Powers D.L. Boundary Value Problems and Partial Differential Equations

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248 Chapter 3 The Wave Equation

6. Find an analytic (integral) solution of this wave problem

∂

∂x

∂

∂t

, −∞ < x < ∞, 0 < t,

u(x, 0) = f (x),

∂u

∂t

(x, 0) = g(x), −∞ < x < ∞,

with g(x) = 0and

f (x ) =



h, |x| <,

0, |x| >.

7. Sketch the solution of the problem in Exercise 6 at times t = 0, /c,2/c,

3/c.

8. Same as Exercise 7, but f (x) =0and

g(x) =



c, |x|<,

0, |x| >.

9. Let u(x, t) be the solution of the problem

∂

∂x

∂

∂t

, 0 < x, 0 < t,

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

u(x, 0) = f (x),

∂u

∂t

(x, 0) = g(x), 0 < x .

Sketch the solution u(x, t) as a function of x at times t = 0, a/6c , a/2c,

5a/6c,7a/6c.Useg(x) ≡ 0and

f (x) =











3hx

, 0 < x <

3h(a −x)

< x < a,

0, a < x.

10. Same task as in Exercise 9 but

f (x ) =



sin(x), 0 < x <π,

0,π<x

and g(x) = 0. Sketch at times t = 0, π/4c, π/2c,3π/4c, π/c,2π/c.

11. Let u(x, t) be the solution of the wave equation on the semi-inﬁnite in-

terval 0 < x < ∞, with both initial conditions equal to zero but with the

Miscellaneous Exercises 249

time-varying boundary condition

u(0, t) =











sin





, 0 < t <

πa

< t.

Sketch u(x, t) as a function of x at various times.

12. Same as Exercise 11, but the boundary condition is u(0, t) = h, for all

t > 0.

13. Same as Exercise 11, but the boundary condition is

u(0, t) =











hct

, 0 < t <

h(2a −ct)

< t <

< t.

14. Estimate the lowest eigenvalue of the problem



αx







+λ

αx

φ = 0, 0 < x < 1,

φ(0) = 0,φ(1) = 0.

(This problem can be solved exactly.)

15. Estimate the lowest eigenvalue of the problem



−x φ +λ

φ = 0, 0 < x < 1,

φ(0) = 0,φ(1) = 0.

16. Show that the nonlinear wave equation

∂u

∂t

∂u

∂x

∂

∂x

(the Korteweg–deVries equation) has, as one solution,

u(x, t) = 12a

sech



ax −4a



A wave of this form is called a soliton or solitary wave.

17. ThesolutioninExercise16isoftheformu(x, t) = f (x −ct). What is the

function f , and what is the wave speed c?

250 Chapter 3 The Wave Equation

18. For t < 0, water ﬂows steadily through a long pipe connected at x = 0

to a large reservoir and open at x = a to the air. At time t = 0, a valve at

x = a is suddenly closed. Reasonable expressions for the conservation of

momentum and of mass are

∂u

∂t

=−

∂p

∂x

, 0 < x < a, 0 < t, (A)

∂p

∂t

=−c

∂u

∂x

, 0 < x < a, all t, (B)

where p is gauge pressure and u is mass ﬂow rate. If the pipe is rigid,

= K/ρ, the ratio of bulk modulus of water to its density. Show that

both p and u satisfy the wave equation. The phenomenon described here

is called water hammer.

19. Introduce a function v with the deﬁnition u = ∂v/∂x, p =−∂v/∂t.

Show that (A) becomes an identity and that (B) becomes the wave equa-

tion for v.

20. Reasonable boundary and initial conditions for u and p are

u(x, 0) = U

(constant), 0 < x < a,

p(x, 0) = 0, 0 < x < a,

p(0, t) = 0, all t,

u(a, t) = 0, t > 0.

Restate these as conditions on v. Show that the ﬁrst and third equations

may be replaced by

v(x, 0) = U

x, 0 < x < a,

v(0, t) = 0, all t.

21. Solve the problem in Exercise 20; ﬁnd a series form for v(x, t).

22. In many problems involving ﬂuid ﬂow, the combination

∂u

∂t

∂u

∂x

(called the Stokes derivative)appears.HereV is the speed of the ﬂuid in

the x-direction. If V equals u or otherwise depends on u,thisoperatoris

nonlinear and difﬁcult to work with. Let us assume that V is a constant,

so that the operator is linear, and deﬁne new variables

ξ =x +Vt,τ= x −Vt, u(x, t) = v(ξ, τ).

Miscellaneous Exercises 251

Show that

∂u

∂t

∂u

∂x

=2V

∂v

∂ξ

23. Assume that u(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),

∂

∂y



∂u

∂t

∂u

∂x



24. A ﬂuid ﬂows between two parallel plates held at temperature 0. At the

inlet, ﬂuid temperature is T

and initially the ﬂuid is at temperature T

If V is the speed of the ﬂuid in the x-direction, a problem describing the

temperature u(x, y, t) is

∂

∂y



∂u

∂t

∂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 < y < b, 0 < t,

u(x, y, 0) = T

, 0 < x, 0 < y < b.

Make a separation of variables as in Exercise 23. State and solve the eigen-

value problem for Y . Show that

(x, y, t) = φ

(x − Vt) exp



−λ

k(x +Vt)/2V



sin(λ

satisﬁes the partial differential equation and boundary conditions at y =

0andy = b, without restriction of φ

(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

the φ

26. Find all functions φ such that u(x, t) =φ(x −ct) is a solution of the heat

equation,

∂

∂x

∂u

∂t

27. Take the constant c = (1 + i)



ωk/2 in Exercise 26 and show that the

functions

−px

sin(ωt −px), e

−px

cos(ωt − px)

252 Chapter 3 The Wave Equation

can be obtained from φ(x −ct) and φ(x −¯ct ).(Herep =



ω/2k and ¯c is

the complex conjugate of c. Refer to Exercise 6 in Chapter 2, Section 10.)

28. Some nonlinear equations can also result in “traveling wave solutions,”

u(x, t) = φ(x −ct). Show that Fisher’s equation,

∂

∂x

∂u

∂t

−u(1 −u),

has a solution of this form if φ satisﬁes the nonlinear differential equa-

tion



+cφ



+φ(1 −φ) = 0.

29. Show that the function u is a solution of the problem

∂

∂x

∂

∂t

, 0 < x < a, 0 < t,

u(0, t) = 0, u(a, t) = sin(ωt), 0 < t,

provided that the parameters are such that sin(ωa/c) = 0.

u(x, t) =

sin(ωx/c) sin(ωt)

sin(ωa/c)

30. If ω = π c/a, the denominator of the function in Exercise 29 is 0. Show

that, for this value of ω, a function satisfying the wave equation and the

given boundary condition is

u(x, t) =−

sin



πx



cos



πct



−

cos



πx



sin



πct



31. A string in a musical instrument is typically not as ﬂexible as assumed

in Section 1. For such a string, the displacement u may satisfy the partial

differential equation

∂

∂x

−

∂

∂x

∂

∂t

, 0 < x < a, 0 < t,

where c

= T/ρ,asinSection1,and = EI/T, with E = Yo u n g ’s m o d -

ulus for the material, I = second moment of area. (See an elasticity ref-

erence.)

Assuming that u(x, t) = φ(x)T(t), carry out a separation of variable

and ﬁnd the eigenvalue problem for φ. Take the boundary conditions to

Miscellaneous Exercises 253

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

∂

∂x

(0, t) = 0,

∂

∂x

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

32. Show that φ(x) =sin(µx) satisﬁes the eigenvalue problem found in Ex-

ercise 31, provided that µ =nπ/a and

=µ

+µ

where −λ



(t)/c

T(t).

33. The values of λ

are related to the frequencies of vibration of the string

mentioned in Exercise 31. Show that λ

approaches nπ/a for any ﬁxed n

as  approaches 0.

34. The longitudinal vibration of a thin rod has been described by Love (see

Bibliography) with the equation

∂

∂x



∂

∂t

−

∂

∂x

∂t



, 0 < x < a, 0 < t.

Here, u(x, t) is the displacement of the points that are at x when there

is no motion;  = νK

,whereν is Poisson’s ratio and K is the radius of

gyration of a cross section of the rod. Take boundary conditions

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

and separate the variables by assuming u(x, t) = φ(x)T(t).Itwillbeuse-

ful to name φ



/φ =−λ

35. The eigenvalue problem in Exercise 34 is routine. Once that is solved,

ﬁnd the differential equation for T(t), solve it, and determine the fre-

quencies of vibration.

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The Potential

Equation

CHAPTER

4.1 Potential Equation

The equation for the steady-state temperature distribution in two dimensions

(see Chapter 5) is

∂

∂x

∂

∂y

=0.

The same equation describes the equilibrium (time-independent) displace-

ments of a two-dimensional membrane, and so is an important common part

of both the heat and wave equations in two dimensions. Many other physical

phenomena — gravitational and electrostatic potentials, certain ﬂuid ﬂows —

and an important class of functions are described by this equation, thus mak-

ing it one of the most important of mathematics, physics, and engineering.

The analogous equation in three dimensions is

∂

∂x

∂

∂y

∂

∂z

=0.

Either equation may be written ∇

u = 0 and is commonly called the pote ntial

equation or Laplace’s equation.

The solutions of the potential equation (called harmonic functions)have

many interesting properties. An important one, which can be understood in-

tuitively, is the maximum principle: If ∇

u = 0inaregion,thenu cannot

have a relative maximum or minimum inside the region unless u is constant.

(Thus, if ∂u/∂x and ∂u/∂y arebothzeroatsomepoint,itisasaddlepoint.)

If u is thought of as the steady-state temperature distribution in a metal plate,

255

256 Chapter 4 The Potential Equation

it is clear that the temperature cannot be greater at one point than at all other

nearby points. For if such were the case, heat would ﬂow away from the hot

point to cooler points nearby, thus reducing the temperature at the hot point.

But then the temperature would not be unchanging with time. We return to

this matter in Section 4.

A complete boundary value problem consists of the potential equation in a

region plus boundary conditions. These may be of any of the three types

u given,

∂u

∂n

given, or αu +β

∂u

∂n

given

along any section of the boundary. (By ∂u/∂n,wemeanthedirectionalderiv-

ative in the direction normal, or perpendicular, to the boundary.) When u is

speciﬁed along the whole boundary, the problem is called Dirichlet’s problem;

if ∂u/∂n is speciﬁed along the whole boundary, it is Neumann’s problem.The

solutions of Neumann’s problem are not unique, for if u is a solution, so is u

plus a constant.

It is often useful to consider the potential equation in other coordinate sys-

tems. One of the most important is the polar coordinate system, in which the

variables are

r =



,θ=tan

−1





x = r cos(θ), y =r sin(θ).

By convention we require r ≥ 0. We shall deﬁne

u(x, y) = u



r cos(θ), r sin(θ)



=v(r,θ)

and ﬁnd an expression for the Laplacian of u,

∇

u =

∂

∂x

∂

∂y

in terms of v and its derivatives by using the chain rule. The calculations are

elementary but tedious. (See Exercise 7.) The results are

∂

∂x

= cos

(θ)

∂

∂r

−

2sin(θ ) cos(θ )

∂

∂θ∂r

sin

(θ)

∂

∂θ

sin

(θ)

∂v

∂r

2sin(θ ) sin(θ)

∂v

∂θ

∂

∂y

= sin

(θ)

∂

∂r

2sin(θ ) cos(θ )

∂

∂θ∂r

cos

(θ)

∂

∂θ

cos

(θ)

∂v

∂r

−

2sin(θ ) sin(θ)

∂v

∂θ

Chapter 4 The Potential Equation 257

From these equations we easily ﬁnd that the Laplacian in polar coordinates is

∇

v =

∂

∂r

∂v

∂r

∂

∂θ

∂

∂r



∂v

∂r



∂

∂θ

In cylindrical (r,θ,z) coordinates, the Laplacian is

∇

v =

∂

∂r



∂v

∂r



∂

∂θ

∂

∂z

EXERCISES

1. Find a relation among the coefﬁcients of the polynomial

p(x, y) = a +bx +cy +dx

+exy +fy

that makes it satisfy the potential equation. Choose a speciﬁc polynomial

that satisﬁes the equation, and show that, if ∂p/∂x and ∂p/∂y are both zero

at some point, the surface there is saddle shaped.

2. Show that u(x, y) = x

−y

and u(x , y) = xy are solutions of Laplace’s equa-

tion. Sketch the surfaces z = u(x, y). What boundary conditions do these

functionsfulﬁllonthelinesx =0, x = a, y = 0, y = b?

3. If a solution of the potential equation in the square 0 < x < 1, 0 < y < 1

has the form u(x, y) = Y(y)sin(πx), of what form is the function Y?Finda

function Y that makes u(x, y) satisfy the boundary conditions u(x, 0) = 0,

u(x, 1) = sin(π x).

4. Find a function u(x), independent of y, that satisﬁes the potential equation.

5. What functions v(r), independent of θ , satisfy the potential equation in

polar coordinates?

6. Show that r

sin(nθ) and r

cos(nθ) both satisfy the potential equation in

polar coordinates (n = 0, 1, 2,...).

7. Find expressions for the partial derivatives of u with respect to x and y in

terms of derivatives of v with respect to r and θ .

8. If u and v are the x-andy-components of the velocity in a ﬂuid, it can be

shown (under certain assumptions) that these functions satisfy the equa-

tions

∂u

∂x

∂v

∂y

= 0, (A)

∂u

∂x

−

∂v

∂x

= 0. (B)