Miscellaneous Exercises 249
time-varying boundary conditionu( 0 ,t)=
sin(
ct
a)
, 0 <t<πca,0 ,πa
c <t.
Sketchu(x,t)as a function ofxat various times.12.Same as Exercise 11, but the boundary condition isu( 0 ,t)=h, for all
t>0.
13.Same as Exercise 11, but the boundary condition is
u( 0 ,t)=
hct
a,^0 <t<a
c,
h( 2 a−ct)
a, a
c<t<^2 a
c,
0 ,^2 ca<t.14.Estimate the lowest eigenvalue of the problem
(
eαxφ′
)′
+λ^2 eαxφ= 0 , 0 <x< 1 ,
φ( 0 )= 0 ,φ( 1 )= 0.(This problem can be solved exactly.)15.Estimate the lowest eigenvalue of the problem
φ′′−xφ+λ^2 φ= 0 , 0 <x< 1 ,
φ( 0 )= 0 ,φ( 1 )= 0.16.Show that the nonlinear wave equation
∂u
∂t+u∂u
∂x+∂^3 u
∂x^3 =^0
(the Korteweg–deVries equation) has, as one solution,u(x,t)= 12 a^2 sech^2(
ax− 4 a^3 t)
.
A wave of this form is called a soliton or solitary wave.17.ThesolutioninExercise16isoftheformu(x,t)=f(x−ct). What is the
functionf, and what is the wave speedc?