Return to computing page for the first course APMA0330
Return to computing page for the second course APMA0340
Return to computing page for the fourth course APMA0360
Return to Mathematica tutorial for the first course APMA0330
Return to Mathematica tutorial for the second course APMA0340
Return to Mathematica tutorial for the fourth course APMA0360
Return to the main page for the course APMA0330
Return to the main page for the course APMA0340
Return to the main page for the course APMA0360
Introduction to Linear Algebra with Mathematica

Glossary

Preface

This section presents basic terminology associated with partial differential equations (PDEs for short) and then describes how PDEs are clasifed.

 

Terminology

Suppose u is a funcction of the spacial variable x and time t so that u = u(x, t). Recall tha the partial derivative of u with respect to x is defined as
\[ \frac{\partial u}{\partial x} = \lim_{h\to 0} \frac{u(x+h, t) - u(x,t)}{h} \]
The partial derivative of u with respect to t is defined in a similar way. Another way of representing the partial derivative of u with respect to x is
\[ \frac{\partial u}{\partial x} = u_x \]
A partial differential equation (PDE for shot) is an equation involving one or more partial derivatives of a dependent vvariable.
An example of a one-dimensional equation is a wave equation
\[ \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} \qquad \mbox{and} \qquad u_{tt} = c^2 u_{xx} , \]
where c os a positive constant. This equation involves second partial derivatives:
\[ \frac{\partial^2 u}{\partial t^2} = \lim_{\tau\to 0} \frac{u(x, t+ 2\tau ) - 2\,u(x, t+ \tau ) + u(x, t)}{\tau^2} , \qquad \mbox{or} \qquad \frac{\partial^2 u}{\partial x^2} = \lim_{h\to 0} \frac{u(x+ 2t,t) - 2\,u(x+h,t) + y(x,t)}{h^2} . \]
This partial differential equation is of the second order because its highest derivative is of order two. It is common for the dependent variable u to be a function of three spacial variables x, y, and z, as well as time t.

 

Classification

The general form of linear partial differential equation with two independent variables of the second order is
\begin{equation} \label{EqTerm.1} A\,u_{xx} + B\,u_{xy} + C\,u_{yy} + D\,u_x + E\, u_y + F\, u = Q . \end{equation}

The partial differential equation \eqref{EqTerm.1} is said to be linear if each coefficients in Eq.\eqref{EqTerm.1} is at most a function of x or y or both. Otherwise, the equation is nonlinear. A nonlinear PDE is quasilinear if it is linear in its highest order derivatives. If the term Q on the right-hand side of Eq.\eqref{EqTerm.1} is zero, the PDE is said to be homogeneous. Otherwise, the equation is classified as nonhomogeneous or inhomogeneous.

Suppose that PDE \eqref{EqTerm.1} is linear. It can be classified further as follows. The equation is said to be parabolic if B² −4AC = 0. Heat flow and diffusion problems are typically described by parabolic forms. Hyperbolic forms equations for whuch B² −4AC > 0. Common eqiations of this type are associated with vibrating system and wave motion. Finally, when B² −4AC < 0, the equation is elliptic. Equations of this type typically represent steady-state (time independent) phenomena.

PDEs: Order and Linearity
PDE Order Linearity
uxuy + 3 ux = u 1st nonlinear
uxx + u ux = x 2nd quasilinear
x ux + y uy = u² + x² 1st quasilinear
x ux + y uy = 1 + x² 1st linear
uxx + uyy = sin(x + y) 2nd linear
u uxx + y uyy = 0 2nd nonlinear
x uxx + u uy = 0 2nd quasilinear

If any of the coefficients A, B or C are functions of x or y, classification of Eq.\eqref{EqTerm.1} as parabolic, elliptic, or hyperbolic may be a function of location in the xy-plane. For instance, consider the PDE

\[ x^2 u_{xx} + x\,u_{xy} + 2\,u_{yy} + u_x + u_y = 0 . \]
The expression B² − 4 AC is equal to x² − 8y². Consequently, the equation is parabolic on the curve x² = 8y², hyperbolic for points (x, y) such that x² > 8y², and elliptic for ordered pairs (x, y) that satisfy x² < 8y².

When the dependent variable u is a function of two independent variables, as indicated in Eq.\eqref{EqTerm.1}, there are three common types of PDEs. The Laplace equation uxx + uyy = 0 is elliptic on the entire xy-plane. The heat or diffusion equation ut − αuxx = 0, where the independent variable t represents time and x represents spacial variable. The heat equation is parabolic on th entire tx-plane when thermal diffusivity α is a positive constant. The wave equation utt − c²uxx = 0 is hyperbolic on the entire tx-plane. Here the dependent variable u represents the displacement or wave height as a function of time t and location x.

It can be shown that with a smooth, nonsingular change of variables, the sign of the discriminant B² − 4 AC remains teh same. Therefore, it is possible to make a change of coordinates in such a way that the elliptic PDE is transforms to a Laplace equation, a parabolic PDE is transformed to the heat equation. and a hyperbolic equation is transformed to the wave equation. Hence, the Laplace, heat, and wave equations are referred to as the canonical form in their representative category.

The following list gives examples of PDEs that are common to physical applications.

  1. One-dimensional wave equation:    utt − c²uxx = 0, where u(x, t) represents the displacement of vibrating string from its equilibrium or initial position, and c is the speed of the wave.
  2. Two-dimensional wave equation:    utt = c²(uxx + uyy), where u(x, y, t) represents the displacement of vibrating membrane from its equilibrium or initial position, and c is the speed of the wave.
  3. One-dimensional heat or diffusion equation:    ut = α uxx, where u(x, t) represents the temperature of a solid body as a function of time t and spacial variable x, and α is the thermal diffusivity.
  4. Two-dimensional (2D) heat equation:    ut = α (uxx + uyy) = α ∇²u, where u(x, t) represents the temperature of a solid body, and ∇² is the Laplace operator.
  5. One-dimensional convective-diffusion equation:    θt = α θxx −uθx −vθy, where the dependent variable θ(x, y, t) may be temperature or perhaps a concentration of a given chemical in a fluid, u(x, y, t) is the velocity of the fluid in the x direction, v(x, y, t) is the velocity of the fluid in the y direction,and α is a diffusion coefficient.
  6. Two-dimensional convective-diffusion equation:    θt = α (θxx + θyy) −uθy, where the dependent variable θ(x, t) may be temperature or perhaps a concentration of a given chemical in a fluid, u(t) is the velocity of the fluid, and α is a diffusion coefficient.
  7. The Schrödinger equation (1926) for a single particle moving in three dimensional space:    jℏ Ψt = Ĥ Ψ, where the Hamiltonian operator for a particle of mass m is \( \hat{H} = \frac{1}{2m} \left( \hat{p}_x^2 + \hat{p}_y^2 + \hat{p}_z^2 \right) + V, \quad \) where j is the imaginary unit vector on the complex plane ℂ so j² = −1, ℏ is reduced Planck’s constant, \( \hat{p}_x = -{\bf j}\,\hbar\,\partial_x \) is the momentum operator in x direction, V is the potential energy of the particle (a function of x, y, z, and t), and Ψ is a wave function.
  8. Two-dimensional Laplace's equation:    ∇²u ≡ uxx + uyy = 0, where u(x, y) nay be a stationary temperature of a solid plate.
  9. Two-dimensional Poisson's equation:    ∇²u ≡ uxx + uyy = f(x, y), where u(x, y) may be an electrostatic field property.
  10. One dimensional Berger's equation:    ut + uux = 0, where u(x, t) is the velocity of a stream of particles or fluid flow with zero viscosity.