460 Answers to Odd-Numbered Exercises
w( 0 ,t)=0, 0 <t,
w(x, 0 )=−C 0 e−ax,0<x;c. w(x,t)=e−a^2 Dt∫∞
0B(λ)sin(λx)e−λ^2 Dtdλ,B(λ)=− 2 C 0 λ/(
π(
λ^2 +a^2))
.
Section 2.11
- Break the interval of integration atx′=0.
3.B(λ)=0,A(λ)=2 T 0 a
π( 1 +λ^2 a^2 ).- The functionu(x,t), as a function ofx, is the famous “bell-shaped” curve.
The smallertis, the more sharply peaked the curve. - In Eq. (3) replace bothf(x′)andu(x,t)by 1.
- Using the integral given, obtain
u(x,t)=π^2∫∞
01
λsin(λx)e−λ^2 ktdλ.Note, however, thatB(λ)= 2 /λπisnotfound using the usual formulas
for Fourier coefficient functions.Section 2.12
- Ast→ 0 +,x/
√
4 πkt→{
+∞ ifx>0,
−∞ ifx<0,so erf(x/√
4 πkt)→{
+1ifx>0,
−1ifx<0.- Make the substitutionx=y^2 .ThenI(x)=√πerf(√x)+c.
- Letzbe defined by erf(z)=−Ub/(Ui−Ub).Thenx(t)=z
√
4 kt.Chapter 2 Miscellaneous Exercises
- SS:v(x)=T 0 ,0<x<a.
EVP:φ′′+λ^2 φ=0,φ( 0 )=0,φ(a)=0,λn=nπ/a,φn=sin(λnx),
n= 1 , 2 ,....