9. Partial Differential Equations (PDEs)

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import mmf_setup; mmf_setup.nbinit()
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9. Partial Differential Equations (PDEs)#

The theory of general PDEs is much less well-developed than that for ODEs, so we will restrict our attention to a few special cases that appear in physics. These tend to include linear PDEs up to second order including the following:

  1. Wave equation:

    \[\begin{gather*} \left(\nabla^2 - \frac{1}{c^2}\pdiff[2]{}{t}\right)\psi = 0. \end{gather*}\]

    This is an example of a hyperbolic equation amenable to solution by the method of characteristics with general solutions that move with speed \(c = \norm{\vect{v}}_{2}\):

    \[\begin{gather*} \psi(x, t) = f(\vect{x} - \vect{v} t) \end{gather*}\]

    for any initial shape \(f(\vect{x})\). I.e., the shape does not change.

  2. Laplace’s equation:

    \[\begin{gather*} \nabla^2\psi = 0. \end{gather*}\]

    This is an example of an elliptic equation has the nice property that specifying the solution \(\phi(\vect{x})\) on the boundary of a region \(\partial V\) defines the solution everywhere in the interior. This solution can be found by relaxation. The inhomogeneous form is Poisson’s equation

    \[\begin{gather*} \nabla^2\psi = -\frac{\rho}{\epsilon_0}. \end{gather*}\]

    Example: \(\psi\) is the electrostatic potential due to a spatial charge distribution \(\rho\).

  3. Diffusion equation:

    \[\begin{gather*} \left(\nabla^2 - \frac{1}{\kappa}\pdiff{}{t}\right)\psi = 0. \end{gather*}\]

    Example: If \(\psi\) is the temperature, this is called the heat equation and \(\kappa\) is the thermal conductivity. This is the prototypical example of a parabolic equation.

  4. Schrödinger equation:

    \[\begin{gather*} \left(\frac{-\hbar^2}{2m}\nabla^2 + V - \I\hbar\pdiff{}{t}\right)\psi = 0. \end{gather*}\]

    This is very closely related to the heat equation, but complex coefficients.

Elliptic, Hyperbolic, and Parabolic#

A big problem is to determine the existence and uniqueness of a solution given set of boundary conditions. For second-order linear equations:

\[\begin{gather*} \sum_{ij} \frac{\partial^2 \psi}{\partial x_i\partial x_j} = A_{ij}\partial_{i}\partial_{j}\psi = A_{ij}\psi_{,ij} = f(\vect{x}, \vect{\nabla}\psi), \end{gather*}\]

the nature of the system is given by the eigenvalues of the matrix \(\mat{A}\).

  • Elliptic: All eigenvalues are non-zero and have the same sign.

  • Parabolic One eigenvalue is zero and all others have the same sign.

  • Hyperbolic All eigenvalues are non-zero and one has the opposite sign of the rest.

  • Ultrahyperbolic All eigenvalues are non-zero and more than one has the opposite sign as the rest. (At least two positive and at least two negative.)

Each of these has different requirements for sufficient boundary conditions (see e.g. Table 9.1 in [Arfken et al., 2013]), but you should use your physical intuition to guide you:

  • Elliptic equations are like Laplace’s equation where specifying the function (Dirichlet or Neumann) on a closed boundary suffices.

  • Parabolic equations are like the heat-equation where specifying the boundary conditions on the outside of a box and at some time in the past is sufficient. (I.e. Dirichlet or Neumann conditions on an open boundary fixes the solution in the time-direction.)

  • Hyperbolic equations are like the wave-equation. If you specify the solution and its derivative along an open surface, then this solution will propagate along the characteristics. (Note that Cauchy conditions are needed here since there are two possible solution traveling along orthogonal characteristics. Thus, we need both values and derivatives (perpendicular to the surface) in order to uniquely specify the linear combination of these.)

Inhomogeneous PDEs#

To solve inhomogeneous linear PDE

\[\begin{gather*} \op{\mathcal{L}}\psi(\vect{x}) = f(\vect{x}), \end{gather*}\]

the method of choice is to find a Green’s functions \(G(\vect{x})\) such that

\[\begin{gather*} \op{\mathcal{L}}G(\vect{x}) = \delta(\vect{x}). \end{gather*}\]

An inhomogeneous solution can then be directly constructed by convolution:

\[\begin{gather*} \psi(\vect{x}) = G*f = \int G(\vect{x} - \vect{y})f(\vect{y})\d\vect{y} \end{gather*}\]

and the general solution found by adding the homogeneous solutions. Note: due to this degeneracy, the choice of Green’s function is not unique and is usually motivated by boundary conditions.

Non-linear PDEs#

Non-linear PDEs can often be approximated locally by a linear PDE, allowing for a local characterization of the solution. A fun example is provided by the Traffic Flow model:

\[\begin{gather*} \pdiff{n}{t} + \pdiff{nu(n)}{x} = 0 \end{gather*}\]

where \(u(n)\) is some function characterizing the speed of traffic give a local traffic density \(n\).

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