## Partial Differential Equations

by AppliedMathematics.info

Partial differential equations (PDEs) are of vast importance in applied mathematics, physics and engineering since so many real physical situations can be modelled by them. PDEs are made up of partial derivatives . PDEs tend to be divided into three categories - hyperbolic, parabolic and elliptic. They can also be categorised in terms of order: the order of a partial differential equation is simply the highest derivative.

Hyperbolic

Standard hyperbolic equations include the wave equation

 ¶2 u ¶t2 = ¶2 u ¶x2 + ¶2 u ¶y2 + ¶2 u ¶z2 (here in 3D space)
and the advection (convection or transport) equation

 ¶u ¶t = ¶u ¶x (here in 1D space) .
These equations are time-dependent; they model the transient movement of signals. The wave equation models acoustic and electromagnetic fields, the advection equation models the translation of waves such as water waves.

Parabolic

The prototypical parabolic partial differential equation is the diffusion equation. As the term suggests, such equations diffusion phenomena in physics, such as the spreading of heat in a conducting material.

 ¶u ¶t = ¶2 u ¶x2 + ¶2 u ¶y2 (here in 2D space)

Worksheet: Diffusion Equation

Elliptic

Elliptic equations are usually used to model steady state phenomena. When the solution to either hyperbolic or parabolic equations are assumed to be invariant with time then they reduce to elliptic equations.

For example the wave equation or the diffusion equation reduce to the Poisson or Laplace equation when the time dependence is removed.
The Laplace equation is

 ¶2 u ¶x2 + ¶2 u ¶y2 + ¶2 u ¶z2 = 0      in 3D space.

Worksheet: Laplace's Equation

The Poisson equation is

 ¶2 u ¶x2 + ¶2 u ¶y2 + ¶2 u ¶z2 = f      in 3D space.

Worksheet: Poisson's Equation

If for example the solution of the wave equation is assumed to be sinusoidal then it reduces to the Helmholtz equation, another important elliptic equation.

The Biharmonic equation is

 ¶4 u ¶x4 + ¶4 u ¶y4 ¶4 u ¶x2 ¶y2 = 0      in 2D space.

Boundary Value Problems and Boundary Conditions

A PDE on its own has a family of solutions. In a practical mathematical model, a PDE often acts as a governing equation within some domain, with conditions prescribed on its boundary (boundary conditions). Such problems are termed boundary value problems, which often have a unique solution.

Order

Observing the equations, we can note that the advection equation is first order. Laplace's equation, Poisson's equation, the Helmholtz equation, the wave equation and the diffusion equation are second order. The biharmonc equation is fourth order.

Laplace Operator or Laplacian

Many of these equations contain the Laplace Operator or Laplacian.

 ¶2 u ¶x2 + ¶2 u ¶y2 + ¶2 u ¶z2 in 3D space.

Worksheet: Laplace Operator

Computational Solution

In some circumstances analytic solutions of partial differential equations can be obtained. However, this is only true of simple forms of the equations and in in simple geometric regions. For other more practical problems it is more useful to seek computational or numerical solutions. There are a number of classes of methods that have been developed for the computational solution of PDEs. Most popular is the finite element method (FEM).

The finite difference method (FDM) is derived more straightforwardly from the PDE and is also very popular. Hyperbolic and parabolic PDEs are often solved using a hybrid of the FEM and FDM; the spatial variables are modelled using the FEM and their variation with time is modelled by the FDM. In transient electromagnetics the FDM is a popular method of solution, but in this application it is usually termed the finite-difference time-domain method (FD-TD).

The boundary element method (BEM) has a much more restricted range of application to PDEs. In general it is mainly applicable to linear elliptic PDEs and it requires the PDE to first be reformulated as an integral equation. However, it has the advantage that the mesh need only cover the boundaries of the domain, rather than the full domain discretisation that is necessary in the FEM and FDM.

The finite volume method (FVM) is an important method in computational fluid dynamics (CFD).