1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

174 Chapter 2 The Heat Equation

(a) (b)

Figure 7 Solution of Eqs. (1)–(4) withT 0 =20,T 1 =100, andf(x)=0. Graphs
(a) and (b) correspond toκ/ha= 0 .1andκ/ha= 1 .0, respectively. In each case,
u(x,t)is graphed as a function ofxfor times chosen so that the dimensionless
timekt/a^2 takes on the values 0.001, 0.01, 0.1, 1. Note that both the temperature
and its slope at the right end(x=a)change with time, so the boundary condition
Eq. (3) is satisfied. See animations on the CD.

EXERCISES

1.Sketchv(x)asgiveninEq.(5)assuming

a. T 1 >T 0 ; b. T 1 =T 0 ; c. T 1 <T 0.

2.IfT 1 >T 0 ,asinFig.7,whatisthemaximumvalueofthetemperature

u(x,t)on the interval 0≤x≤aat any fixed timet?Thesolutionwillbea

function ofT 0 ,T 1 andz=κ/ha.

3.Whyhaveweignoredthenegativesolutionsoftheequation

tan(λa)=−κλ

h

?

4.Derive the formula Eq. (12) for the coefficientsbm.

5.Sketch the first two eigenfunctions of this example takingκ/h= 0 .5(λ 1 =

2. 29 /a,λ 2 = 5. 09 /a).

6.Ve r i f y t h a t

∫a

0

sin^2 (λmx)dx=a

2

+κ

h

cos^2 (λma)

2

.