1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

Miscellaneous Exercises 207

∂u

∂x(^0 ,t)=0,

∂u

∂x(a,t)=0,^0 <t,

u(x, 0 )=Ta^1 x,0<x<a.

6. ∂

(^2) u
∂x^2

=^1

k

∂u

∂t

,0<x<a,0<t,

u( 0 ,t)=0, u(a,t)=T 0 ,0<t,

u(x, 0 )=0, 0 <x<a.

7.

∂^2 u

∂x^2 =

1

k

∂u

∂t,0<x<a,0<t,

u( 0 ,t)=T 0 , u(a,t)=T 0 ,

u(x, 0 )=T 0 ,0<x<a.

8. ∂

(^2) u
∂x^2

=^1

k

∂u

∂t

,0<x<a,0<t,

∂u

∂x

( 0 ,t)=^ T

a

, ∂u

∂x

(a,t)=^ T

a

,0<t,

u(x, 0 )=T 0 ,0<x<a.

9.

∂^2 u

∂x^2 =

1

k

∂u

∂t,0<x<a,0<t,

u( 0 ,t)=T 0 , ∂u

∂x

(a,t)=0, 0 <t,

u(x, 0 )=T 1 ,0<x<a.

10. 
    

∂^2 u

∂x^2 =

1

k

∂u

∂t,0<x<a,0<t,

∂u

∂x(^0 ,t)=0,

∂u

∂x(a,t)=0,^0 <t,

u(x, 0 )=







T 0 , 0 <x<

a

2 ,

T 1 , a 2 <x<a.

11. ∂

(^2) u
∂x^2 =

1

k

∂u

∂t,0<x<∞,0<t,

u( 0 ,t)=T 0 ,0<t,

u(x, 0 )=T 0

(

1 −e−αx

)

,0<x.

12. 
    

∂^2 u

∂x^2 =

1

k

∂u

∂t,0<x<∞,0<t,