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 $u,u_{x},u_{y},\dots ,u_{xx},u_{yy},\dots$.

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.

Analytical solutions usually involve tricks to reduce a single PDE into a set of ODEs, or to somehow reduce the number of independent variables. This usually requires you to make a LOT of assumptions about your system. If you're looking at a model for a flame, for example, you basically have to throw out all the interesting stuff to get a PDE that you can solve analytically.

Most of the time, modeling realistic problems requires Numerical solution of PDEs. This involves approximating the terms in the differential equation (discretization), and solving the resulting (discretized) equation using a computer program.

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.

Linear, Quasi-Linear, and Non-Linear PDEs

For a given PDE, e.g.

$a\phi _{xx}+b\phi _{xy}+c\phi _{yy}+d\phi _{x}+e\phi _{y}+f\phi =g$

the PDE can be classified as linear, quasi-linear, or non-linear according to the following criteria.

Linear

A PDE is linear if the unknown function, and its derivatives, appear with only powers of 1 (products of the function and partial derivatives are not allowed). This means the coefficients $a(x,y)$, $b(x,y)$, etc. are either constant, or are functions of the PDE's independent variables $x$ and $y$, but do not contain the unknown function itself.

An example of this would be the constant-velocity Burger's equation:

$u_{t}+cu_{x}=0$

The coefficient of the time derivative $u_{t}$ is $1$, and the coefficient of the spatial derivative $u_{x}$ is the constant $c$. $c$ could be a function of space and time, but cannot be a function of $u$.

Quasi-Linear

A PDE is quasi-linear if the coefficients are functions of the independent variables or the unknown function, i.e.

${\begin{aligned}a(x,y,u(x,y))\\b(x,y,u(x,y))\\\dots \\f(x,y,u(x,y))\end{aligned}}$

Nonlinear

Nonlinear PDEs are those which do not fall in either of the above two categories. These can be expressed using the truly general notation,

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

In practice, nonlinear partial differential equations usually contain powers of partial derivatives, e.g. $u_{xx}^{3}$, products of partial derivatives, etc. An excellent example of a nonlinear equation is the Navier-Stokes equation.

Homogeneity and Non-Homogeneity

A PDE is said to be homogeneous of $g=0$; that is, if the constant right hand side (RHS) term is zero.

If $g\neq 0$, then the PDE is non-homogeneous.

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).

First, consider a general second-order PDE of the form:

$a\phi _{xx}+b\phi _{xy}+c\phi _{yy}+d\phi _{x}+e\phi _{y}+f\phi =g$

The classification of this second-order PDE depends only on the second-derivative terms of the equation, so this equation can be rearranged as:

$a\phi _{xx}+b\phi _{xy}+c\phi _{yy}=-d\phi _{x}-e\phi _{y}-e\phi +g=H$

Now, we will consider the characteristic curves

These are real curves through the solution domain, and along these characteristic curves, the second derivatives may not be continuous.

Shock waves and discontinuities may appear along these curves

Now, for a second order PDE, the initial and boundary conditions are specified in terms of the function $u$, and first derivatives of $u$

For a continuous $u$ and for continuous derivatives of $u$:

• Are there any locations where this information would not uniquely determine the solution?
• Are there locations where the second derivatives are discontinuous?

If there is a characteristic curve $C$, it can be parameterized, e.g. using $\tau$. Defining derivatives as parameterized by $\tau$:

${\begin{array}{c}\phi _{x}=p(\tau )\\\phi _{y}=q(\tau )\end{array}}$

and the second derivatives:

${\begin{array}{c}\phi _{xx}=u(\tau )\\\phi _{xy}=v(\tau )\\\phi _{yy}=w(\tau )\end{array}}$

This makes the original differential equation:

$au(\tau )+bv(\tau )+cw(\tau )=H$

and using the chain rule, the derivatives of $p(\tau )$ and $q(\tau )$ can be written along the curve as:

${\begin{array}{c}{\frac {dp}{d\tau }}=u{\frac {dx}{d\tau }}+v{\frac {dy}{d\tau }}\\{\frac {dq}{d\tau }}=v{\frac {dx}{d\tau }}+w{\frac {dy}{d\tau }}\end{array}}$

These differential equations can be written in matrix form, $\mathbf {Ax} =\mathbf {B}$, as follows:

$\left[{\begin{array}{ccc}a&b&c\\{\frac {dx}{d\tau }}&{\frac {dy}{d\tau }}&0\\0&{\frac {dx}{d\tau }}&{\frac {dy}{d\tau }}\end{array}}\right]\left[{\begin{array}{c}u\\v\\w\end{array}}\right]=\left[{\begin{array}{c}H\\{\frac {dp}{d\tau }}\\{\frac {dq}{d\tau }}\end{array}}\right]$

A solution to these equations along the curve $C$ only exists if the matrix has a non-zero determinant. Since we're trying to find the characteristic curves, let the determinant equal zero.

The condition fr a non-unique solution is then:

$a\left({\frac {dy}{d\tau }}\right)^{2}-b\left({\frac {dx}{d\tau }}\right)\left({\frac {dy}{d\tau }}\right)+c\left({\frac {dx}{d\tau }}\right)^{2}=0$

Multiplying by $d\tau ^{2}$ and dividing by $dx^{2}$ leads to the equation:

$ah^{2}-bh+c=0$

where $h={\frac {dy}{dx}}$. This can be solved to yield the condition for non-uniqueness:

$h={\frac {dy}{dx}}=\displaystyle {\frac {b\pm {\sqrt {b^{2}-4ac}}}{2a}}$

The quantity $b^{2}-4ac$ is an important quantity that groups conditions for non-uniqueness into three groups:

1. $b^{2}-4ac>0$ - hyperbolic equation, 2 real characteristic curves exist
2. $b^{2}-4ac=0$ - parabolic equation, 1 real characteristic curve exists
3. $b^{2}-4ac<0$ - elliptic equation, no real characteristic curves exist (only imaginary curves)

Question: can we have an elliptic-hyperbolic equation? Or, a parabolic-hyperbolic equation? Or a hyperbolic-elliptic-parabolic equation?

Another way to ask this question: going back to the original equation...

$a\phi _{xx}+b\phi _{xy}+c\phi _{yy}=-d\phi _{x}-e\phi _{y}-e\phi +g=H$

Are $a,b,c$ constant?

The answer is, NO - we can have mixed partial differential equation types

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,

$u_{t}=ku_{xx}$

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.

Systems of Differential Equations

• equivalence of systems of differential equations and single differential equations
• elimination from linear system with constant coefficients
• determined, overdetermined, underdetermined systems

Applications

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

An excellent example of this is the set of governing equations for combustion.

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

References

• Wikipedia: http://en.wikipedia.org/wiki/Differential_equations

• Solving quasi-linear PDEs - some examples: http://staff.um.edu.mt/jmus1/PDE.pdf

• Lots of PDE solution methodology info: http://eqworld.ipmnet.ru/en/methods/meth-pde.htm