Suppose that represents a one dimensional population density and assume that the individuals in this population move in the x direction so as to avoid crowding. That is, the flux of population is proportional to the negative of the gradient of population density,
In this flux expression, the proportionality factor determines the rate at which population will relocate in response to a unit gradient in population density. For this example, we have , so the conservation law Equation 7 takes the form
For different physical settings, a variety of different expressions for might be assumed. For example, if the rate at which individuals respond to crowding is independent of position, time, and population density, then we might assume that is a constant. If individuals in the population move at a rate that is dependent on position x and time t, then we could take . If individuals respond differently to crowding at low population densities than at high population densities, then it would be appropriate to make a function of density.
Consider one dimensional heat flow in the direction of the x-axis in a heat conducting material with density and specific heat s. The specific heat represents the number of calories required to raise a unit mass of the material a unit degree of temperature. The units of are mass/volume and the units for s are calories/degree/mass. For simplicity, we assume that s and can be treated as functions of at most only position: and . In reality, s and generally depend on temperature u as well. Let r denote the heat density measured in calories/volume. In accordance with Fourier's law of heat conduction, let be the heat flux vector. The units of q are calories/time/area. Here is called the thermal conductivity, and it represents the rate at which heat moves from a hot region to a cold region in response to a unit gradient of temperature. If we assume that is a function of position and temperature, , then the conservation law
leads to the parabolic conservation law
If the material density , the specific heat s, and the thermal conductivity can be treated as constants, then our equation is a constant coefficient linear PDE of second order. If these quantities depend on x or t, but not on u, then this PDE is linear but has variable coefficients. If any of the coefficients are allowed to depend on u, then the PDE is nonlinear.
Analytical methods, using Fourier series integral transforms and superposition, can often be used to develop exact solutions to constant coefficient, linear PDE problems. If the PDE is nonlinear or has variable coefficients, then numerical methods generally must be used to approximate the solution of the PDE. One popular numerical method for dealing with PDEs is introduced in the next section.