Introduction to partial differential equations

See also Introduction to ordinary differential equations

Introduction

What is a differential equation?

Equation that describes rates of change (derivatives) of a function of one or more variables

http://en.wikipedia.org/wiki/Partial_differential_equation - a type of differential equation involving an unknown function or functions of several independent variables and their partial derivatives with respect to those variables

http://mathworld.wolfram.com/PartialDifferentialEquation.html - an equation involving functions and their partial derivatives

Formal definition:

An equation of the form

$ F(x,y,\dots, u, u_{x}, u_{y}, \dots, u_{xx}, u_{xy}, \dots ) = 0 $

where $ F $ is a function of the variables $ x,y,\dots,u,u_{x},u_{y},\dots,u_{xx},u_{yy},\dots $.

We want to find a function $ u(x,y,\dots) $ that will satisfy this equation. The function $ u $ is called the solution of the partial differential equation.

The differential equation is $ n^{th} $ order if the highest order derivative appearing in $ F $ is of degree $ n $.

The differential equation is linear if $ F $ is linear in the variables $ x,y,u,\dots,u_{x},u_{y},\dots,u_{xx},u_{yy},\dots $ and the coefficients depend only on the independent variables $ x,y,\dots $.

The differential equation is quasi-linear if $ F $ is linear in the highest order derivatives (degree $ n $), but the coefficients for all other derivatives (up to degree $ n-1 $) depend on $ u,u_{x},u_{y},\dots,u_{xx},u_{yy},\dots $. (NOT SURE IF I UNDERSTOOD/EXPRESSED THIS CORRECTLY)

Variety of Solutions

An ordinary differential equation does not have a single solution - there are families of solutions, each of which have a different set of integration constants.

For an $ n^{th} $ order ODE, the family of solutions is given by

$ u = \phi(x; c_1, c_2, \dots, c_n) $

and when this is plugged into the differential equation, each of the integration constants disappear.

The way that one reduces the family of solutions to a single solution is by specifying initial and boundary conditions for the differential equation.

Partial differential equations are more complex. Because the solution does not depend on a single independent variable, the family of solutions are distinguished by arbitrary functions.

Example:

Solve the differential equation

$ u_{y} = 0 $

where $ u = u(x,y) $.

So if we're looking for a solution $ u(x,y) $ to this differential equation... This differential equation tells us that the solution doesn't vary with respect to $ y $.

Hence,

$ u = w(x) $

where $ w(x) $ is one of these arbitrary functions we just talked about. If this were an ordinary differential equation, we would know that $ u $ is an arbitrary constant. But since it's a partial differential equation, we know that $ u $ is an arbitrary function.

Methods for Solution

Differential equations can either be solved analytically, or they can be solved numerically.

Analytical solution of PDEs are difficult, and solution techniques often fail for non-trivial PDEs.

Most of the time, modeling realistic problems requires Numerical solution of PDEs.

Classification of PDEs

Partial differential equations can be classified two ways.

Physical classification

Transient problems - these are described by PDEs that have a temporal independent variable, and therefore have a solution that varies with time.

Equilibrium problems - these are described by PDEs that have no temporal independent variable, and the solution is fixed and is determined by the boundary conditions.

Mathematical classification

There are three classes of PDEs important to the discussion: hyperbolic, parabolic, and elliptic.

Hyperbolic Equations

Hyperbolic PDEs have wave-like solutions. If a disturbance is made in the initial data, it is not immediately felt in the entire domain (and after some period of time, the disturbance will no longer influence the solution at certain locations in the domain). The disturbance travels along the characteristics of the equation.

An example of a model hyperbolic PDE is the wave equation.

Parabolic Equations

Parabolic PDEs are associated with diffusion processes. The solutions to parabolic PDEs exhibit diffusion-like behavior. These differ from hyperbolic equations in their range of influence - anything that has happened, anywhere in the domain, may influence the solution at the current time.

An example of a model parabolic PDE is the heat (diffusion) equation.

Elliptic Equations

Elliptic PDEs are used to model equilibrium problems. These problems describe a domain, and the problem solution must satisfy the boundary conditions at all boundaries.

An example of a model elliptic PDE is the Laplace equation or the Poisson equation.

Rigorous Mathematical Criteria

The mathematical classification of PDEs is based on the concept of characteristic curves (see wikipedia:Method of characteristics).

Consider a general second-order PDE of the form:

$ A u_{xx} + B u_{xy} + C u_{yy} + D u_{x} + E u_{y} + F u = G $

and a matrix $ Z $,

$ Z = \left| \begin{array}{cc} A & B \\ B & C \end{array} \right| $

The equation is hyperbolic if $ det(Z) < 0 $.

The equation is parabolic if $ det(Z) = 0 $.

The equation is elliptic if $ det(Z) > 0 $.

It is important to note that the coefficients $ A,B,C $ are not constant and may change throughout the domain of the problem.

Model Partial Differential Equations

There are a set of canonical PDEs that serve as models for each of the three mathematical classifications above.

Model Hyperbolic PDEs

A model hyperbolic PDE is the second order wave equation, written for a scalar function $ u(x_{1}, x_{2}, x_{3}, \dots, x_{n}, t) $ that satisfies:

$ u_{tt} = c^{2} \nabla^{2} u $

where c is the (constant) propagation speed of the wave.

An analytical solution can be found by combination of variables. See Analytical solution of PDEs.

Model Parabolic PDEs

A model parabolic PDE is the 1-D heat equation,

$ u_{t} = k u_{xx} $

An analytical solution to this equation can be found by combination of variables. See Analytical solution of PDEs.

Model Elliptic PDEs

A model elliptic PDE is the Laplace equation,

$ u_{xx} + u_{yy} = 0 $

An analytical solution to this equation can be found by separation of variables. See Analytical solution of PDEs.

Applications

Partial differential equations, needless to say, are extremely useful for describing physical phenomena.

An excellent example of this are the governing equations for combustion.

Using this set of partial differential equations, it is possible to describe the dynamics of a combusting system.