162 Chapter 2 The Heat Equation
c. Show that the functionu(x,t)=A(kt+x^2 / 2 )+Bxsatisfies the heat
equation for arbitraryAandBand thatAandBcan be chosen to satisfy
the boundary conditions
∂u
∂x( 0 ,t)=S 0 , ∂u
∂x(a,t)=S 1 , 0 <t.What happens tou(x,t)astincreases ifS 0 =S 1?
6.Ve r i f y t h a tun(x,t)in Eq. (8) satisfies the partial differential equation (1)
and the boundary conditions, Eq. (2).
7.State the eigenvalue problem associated with the solution of the heat prob-
lem in Section 3. Also state its solution.
8.Suppose that the functionφ(x)satisfies the relation
φ′′(x)
φ(x)=p^2 > 0.Show that the boundary conditionsφ′( 0 )=0,φ′(a)=0, then forceφ(x)
to be identically 0. Thus, a positive “separation constant” can only lead to
the trivial solution.
9.Refer to Eqs. (9) and (10), which give the solution of the problem stated
in Eqs. (1)–(3). Iffis sectionally continuous, the coefficientsan→0as
n→∞.Fort=t 1 >0, fixed, the solution isu(x,t 1 )=a 0 +∑∞
1anexp(
−λ^2 nkt 1)
cos(λnx)and the coefficients of this cosine series are
An(t 1 )=anexp(
−λ^2 nkt 1)
.
Show thatAn(t 1 )→0 so rapidly asn→∞that the series given in the
preceding converges uniformly 0≤x≤a. (See Chapter 1, Section 4, The-
orem 1.) Show the same for the series that represents
∂^2 u
∂x^2 (x,t^1 ).
10.Sketch the functionsφ 1 ,φ 2 ,andφ 3 and verify graphically that they satisfy
the boundary conditions of Eq. (7).
11.The boundary conditions Eq. (2) require that
∂u
∂x( 0 ,t)= 0 , 0 <t,and similarly atx=a.Doesthismeanthatuis constant atx=0?