Miscellaneous Exercises 251
Show that
∂u
∂t+V∂u
∂x= 2 V∂v
∂ξ.
23.Assume thatu(x,y,t)has the product form shown in what follows. Sep-
arate the variables in the given partial differential equation.
u(x,y,t)=ψ(x+Vt)φ(x−Vt)Y(y),
∂^2 u
∂y^2=^1
k(∂u
∂t+V∂u
∂x)
.
24.A fluid flows between two parallel plates held at temperature 0. At the
inlet, fluid temperature isT 0 and initially the fluid is at temperatureT 1.
IfVis the speed of the fluid in thex-direction, a problem describing the
temperatureu(x,y,t)is
∂^2 u
∂y^2 =1
k(∂u
∂t+V∂u
∂x)
, 0 <y<b, 0 <x, 0 <t,u(x, 0 ,t)= 0 , u(x,b,t)= 0 , 0 <x, 0 <t,
u( 0 ,y,t)=T 0 , 0 <y<b, 0 <t,
u(x,y, 0 )=T 1 , 0 <x, 0 <y<b.Make a separation of variables as in Exercise 23. State and solve the eigen-
value problem forY. Show thatun(x,y,t)=φn(x−Vt)exp(
−λ^2 nk(x+Vt)/ 2 V)
sin(λny)satisfies the partial differential equation and boundary conditions aty=
0andy=b, without restriction ofφn(except differentiability).25.Show how to satisfy the initial and inlet conditions in the problem of Ex-
ercise 24, by forming a sum of product solutions and correctly choosing
theφn.
26.Find all functionsφsuch thatu(x,t)=φ(x−ct)is a solution of the heat
equation,
∂^2 u
∂x^2 =
1
k∂u
∂t.27.Take the constantc=( 1 +i)
√
ωk/2 in Exercise 26 and show that the
functions
e−pxsin(ωt−px), e−pxcos(ωt−px)