1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

486 Answers to Odd-Numbered Exercises

5. a.

ω

ω^2 −π^2

( 1

πsin(πt)−

1

ωsin(ωt)

)

sin(πx);

b.

1

2 π^2

(

sin(πt)−πtcos(πt)

)

sin(πx).

7. a.u(x,t)=x−sin(

√

ax)

sin(√a)e

−at+^2 a

π

∑∞

1

sin(nπx)exp(−n^2 π^2 t)

n(a−n^2 π^2 )cos(nπ);

b. The term−

xcos(nπx)

cos(nπ) exp

(

−n^2 π^2 t

)

arises.

Chapter 6 Miscellaneous Exercises

1.U(s)=

T 0

γ^2 +s+

γ^2 T

s(γ^2 +s),

u(x,t)=T 0 exp

(

−γ^2 t

)

+T

(

1 −exp

(

−γ^2 t

))

.

3.U(s)=

cosh(√sx)

s^2 cosh(√s),

u(x,t)=t−^1 −x

2

2

+

∑∞

n= 1

2cos(ρnx)

ρn^3 sin(ρn)

exp

(

−ρn^2 t

)

,

whereρn=(^2 n− 21 )π.

5.u(x,t)=x(^12 −x)−

∑∞

n= 1

4cos

(

ρn(x−^12 )

)

ρnsin(ρn/ 2 ) exp

(

−ρn^2 t

)

,

whereρn=( 2 n− 1 )π.

7.u(x,t)=x+

∑∞

1

2sin(nπx)

nπcos(nπ)exp

(

−n^2 π^2 t

)

.

9.U(x,s)=^1

s

(

1 −exp

(

−√sx

))

.

11.f(t)=√x

4 πt^3

exp

(

−x^2

4 t

)

.

13.u(x,t)=

∑∞

n= 0

[

erfc

( 2 n+ 1 −x

√

4 t

)

−erfc

( 2 n+ 1 +x

√

4 t

)]

.

15.F(s)= 2

∑∞

n= 1

1

s^2 +n^2

.