1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

Miscellaneous Exercises 209

are solutions of the heat equation. (These are sometimes called heat

polynomials.) Find a linear combination of them that satisfies the

boundary conditionsu( 0 ,t)=0,u(a,t)=t.

20.Suppose thatu(x,t)is a positive function that satisfies

∂^2 u

∂x^2 =

∂u

∂t.

Show that the function

w(x,t)=−^2

u

∂u

∂x

satisfies the nonlinear partial differential equation calledBurgers’ equa-

tion:

∂w

∂t +w

∂w

∂x=

∂^2 w

∂x^2.

21.Find a solution of the Burgers’ equation that satisfies the conditions

w( 0 ,t)= 0 ,w( 1 ,t)= 0 , 0 <t,

w(x, 0 )= 1 , 0 <x< 1.

22.Taking the functionu(x,t)given here as a solution of the heat equation
(withk=1), find a solutionwof Burgers’ equation. Verify thatwsatis-
fies Burgers’ equation.

u(x,t)=√^1

4 πt

exp

(

−x^2

4 t

)

.

23.Consider a solid metal bar surrounded by a finite quantity of water con-
fined in a water jacket. If the bar and the water are at different tempera-
tures, they will exchange heat. Letu 1 andu 2 be the temperatures in the
bar and in the water, respectively. Heat balances for the water and the bar
give these two equations:

c 1

du 1

dt =h(u^2 −u^1 ),

c 2 dudt^2 =h(u 1 −u 2 ).

Here,c 1 andc 2 are the heat capacities of the bar and the water, respec-

tively, andhis the product of the convection coefficient with the area

of the bar–water interface. Find temperaturesu 1 andu 2 assuming initial

conditionsu 1 ( 0 )=T 0 ,u 2 ( 0 )=0.