204 Chapter 2 The Heat Equation
u( 0 ,t)=Ub, 0 <t,
u(x, 0 )=Ui, 0 <x.
(Hint: What conditions doesu(x,t)−Ubsatisfy?)
9.Assuming thatUb<0andUi>0, the problem in Exercise 8 might be
interpreted as representing the temperature in a freezing lake. (Think of
xas measuring depth from the surface.) Definex(t)as the depth of the
ice–water interface; thenu(x(t),t)=0. Now findx(t)explicitly.
10.Use error functions to solve the problem
∂^2 u
∂x^2 =1
k∂u
∂t, −∞<x<∞,^0 <t,
u(x, 0 )=f(x), −∞<x<∞,
wheref(x)=U 0 forx<0andf(x)=U 1 forx>0.2.13 Comments and References
In about 1810, Fourier made an intensive study of heat conduction problems,
in which he used the product method of solution and developed the idea of
Fourier series. Sturm and Liouville made their clear and simple generalization
of Fourier series in the 1830s. Among modern works,Conduction of Heat in
Solids, by Carslaw and Jaeger, is the standard reference.The Mathematics of
Diffusion,byCrank,andThe Heat Equation, by D.V. Widder, are also useful
references. (See the Bibliography.)
Although we have motivated our study in terms of heat conduction and,
to a lesser extent, by diffusion, many other physical phenomena of interest in
engineering are described by the heat/diffusion equation: for example, voltage
and current in an inductance-free cable and vorticity transport in fluid flow.
The heat/diffusion equation and allied equations are being employed in biol-
ogy to model cell physiology, chemical reactions, nerve impulses, the spread of
populations, and many other phenomena. Two good references areDifferen-
tial Equations and Mathematical Biology, by D.S. Jones and B.D. Sleeman, and
Mathematical Biology,byJ.D.Murray.
The diffusion equation also turns up in some classical problems of probabil-
ity theory, especially the description of Brownian motion. Suppose a particle
moves exactly one step of length xin each time interval t.Thestepmay
be either to the left or to the right, each equally likely. Letui(m)denote the
probability that, at timem t, the particle is at pointi x(m= 0 , 1 , 2 ,...,
i= 0 ,± 1 ,± 2 ,...). In order to arrive at pointi xat time(m+ 1 ) t,the
particle must have been at one of the adjacent points(i± 1 ) xat the preced-
ing timem tand must have moved towardi x. From this, we see that the