There are a few standard examples of partial differential equations. You must know these by heart.
The Laplace equation governs basic steady heat conduction, among much else.
An example problem is shown in figure 1.1. Physically it is steady heat conduction in a rectangular plate of dimensions . The unknown in this example is the temperature. The independent variables are the Cartesian coordinates and . The domain is the two-dimensional interior of the plate. The boundary is the one-dimensional perimeter of the plate. (The boundary might still be indicated by instead of even though here it is not a surface.)
The Laplace equation also describes ideal flows, unidirectional flows, membranes, electrostatics and magnetostatics, complex functions, and countless other problems.
In any number of dimensions, the Laplace equation reads
Some important properties of the Laplace equation are:
(For domains that extent to infinity, various rules above assume that you consider the infinite domain as the limit of a finite one.)
The Laplace equation is the basic example of what is called an “elliptic” partial differential equation. Solutions of the Laplace equation are called “harmonic functions.”
Derive the Laplace equation for steady heat conduction in a two-dimensional plate of constant thickness . Do so by considering a little Cartesian rectangle of dimensions . A sketch is shown below:
Assume Fourier’s law:
If you want the heat flow through an area element that is not normal to the direction of heat flow, the expression is
Assume that no heat is added to the little rectangle from external sources.
Derive the Laplace equation for steady heat conduction using vector analysis. Assume Fourier’s law as given in the previous question. In vector form
Assume that no heat is added to the solid from external sources.
Consider the Laplace equation within a unit circle:
To find the value of at the point (0.1,0.2), can I just plug in the coordinates of that point into the boundary condition?
There is a symmetry argument that you can give to show that is zero on the entire -axis . What?
Consider the Laplace equation within a unit circle:
Consider the Laplace equation within a unit circle, but now in polar coordinates:
The solution is the Poisson integral formula
Now suppose that function is increased slightly, by an amount , and only in a very small interval .
Does the solution change everywhere in the circle, or only in the immediate vicinity of the interval on the boundary at which was changed. What is the sign of the change in if is positive?
Consider the following Laplace equation problem in a unit square:
The problem as shown has a unique solution. It is relevant to a case of heat conduction in a square plate, with the temperature. Someone proposed that the solution should be simple: in the upper triangle the solution is 0, and in the lower triangle, it is 1.
Thoroughly discuss this proposed solution. Find out whether the boundary conditions and initial conditions are satisfied. Is the partial differential equation satisfied in both triangles?
Explain why all isotherms except 0 and 1 coincide with the 45 line. And why the zero and 1 isotherms are indeterminate.
Finally discuss whether the solution is right.
If the problem of the previous question does not have the proposed solution, then the isotherms are not right either.
Consider the following simpler problem, in which the top and right boundaries have been distorted into a quarter circle:
Also neatly draw versus the polar angle at . In a separate graph, draw the solution proposed in the previous section, for and for , again against at .
Now go back to the problem of the previous question and very neatly sketch the correct , 0.25, 0.5, 0.75, and 1 isotherms for that problem. Pay particular attention to where the 0.25, 0.5, and 0.75 isotherms meet the boundaries and under what angle.
Return once again to the problem of the second-last question.
The correct solution to this problem, that you would find using the so-called method of separation of variables, is:
Now shed some light on the question why this solution is smooth for any arbitrary . To do so, first explain why any sum of sines of the form
Next, you are allowed to make use of the fact that the function is still smooth if the coefficients go to zero quickly enough. In particular, if you can show that
Use this to show that above is indeed infinitely smooth for any . And show that it is not true for , where the solution jumps at the origin.
The heat equation governs basic unsteady heat conduction, among much else.
An example problem is shown in figure 1.2. Physically it is unsteady heat conduction in a bar of length . The unknown is the temperature. The independent variables in this case are the coordinate along the bar and the time . The domain in this example is the bar. Mathematically, that is the line segment with the length of the bar. The boundary consists in this case of a mere two points: and .
The heat equation also describes unsteady viscous unidirectional flows and many other diffusive phenomena.
In any number of dimensions, the heat equation reads
Some important properties of the heat equation are:
The heat equation is the basic example of what is called a “parabolic” partial differential equation.
This is a continuation of a corresponding question in the subsection on the Laplace equation. See there for a definition of terms.
Derive the heat equation for unsteady heat conduction in a two-dimensional plate of thickness , Do so by considering a little Cartesian rectangle of dimensions .
In particular, derive the heat conduction coefficient in terms of the material heat coefficient , the plate thickness , and the specific heat of the solid .
This is a continuation of a corresponding question in the subsection on the Laplace equation. See there for a definition of terms.
Derive the heat equation for unsteady heat conduction using vector analysis.
This equation governs basic vibrations, among much else.
An example problem, vibrations of a string, is shown in figure 1.3. The unknown is the transverse deflection of the string. The independent variables are again and like for the heat equation example. The domain is again the -interval along the string and the boundary is the two end points.
The heat equation also describes acoustics, steady supersonic flow, water waves, optics, electromagnetic waves, and many other basic phenomena characterized by wave propagation.
In any number of dimensions, the wave equation reads
Some important properties of the wave equation are:
The wave equation is the basic example of what is called a “hyperbolic” partial differential equation.
Derive the wave equation for small transverse vibrations of a string by considering a little string segment of length .
Maxwell’s equations for the electromagnetic field in vacuum are
Show that if you know how to solve the standard wave equation, you know how to solve Maxwell’s equations. At least, if the charge and current densities are known.
Identify the wave speed.
Consider the following wave equation problem in a unit square:
This is basically identical to a Laplace equation problem in the first subsection. Like that problem, the above wave equation problem has a unique solution. It is relevant to a case of acoustics in a tube, with the pressure. Someone proposed that the solution should be simple: in the upper triangle the solution is 0, and in the lower triangle, it is 1.
Thoroughly discuss this proposed solution. Find out whether the boundary conditions and initial conditions are satisfied. Is the partial differential equation satisfied in both triangles? Finally discuss whether the solution is right. Consider the value of the wave speed in your answer.
Sketch the isobars of the correct solution. In particular, sketch the 0.25, 0.5, 0.75, and 1 isobars, if possible. Sketch both the case that and that .
Return again to the problem of the last question. Assume .
The correct solution to this problem, that you would find using the so-called method of separation of variables, is:
Explain that it produces the moving jump in the solution as given in the previous question.
The discontinuous solution given in the previous question is right in this case. It is right because it is the proper limiting case of a smooth solution that everywhere satisfies the partial differential equation. In particular, if you sum the above sum for up to a very high, but not infinite value of , you get a smooth solution of the partial differential equation that satisfies all initial and boundary conditions, except that the value of at still shows small deviations from . The more terms you sum, the smaller those deviations become. (There will always be some differences right at the singularity, but these will be restricted to a negligibly small vicinity of .)
Find the possible plane wave solutions for the two-dimensional wave equation
Also find the possible standing wave solutions. Assume homogeneous Dirichlet or Neumann boundary conditions on some rectangle , . What is the frequency?
Repeat for the generalized equation