Skip to the content of the web site.

Lecture 31

Lecture 30 | Lecture 32

The last six lectures of this course are dedicated to finding solutions to partial differential equations. An ordinary differential equation is an equation which has only one independent variable and therefore it is reasonably easy to work with. A partial differential equation, however, equations partial derivatives with multiple variables and therefore there it becomes much more difficult to solve such equations or to even determine if a solution exists.

With ordinary differential equations, we began with 1st-order ODEs followed by 2nd-order ODEs we saw that the tools we used for 2nd-order linear ODEs with constant coefficients could be used for similar ODEs with an arbitrary order; however, for partial differential equations, it is more interesting to look at very specific 2nd-order PDEs; specifically, we will be looking at solutions to

  • Laplace's equation,
  • The heat-conduction/diffusion equation, and
  • The wave equation.

Each of these are 2nd-order PDEs, but to begin, we must begin by defining the Laplacian operator.

Note: to this point, we have used the function tex:$$u$$ to represent the unit step function. By common convention and by the convention in the text book, we will use tex:$$u$$ to represent the unknown solution to our differential equation.

When we consider other forms of energy, again, in the absence of a forcing function, there is a natural response can at most affect the acceleration of a particle and therefore we are, again, required to consider the total acceleration. When we are no longer restricting ourselves to a single dimension, we must consider the sums of the second derivatives in all directions, or

tex:$$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}$$.

Rather than always having to write out the number of required partial derivatives, we abbreviate note that the gradient operator is

tex:$$\vec{\nabla} = \left( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right)$$

and therefore

tex:$$\vec{\nabla}\cdot\vec{\nabla} = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}$$.

We abbreviate the inner product tex:$$\vec{\nabla}\cdot\vec{\nabla} = \nabla^2$$ removing the arrow to emphasize that the result tex:$$\nabla^2 u$$ is a scalar value. This operator is called the Laplacian operator.

Laplace's Equation

The simplest form of Laplacian operator is to have a steady-state solution where the sum of the forces in all directions is zero:

tex:$$\nabla^2u = 0$$.

Note that this is not the equation of a moving particle, but rather, it is the description of a scalar field: something, perhaps a temperature or an electric field, satisfies this equation at each point. For example, consider two infinitely long parallel plates set a unit distance apart where one is maintained at 10°C and the other is maintained at 30°C as shown in Figure 2.

Figure 2. Two infinitely large plates with different temperatures.

Assuming the temperature has settled (or maintained a steady state) we would expect a uniform transition of temperature from 10°C to 30°C across this boundary and we would expect the transition to be the same with any cross section of the plate. Consequently, we may consider such a situation to be one-dimensional tex:$$u(x, y, z) = u(x)$$ (in which case, tex:$$\frac{\partial^2 u}{\partial y^2} = \frac{\partial^2 u}{\partial z^2} = 0$$), in a steady state, the temperature obeys Laplace's equation, or in this case, tex:$$\frac{\partial^2 u}{\partial x^2} = 0$$ which must satisfy the boundary conditions tex:$$u(0) = 10$$ and tex:$$u(1) = 30$$. The obvious solution is the expected solution tex:$$u(x) = 10 + 20x$$.

Unfortunately, things become much more complex in higher dimensions. Suppose you have two concentric pipes, one within the other, and the inner pipe is kept at 100°C while outer pipe is kept at 0°C as shown in Figure 3.

Figure 2. Two infinitely long pipes with different temperatures.

It is no longer reasonable to assume that the temperature will drop linearly as we move from away from the centre of the pipe, and yet physics indicates that the temperature must satisfy Laplace's equation: the concavity in one direction must perfectly match the concavity in the other.

Other more complicated situations, for example, consider a buried pipe where the walls of the pipe are kept at 5°C and through this pipe is flowing water heated at 20°C. What is the air temperature at any point in the pipe? (Assume that there is no turbulence in the water or the air flow.)

In general, Laplace's equation is satisfied by a scalar property within a region and the boundary of that region has the property constrained to specific values. In one dimension, this is nothing more than the boundary-value problems seen previously. In two dimensions, the boundary (curves) defines a enclosed area in the plane, and in three dimensions, the boundary (surfaces) defines an enclosed volume.

As you may have guessed by now, it is impossible in general to find solutions to partial differential equations and there are many numerical tools to assist you in solving such problems, but we will look at some specific examples where we can find solutions so that we can examine and understand the behaviours.

The Heat-conduction/Diffusion Equation

Consider the example of two plates shown in Figure 2. Suppose now that we add a time-varying element. For example, suppose at time tex:$$t = 0$$, we fill the volume between the two plates with air that is 0°C and then we wait. Initially, the air near the plates will begin to heat up, but after some time, the temperature will approach the steady-state scenario.

The equation which describes such a phenomena is the heat-conduction equation or also known as the diffusion equation:

tex:$$\frac{1}{\kappa}\frac{\partial u}{\partial t} = \nabla^2u$$.

What this says is that the rate of change of u is proportional to how far the system is from stability as defined by Laplace's equation. If the system already satisfies Laplace's equation, that is, tex:$$\nabla^2u = 0$$, then tex:$$\frac{1}{\kappa}\frac{\partial u}{\partial t} = 0$$ and therefore there u will be constant with respect to time.

James derives this equation in the one-dimensional case in Section 9.2.2 around page 666.

The Wave Equation

Finally, a third equation which appears often is the wave equation which describes the motion of a wave through a medium. This includes:

  • The plucking of a string (one dimensional),
  • Dropping a stone in the water (two dimensional), and
  • Making a sound wave in air (three dimensional).

In each of these cases, a medium at rest satisfies Laplace's equation: a string is straight, the pond is smooth, and air is uniform and motionless. The introduction of a disturbance causes the medium to begin to move and the propagation of that disturbance is, under appropriate circumstances, called a wave and they usually satisfy the equation

tex:$$\frac{1}{c^2}\frac{\partial^2 u}{\partial t^2} = \nabla^2u$$.

Again, what this equation says is that the acceleration of u is proportional to how far the system is away from stability as defined by Laplace's equation tex:$$\nabla^2u = 0$$.