Myint Tyn U., Debnath L. Linear Partial Differential Equations for Scientists and Engineers

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10.13 Exercises 401

(x, y, z, 0) = 0,

u (0,y,z,t)=u (1,y,z,t)=0,

u (x , 0,z,t)=u (x, 1,z,t)=0,

u (x , y, 0,t)=u (x, y, 1,t)=0.

17. Solve

+ ku

= c

∇

u, 0 <x<a, 0 <y<b, 0 <z<d, t>0,

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

(x, y, z, 0) = g (x, y, z) ,

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

u (x , 0,z,t)=u (x, b, z, t)=0,

u (x , y, 0,t)=u (x, y, d, t)=0.

18. Obtain the solution of the problem for t>0,

= c



θθ

+ u



,r<a,0 <θ<2π, 0 <z<l,

u (r, θ, z, 0) = f (r, θ, z) ,u

(r, θ, z, 0) = g (r, θ, z) ,

u (a , θ, z, t)=0,u(r, θ, 0,t)=u (r, θ, l, t)=0.

19. Determine the solution of th e heat conduction problem

= k ∇

u, 0 <x<a, 0 <y<b, 0 <z<c, t>0,

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

(0,y,z,t)=u

(a, y, z, t)=0,

(x, 0,z,t)=u

(x, b, z, t)=0,

(x, y, 0,t)=u

(x, y, c, t)=0.

402 10 Higher-Dimensional Boundary-Value Problems

20. Solve the problem

= k ∇

u, r < a, 0 <θ<2π, 0 <z<l, t>0,

u (r, θ, z, 0) = f (r, θ, z) ,

(a, θ, z, t)=0,

u (r, θ, 0,t)=u (r, θ, l, t)=0.

21. Find the temperature distribution in the section of a sphere cut out

by the cone θ = θ

. The surface temperature is zero while the initial

temperature i s given by f (r, θ, ϕ).

22. Solve the initial boundary-value problem

= c

∇

u + F (x, y, t) , 0 <x<a, 0 <y<b, t>0,

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

(x, y, 0) = g (x, y) ,

(0,y,t)=u

(a, y, t) = 0 for all t>0,

(x, 0,t)=u

(x, b, t) = 0 for all t>0.

23. Solve the problem

= c

∇

u + xy sin t, 0 <x<π, 0 <y<π, t>0,

u (x , y, 0) = 0,u

(x, y, 0) = 0,

u (0,y,t)=u (π, y, t)=0,

u (x , 0,t)=u (x, π, t)=0.

24. Solve

= k∇

u + F (x, y, z, t) , 0 <x<a, 0 <y<b, 0 <z<c,

t>0,

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

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

10.13 Exercises 403

u (x , 0,z,t)=u (x, b, z, t)=0,

(x, y, 0,t)=u

(x, y, c, t)=0.

25. Solve the nonhomogeneous diﬀusion problem

= k ∇

u + A, 0 <x<π, 0 <y<π, t>0,

u (x , y, 0) = 0,

u (0,y,t)=u (π, y, t)=0,

(x, 0,t)+u (x, 0,t)=0,

(x, π, t)+u (x, π, t)=0,

where k and A are constants.

26. Find the temperature distribution of the composite cylinder consisting

of an inner cylin der 0 ≤ r ≤ r

and an outer cylin dri cal tube r

≤ r ≤ a.

The surface temperature is maintained at zero degrees, and the initial

temperature d istrib uti on is given by f (r, θ, z) .

27. Solve the initial boundary-value problem

− c

∇

u =0, 0 <x<π, 0 <y<π, t>0,

u (x , y, 0) = 0,

u (0,y,t)=u (π, y, t)=0,

u (x , 0,t)=x (x − π)sint, u (x, π, t)=0, 0 ≤ x ≤ π, t ≥ 0.

28. Solve the problem

= c

∇

u, r < a, 0 <θ<2π, t > 0,

u (r, θ, 0) = f (r, θ) ,

(r, θ, 0) = g (r, θ) ,

u (a , θ, t)=p (θ, t) .

404 10 Higher-Dimensional Boundary-Value Problems

29. Solve

= c

∇

u, r < a, t > 0,

u (r, θ, 0) = f (r, θ) ,u

(a, θ, t)=g (θ, t) , 0 <θ<π.

30. Determine the solution of th e biharmonic equation

∇

u = q/D

with the boundary conditions

u (x , 0) = u (x, b)=0,



−



= u





=0,



−



= u





=0,

(x, 0) = u

(x, b)=0,

where q is the load per unit area and D is the ﬂexural rigidity of the

plate. This is the problem of the deﬂection of a uniformly loaded plate,

the sides of which are simply supported.

31. (a) Show that the solution of the one-dimensional Schr¨odinger equation

for a free particle of mass M



i



ψ (x, t)=





exp



−



,b=



it



where a is an integrating constant that can be determined from the

initial value of the wave function ψ (x, t), and N is also a constant

that can be determined from the normalization of the pr obab ility (wave

function) of ﬁnding the particle.

(b) Show that the Gaussian probability density is

|ψ|

= ψψ

∗

|N|

exp



−



and its mean width is

δ =

√

,C=







10.13 Exercises 405

32. Analogous to Example 10.10.1, solve the problem for a ﬁnite square

well potential (see Figure 10.10.1) with a ﬁnite valu e for the height of

the potential given as

V (x)=

⎧

⎨

⎩

0, for −a ≤ x ≤ a

, for x ≤−a, x ≥ a.

33. Consider the quantum mechanical problem described by the one-dimensional

Schr¨odinger equation

+ k

ψ =0

where the wavenumber k =





2M (E − V ) in the rectangular poten-

tial barrier of height V

and width 2a,and

V (x)=V

H (a −|x|) ,

where H is the Heaviside unit step function. The particle is free f or

x<−a and x>a,andV (x) is an even function; the case V

>Eis of

great interest here.

Show that the general solution of the Schr¨odinger equation for V

ψ (x)=

⎧

⎪

⎨

⎪

⎩

ikx

+ Be

−ikx

,x≤−a,

−κx

+ De

+κx

, −a ≤ x ≤ a,

ikx

+ Ge

−ikx

,x≥ a,

where k =

√

2ME and κ =



2M (V

− E).

Matching the boundary conditions at x = −a, show that

⎡

⎣

⎤

⎦

⎡

⎣



iκ



exp (κa + ika)



1 −

iκ



exp (−κa + ika)



1 −

iκ



exp (κa − ika)



iκ



exp (−κa − ika)

⎤

⎦

⎡

⎣

⎤

⎦

where [ ] denotes a matrix.

Using the matching conditions at x = a, show that

⎡

⎣

⎤

⎦

⎡

⎣



1 −



exp (aκ + iak)





exp (aκ − iak)





exp (−aκ + iak)



1 −



exp (−aκ − ika)

⎤

⎦

⎡

⎣

⎤

⎦

Hence, deduce

⎡

⎣

⎤

⎦

⎡

⎣



cosh 2aκ +

iε

sinh 2aκ



2iak

(iη)sinh2aκ

−

(iη)sinh2aκ



cosh 2aκ −

iε sinh 2aκ



−2ika

⎤

⎦

⎡

⎣

⎤

⎦

where ε =



−



and η =





Green’s Functions and Boundary-Value

Problems

“Potential theory has developed out of the vector anal ysis created by Gauss,

Green, and Kelvin for the mathematical theories of gravitational attraction,

of electrostatics and of th e hydrodynamics of perfect ﬂuids (i.e., incom-

pressible and inviscid ﬂuids). The ﬁrst stage of abstraction was the study

of harmonic functions, i.e., potential functions in space free from masses,

charges, sources, or sinks. This led to the inspired intuition of Dirichlet and

the early attempts to justify his ‘principle’.”

George Temple

11.1 Introduction

Boundary-value problems associated with either ordinary or partial dif-

ferential equations arise most frequently in mathematics, mathematical

physics and engineering science. The linear superposition principle is one of

the most elegant and eﬀective methods to represent solutions of boundary-

value problems in terms of an auxiliary function known as Green’s function.

Such a function was ﬁrst introduced by George Green as early as 1828. Sub-

sequently, the method of Green’s functions became a very useful analytical

method in mathematics and in many of the applied sciences.

In previous chapters, it h as been shown that the eigenfunction method

can eﬀectively b e used to express the solutions of diﬀerential equations as

inﬁnite series. On the other hand, solutions of diﬀerential equations can be

obtained as an integral superposition in terms of Green’s functions. So the

method of Green’s functions oﬀers several advantages over eigenfunction ex-

pansions. First, an integral representation of solutions provides a direct way

of describing the general analytical structure of a solution that may be ob-

scured by an inﬁnite series representation. Second, from an analytical point

of view, the evaluation of a solution from an integral representation may

prove simpler than ﬁnding the sum of an inﬁnite series, particularly near

408 11 Green’s Functions and Boundary-Value Problems

rapidly-varying features of a function, where the convergence of an eigen-

function expansion may be slow. Third, in view of the Gibbs phenomenon

discussed in Chapter 6, the integral representation seems to impose less

stringent requirements on the functions that describe the values that the

solution must assume on a given boundary than the expansion based on

eigenfunctions.

Many physical problems are described by second-order nonhomogeneous

diﬀerential equations with homogeneous boundary conditions or by second-

order homogeneous equations with nonhomogeneous boundary conditions.

Such problems can be solved by the method of Green’s functions.

We consider a nonhomogeneous partial diﬀerential equation of the form

u (x)=f (x) , (11.1.1)

where x =(x, y, z) is a vector in three (or higher) dimensions, L

is a

linear partial diﬀerential operator in three or more independent variables

with constant coeﬃcients, and u (x)andf (x) are fu nctions of th ree or

more independent variables. The Green’s function G (x,ξ) of this problem

satisﬁes the equation

G (x,ξ)=δ (x − ξ) (11.1.2)

and represents the eﬀect at the point x of the Dirac delta function source

at the point ξ =(ξ, η, ζ ).

Multiplying (11.1.2) by f (ξ) and integrating over the volume V of the

ξ space, so that dV = dξ dη dζ, we obtain



G (x,ξ) f (ξ) dξ =



δ (x − ξ) f (ξ) dξ = f (x) . (11.1.3)

Interchanging the order of the operator L

and integral sign in (11.1.3)

gives





G (x,ξ) f (ξ) dξ



= f (x) . (11.1.4)

A simple comparison of (11.1.4) with (11.1.1) leads to the solution of

(11.1.1) in the form

u (x)=



G (x,ξ) f (ξ) dξ. (11.1.5)

Clearly, (11.1.5) is valid for any ﬁnite number of components of x.Ac-

cordingly, the Green’s function method can be applied, in general, to any

linear, constant coeﬃcient, nonhomogeneous partial diﬀerential equation in

any number of independent variables.

Another way to approach the p rob lem is by looking for the inverse

operator L

−1

. If it is possible to ﬁ nd L

−1

, then the solution of (11.1.1) can

11.2 The Dirac Delta Function 409

be obtained as u (x)=L

−1

(f (x)). It turns out that in many important

cases it is possible, and the inverse operator can be expressed as an integral

operator of the form

u (x)=L

−1

(f (ξ)) =



G (x,ξ) f (ξ) dξ. (11.1.6)

The kernel G (x,ξ) is called the Green’s function which is, in fact, the char-

acteristic of the operator L

for any ﬁnite number of independent variables.

In our study of partial diﬀerential equations with the aid of Green’s

functions, special attention will be given to those three partial diﬀeren-

tial equations which occur most frequently in mathematics, mathematical

physics and engineering science; the wave equation

− c

∇

u = f (x) , (11.1.7)

the heat or diﬀusion equation

− κ∇

u = f (x) , (11.1.8)

and the potential or the Laplace equation

∇

u = f (x) , (11.1.9)

where the Laplacian ∇

in an n-dimensional Euclidean space is given by

∇

≡

∂

∂x

∂

∂x

+ ...+

∂

∂x

, (11.1.10)

and x =(x

,...,x

Clearly, the solutions of the wave and heat equations are functions of

(n + 1) coordinates consisting of n space dimensions, x =(x

,...,x

)

and one time dimension t, whereas the solutions of the Laplace equation

are functions of n space dimensions.

This chapter deals with the basic idea and properties of Green’s func-

tions and how to construct such functions for ﬁnding solutions of partial

diﬀerential equations. Some examples of applications are provided i n this

chapter and in the next chapter.

11.2 The Dirac Delta Function

The application of Green’s functions to boundary-value problems in ordi-

nary diﬀerential equations was described earlier in Chapter 8. The Green’s

function method is applied here to boundary-value problems in partial dif -

ferential equation s. The method provides solutions in integral form and is

applicable to a wide class of problems in applied mathematics and mathe-

matical physics.

410 11 Green’s Functions and Boundary-Value Problems

Before developing the method of Green’s functions, we will ﬁrst deﬁne

the Dirac delta function δ (x − ξ, y − η) in two dimensions by

(a) δ (x − ξ, y − η)=0,x= ξ, y = η, (11.2.1)

(b)



δ (x − ξ, y − η) dx dy =1,R

:(x − ξ)

+(y − η)

<ε

(11.2.2)

(c)



F (x, y) δ (x − ξ,y − η) dx dy = F (ξ, η) , (11.2.3)

for arbitrary continuous function F in the region R

The delta function is not a function in the ordinary sense. For an elegant

treatment of the delta function as a generalized function, see L. Schwartz,

Th´eorie des Dis tributions (1950, 1951). It is a symbolic function, and is

often viewed as the limit of a distribution.

If δ (x − ξ)andδ (y − η) are one-dimensional delta functions, we have



F (x, y) δ (x − ξ) δ (y − η) dx dy = F (ξ,η) . (11.2.4)

Since (11.2.3) and (11.2.4) hold for an arbitrary continuous function F ,we

conclude that

δ (x − ξ, y − η)=δ (x − ξ) δ (y − η) . (11.2.5)

Thus, we may state that the two-dimensional delta function is the product

of two one-dimensional delta fun ctions.

Higher dimensional delta functions can be deﬁned in a similar manner.

δ (x

,...,x

)=δ (x

) δ (x

) ...δ(x

) . (11.2.6)

The expression for the δ-function become much more complicated when

we introduce curvilinear coordinates. However, for simplicity, we transform

the two-dimensional delta function from Cartesian coordinates x, y to curvi-

linear coordinates α, β by means of the transformation

x = u (α, β)andy = v (α, β) , (11.2.7)

where u and v are single-valued continuous and diﬀerentiable functions

of their arguments. We assume that under this transformation α = α

and β = β

correspond to x = ξ and y = η respectively. Changing the

coordinates according to (11.2.7), we reduce equation (11.2.4) to



F (u, v) δ (u − ξ) δ (v − η) |J|dα dβ = F (ξ, η) , (11.2.8)

where J is the Jacobian of the transformation deﬁned by