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

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 $.

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

Differential equations can either be solved analytically, i.e. for the general case, or they can be solved numerically

Analytical solutions 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

Ordinary differential equation (ODE) - function of one variable

Partial differential equation (PDE) - function of multiple variables

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. The disturbance travels along the characteristics of the equation.

Parabolic -

Elliptic

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 PDE Solution Techniques.