Introduction to partial 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 $. It is desirable 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 said to be of $ 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 $.

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 - 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.

Parabolic - 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.

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

The definitions are not quantitative, but there are rigorous mathematical conditions that define each class (see wikipedia:differential operators).

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.