Heat equation examples The Heat equation is discussed in depth in http://tutorial.math.lamar.edu/classes/de/intropde.aspx, starting on page 6. You may recall Newton s Law of Cooling from Calculus. Just kidding. In case you don t, here goes: An object changes temperature at a rate proportional to the difference between its temperature and the temperature of the surrounding medium. As an equation: dy kt ( 0 y) dt for an object with temperature y(t) in a medium of constant temperature T 0 and thermal constant k. And in case you don t recall how to solve this, you might want to seek help from a professional (hint: better be a professional math teacher, because ln is involved). This N s LoC treats the object as a point body (aren t all Newtonian objects point bodies?) immersed in a uniform medium (aren t all media uniform?) What s different about the heat equation we write as a PDE? The temperature function u(x, t) (or u(x, y, t) in two spatial dimensions) is the temperature within an object as a function of both position and time. Heat flows within an object in a process known as conduction, in response to an internal temperature gradient, which we represent in one dimension by du/dx and in two dimensions as some combination of du/dx and du/dy: Let q(x, t) be the heat flow function (aka heat flux) within the object: In one u dimension, we have the flux equation qxt (, ) for a constant. x Heat flows down the temperature gradient: When the gradient is +, temperature increases with increasing x position and heat flows to the left (towards decreasing x). When the gradient is negative, temperature decreases with increasing x and heat flows to the right.
This gives rise to Fourier s Law of Conduction, which states that the time rate of change in the temperature function u is proportional to the negative gradient of the heat flow, -dq/dx. Putting it all together, we obtain the Heat Equation: uxt (, ) q u k x x or u t = - ku xx. See http://www.math.duke.edu/education/ccp/materials/engin/pdeintro/pde1.html and subsequent pages (accessed from the intro page with the button). Be sure you look through all 4 parts of the exercise; answer the two questions in the summary. In more general terms, the function u(x,t) is a solution to the heat equation if it satisfies u the PDE k q( x, t). The constant k is the thermal diffusivity and the function x q(x, t) represents any external heat source (positive for heat in) or sink (negative for heat out). The function u(x,t) describes the temperature distribution in the one-dimensional material as functions of position and time. Here is a Mathematica expression for solving the one dimensional heat equation for a rod of length 1 over a time interval of 5 sec. In this expression, the initial condition is the temperature function at t = 0, f[x]. A heat supply/sink function q[x,t] (positive for heat flow in, negative for heat flow out) can be specified outside NDSolve[ ]. The thermal diffusivity constant is arbitrarily set at 0.01. The ends of the rod are kept at temperature = 0, as specified by the twin BCs u[0,t] == 0 and u[len,t] == 0. Sample output (q[x] = 0): To produce this output, the initial temperature function f[x] was a half cycle of a sinusoid with a peak temperature of 100 o at x = 0.5. Note how the time-decaying exponential takes over and the temperature
function flattens out. Note the snazzy color scheme. You can also try a different looking color scheme by specifying We can obtain the exponential decay rate, even though u[x, t] is an output of NDSolve[ ]: FindFit[ ] neatly generates the required exponential decay with time, using the temperatures at x =.5, providing a value for the exponent b. Now suppose there is a nonzero q[x] external source function. Let s put a gentle flame at the middle of the bar: Here is the resulting 3D plot of the solution: Not surprisingly, the heat source in the center kept the central temperature from decreasing as rapidly as above. 3
And here our center heat source is turned into a sink, representing heat flow out of the onedimensional system: The temperature now rapidly cools to well below 0! Make sure you see that this is one dimensional (x only) as a function of time! Problems 1. Try a similar heat flow model (cold boundaries, hot middle) for a variety of thermal diffusivities (it was 0.01 in the above).. Investigate different initial temperature distributions with no external heat flow. 3. What happens if there are nonzero q functions? Start with a hot middle and allow positive heat flow. Try one side with positive heat flow, the other side with negative heat flow. Note that we also can produce slices of temperature vs. time: 4
Samples shown; the actual result of GraphicsRow would be three temp vs time graphs, side-by-side. Here is a slice at a constant time: temp as a function of x This is a good place for Animate[ ]. See the worked examples 1-5 on pages 9-44 in Dawkins PDE chapter. Heat equation with a varying boundary condition Evaluate and plot the following pair of two heat equations, paying particular attention to the behavior of solutions u(t, x) at x = 0. sol1= NDSolve[{D[u[t,x],t]==D[u[t,x],{x,}], u[0,x]==0, u[t,0]==sin[t], u[t,5]==0}, u[t,x],{t,0,10},{x,0,5}]//flatten sol=ndsolve[{d[u[t,x],t]==d[u[t,x],{x,}], u[0,x]==1, u[t,0]==cos[t], u[t,5]==1}, u[t,x],{t,0,10},{x,0,5}]//flatten 5
Diffusion The same PDE is also known as the diffusion relationship, governing the rate of spread of a substance throughout a medium: Mixtures of different chemicals, osmosis, as well as the spread of biological characteristics in otherwise stable populations. u D, where D is the diffusivity constant and u(x,t) is the concentration of the x diffusing substance. Note that if the concentration increases in time ( 0 ), the concentration gradient ( x ) must also be increasing ( u ( ) 0). At steady state x x x u (equilibrium), 0 0 const x x Diffusion is usually a smoother-out-er: if you add salt to water, the salt diffuses throughout the water until equilibrium is reached. Coupled diffusion models or How the leopard got its spots (ok, they re giraffes, not leopards) In a two-chemical system, diffusion can produce feedback, leading to unstable systems (or the evolutionary development of zebra stripes or spots on leopards or patches on giraffes). These are known as stimulator-inhibitor models, which have common characteristics: Suppose chemical U 1 produces melanin in the skin and chemical U inhibits it; excess of stimulator also stimulates production of inhibitor; diffusion rates for the two chemicals differ. Let the concentrations of stimulator and inhibitor be given by u 1 = u 1e + 1, u = u e +, where the subscript e represents equilibrium concentration (when 1 = = 0 and 1 0 ). In a linear model, we could specify that for small deviations from 1 equilibrium, au1 bu, cu1 du (for constants a, b, c, d), but this neglects 6
the effect of diffusion. To include differences in diffusion, we have two additional constants and the stimulator-inhibitor diffusion relationship: u u u u x x 1 1 au1 bu D1, cu 1 du D These kinds of equations were first described by Alan Turing (the Alan Turing if you don t know the name, rent a copy of Enigma or go see The Imitation Game) and elaborated by J. Maynard Smith. See http://www.genetics.org/cgi/content/full/168/3/1105. An interesting case occurs when all of the following conditions exist: a > 0 and c > 0: As stimulator concentration rises above equilibrium, the rates of production of both stimulator and inhibitor also increase; b < 0: As inhibitor concentration rises, the rate of production of stimulator decreases; and D > D 1 : Inhibitor diffuses faster than stimulator. Given the right parameters, a small change from equilibrium will lead to pronounced peaks and troughs in the concentrations that do not smooth out with time. Hence: spots on a giraffe! But the solutions of these coupled systems have to wait for finite difference techniques.. 7