model most complex phenomena and processes: diffusion, hydrodynamics, quantum mechanics, ecology, and so on. There is a widely held opinion that these equations can be used to explain everything without exception, and that all future world science successes will be directly related to correctly writing out and solving partial derivative equations.

As a rule, different types of partial differential equations require using different, and sometimes quite specific, numerical algorithms. Moreover, some equations can be solved using substantially different numerical methods for different parameters, in particular for those that define the equation s nonlinearity. It is, therefore, almost impossible...