140 Chapter 2 The Heat Equation
u( 0 ,t)=u(a,t), t> 0 , (11)
∂u
∂x(^0 ,t)=∂u
∂x(a,t), t>^0 , (12)both of mixed type.
Many other kinds of boundary conditions exist and are even realizable, but
the four kinds already mentioned here are the most commonly encountered.
Animportantfeaturecommontoallfourtypesisthattheyinvolvealinear
operation on the functionu.
The heat equation, an initial condition, and a boundary condition for each
endformwhatiscalledaninitial value–boundary value problem. For instance,
onepossibleproblemwouldbe
∂^2 u
∂x^2 =1
k∂u
∂t,^0 <x<a,^0 <t, (13)
u( 0 ,t)=T 0 , 0 <t, (14)−κ∂u
∂x(a,t)=h(
u(a,t)−T 1)
, 0 <t, (15)
u(x, 0 )=f(x), 0 <x<a. (16)Notice that the boundary conditions may be of different kinds at different
ends.
Although we shall not prove it, it is true that there is one, and only one,
solution to a complete initial value–boundary value problem.
We have derived the heat equation (4) as a mathematical model for the tem-
perature in a “rod,” suggesting an object that is much longer than it is wide.
The equation applies equally well to a “slab,” an object that is much wider than
it is thick. The important feature is that we may assume in either case that the
temperature varies in only one space direction (along the length of the rod
or the thickness of the slab). In Chapter 5, we derive a multidimensional heat
equation.
It may come as a surprise that the partial differential equations of this sec-
tion have another completely different but equally important physical inter-
pretation. Suppose that a static medium occupies a region of space between
x=0andx=a(a slab!) and that we wish to study the concentrationu,mea-
sured in units of mass per unit volume, of another substance, whose molecules
or atoms can move, or diffuse, through the medium. We assume that the con-
centration is a function ofxandtonly and designateq(x,t)to be the mass
flux([q]=m/tL^2 ). Then the principle of conservation of mass may be applied
to a layer of the medium betweenxandx+ xto obtain the equation
q(x,t)+ xg=q(x+ x,t)+ x∂∂ut(x,t). (17)