Chapter 2 The Heat Equation 139
whereαis a function of time. Of course, the case of a constant function is
included here. This type of boundary condition is called aDirichlet condition
orcondition of the first kind.
Another possibility is that the heat flow rate is controlled. Since Fourier’s
law associates the heat flow rate and the gradient of the temperature, we can
write
∂u
∂x(x 0 ,t)=β(t), (7)whereβis a function of time. This is called aNeumann conditionorcondition
of the second kind.Wemostfrequentlytakeβ(t)to be identically zero. Then
the condition
∂u
∂x(x 0 ,t)= 0corresponds to aninsulatedsurface, for this equation says that the heat flow is
zero.
Still another possible boundary condition is
c 1 u(x 0 ,t)+c 2 ∂u
∂x(x 0 ,t)=γ(t), (8)calledthird kindor aRobin condition. This kind of condition can also be real-
ized physically. If the surface atx=ais exposed to air or other fluid, then the
heat conducted up to that surface from inside the rod is carried away by con-
vection. Newton’s law of cooling says that the rate at which heat is transferred
from the body to the fluid is proportional to the difference in temperature
between the body and the fluid. In symbols, we have
q(a,t)=h(
u(a,t)−T(t))
, (9)
whereT(t)is the air temperature. After application of Fourier’s law, this be-
comes
−κ∂u
∂x(a,t)=hu(a,t)−hT(t). (10)This equation can be put into the form of Eq. (8). (Note:his called the con-
vection coefficient or heat transfer coefficient; [h]=H/L^2 tT.)
AlloftheboundaryconditionsgiveninEqs.(6),(7),and(8)involvethe
functionuand/or its derivative at one point. If more than one point is in-
volved, the boundary condition is calledmixed. For example, if a uniform rod
is bent into a ring and the endsx=0andx=aare joined, appropriate bound-
ary conditions would be