The main entity of the PDE analysis is not a separate equation but a complete PDE problem consisting of an equation, domain of solution, boundary and initial conditions. The problem has to be solved in a spatial domain and, in the case of time-dependency, in a time interval between t(0) and t(end) . tend can be a finite number or infinity. Similarly, the domain may have a finite size or extend to infinity in one or several directions. In numerical simulations, the infinite limits of the spatial or time domain are replaced by sufficiently large finite numbers.Boundary conditions have to be imposed at the boundaries of the spatial domain Ω,
This is necessary not only to account for the effect of real physical boundaries but also from a purely mathematical viewpoint.
Only a problem with properly set boundary conditions is well-posed (i.e., consistent and having a unique solution).The physical boundary conditions are usually expressed in terms of the boundary values of the unknown field u or its normal derivative. Mathematically, the possibilities are: the Dirichlet boundary condition:
the Neumann boundary condition,
the Robin (mixed) boundary condition
and the periodicity condition
g is a known function of space and time defined at the boundary, and L is the length of periodicity, which we, as an example, assume to be in the x-direction. It is mathematically allowed and sometimes required by the physics that the boundary conditions of different types are applied at different parts of the boundary.
If the domain is infinite, special boundary conditions have to be applied at infinity. For example, if the domain extends to Ω, in the
x-direction, the conditions may be
In the problems where the solution is a function of time, initial conditions have to be imposed at t = t(0). Depending on the type of the equation, one or two conditions are necessary. The most common situations are when the solution itself is known:
and when its first time-derivative is known:
Among our model equations, the heat equation, linear convection equation , Burgers equation , and generic transport equation require the initial condition , while the wave equation needs both above two equation.