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

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7.9 Exercises 271

29. Determine the solution of th e initial boundary-value problem:

− c

= h (x, t) , 0 <x<l, t>0,

u (x , 0) = f (x) , 0 ≤ x ≤ l,

(x, 0) = g (x) , 0 ≤ x ≤ l,

u (0,t)=p (t) ,u

(l, t)=q (t) ,t≥ 0.

30. Determine the solution of th e initial boundary-value problem:

− c

= h (x, t) , 0 <x<l, t>0,

u (x , 0) = f (x) , 0 ≤ x ≤ l,

(x, 0) = g (x) , 0 ≤ x ≤ l,

(0,t)=p (t) ,u

(l, t)=q (t) ,t≥ 0.

31. Solve the problem:

− u

=0, 0 <x<1,t>0,

u (x , 0) = x , u

(x, 0) = 0, 0 ≤ x ≤ 1,

u (0,t)=t

,u(1,t)=cost, t ≥ 0.

32. Solve the problem:

− 4 u

= xt, 0 <x<1,t>0,

u (x , 0) = x , u

(x, 0) = 0, 0 ≤ x ≤ 1,

u (0,t)=0,u

(1,t)=1+t, t ≥ 0.

33. Solve the problem:

− 9 u

=0, 0 <x<1,t>0,

u (x , 0) = sin



πx



(x, 0) = 1 + x, 0 ≤ x ≤ 1,

(0,t)=π/2,u

(1,t)=0,t≥ 0.

34. Find the solution of the problem:

+2ku

− c

=0, 0 <x<l, t>0,

u (x , 0) = 0,u

(x, 0) = 0, 0 ≤ x ≤ l,

(0,t)=0,u(l, t)=h, t ≥ 0,h= constant.

35. Solve the problem:

− c

+ hu = hu

, −π<x<π, t>0,

u (x , 0) = f (x) , −π ≤ x ≤ π,

u (−π, t)=u (π, t) ,u

(−π, t)=u

(π, t) ,t≥ 0,

where h and u

are constants.

272 7 Method of Separation of Variables

36. Prove the uniqueness theorem for the boundary-value problem involving

the Laplace equation:

+ u

=0, 0 <x<a, 0 <y<b,

u (x , 0) = f (x) ,u(x, b)=0, 0 ≤ x ≤ a,

(0,y)=0=u

(a, y) , 0 ≤ y ≤ b.

37. Consider the telegraph equation problem:

− c

+ au

+ bu =0, 0 <x<l, t>0,

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

(x, 0) = g (x) for 0 ≤ x ≤ l,

u (0,t)=0=u (l, t)fort ≥ 0,

where a and b are positive constants.

(a) Show that, for any T>0,





+ c

+ bu



t=T

dx ≤





+ c

+ bu



t=0

dx.

(b) Use th e above integral inequality from (a) to show that the initial

boundary-value problem for the telegraph equation can have only one

solution.

Eigenvalue Problems and Special Functions

“The tool which serves as intermediary between theory and practice, be-

tween thought and observation, is mathematics; it is mathematics which

builds the linking bridges and gives the ever more reliable forms. From this

it has come about that ou r entire contemporary culture, in as much as it

is based on the intellectual penetration and the exploitation of nature, has

its foundations in mathematics.”

David Hilbert

“In 1836/7, he published some important joint work with his friend Sturm

on what became known as Sturm–Liouville theory, which became important

in physics.”

Ioan James

8.1 Sturm–Liouville Systems

In the preceding chapter, we determined the solutions of partial diﬀeren-

tial equations by the method of separation of variables. In this chapter, we

generalize the method of separation of variables and the associated eigen-

value problems. This generalization, usually known as the Sturm–Liouville

theory, greatly extends the scope of the method of separation of variables.

Under separable conditions we transformed the second-order homoge-

neous partial diﬀerential equation into two ordinary diﬀerential equations

(7.2.11) and (7.2.12) which are of the form

(x)

+ a

(x)

+[a

(x)+λ] y =0. (8.1.1)

If we introduce

274 8 Eigenvalue Problems and Special Functions

p (x)=exp





(t)



,q(x)=

(x)

p (x) ,s(x)=

p (x)

(x)

, (8.1.2)

into equation (8.1.1), we obtain





+(q + λs) y =0, (8.1.3)

which is known as the Sturm–Liouville equation. In terms of the Sturm–

Liouville operator

L ≡





+ q,

equation (8.1.3) can be written as

L [y]+λs(x) y =0, (8.1.4)

where λ is a parameter independent of x,andp, q,ands are real-valued

functions of x. To ensure the existence of solutions, we let q and s be

continuous and p be continuously diﬀerentiable in a closed ﬁnite interval

[a, b].

The Stu rm–Liouvi lle equation is called regul ar in the interval [a, b]if

the functions p (x)ands (x) are positive in the interval [a, b]. Thus, for a

given λ, there exist two linearly independent solutions of a regular Sturm–

Liouville equation in the interval [a, b].

The Sturm–Liouville equation

L [y]+λs(x) y =0,a≤ x ≤ b,

together with the separated end conditions

y (a)+a

′

(a)=0,b

y (b)+b

′

(b)=0, (8.1.5)

where a

and a

, and likewise b

and b

, are not both zero and are given

real numbers, is called a regular Sturm–Liouville (RSL) system.

Thevaluesofλ for which the Sturm–Liouville system has a nontrivial

solution are called the eigenvalues, and the corresponding solutions are

called the eigenfunctions. For a regular Stur m–Liouvill e problem, we denote

the domain of L by D (L), that is, D (L) is the space of all complex-valued

functions y deﬁned on [a, b] for which y

′′

∈ L

([a, b]) and which satisfy

boundary conditions (8.1.5).

Example 8.1.1. Consider the regular Sturm–Liouville problem

′′

+ λy =0, 0 ≤ x ≤ π,

y (0) = 0,y

′

(π)=0.

8.1 Sturm–Liouville Systems 275

When λ ≤ 0, it can be readily shown that λ is not an eigenval ue. How-

ever, when λ>0, the solution of the Sturm–Liouville equation is

y (x)=A cos

√

λx+ B sin

√

λx.

Applying the condition y (0) = 0, we obtain A = 0. The condition y

′

(π)=0

yields

√

λ cos

√

λπ=0.

Since λ = 0 and B = 0 yields a trivial solution, we must have

cos

√

λπ=0,B=0.

This equation is satisﬁed if

√

λ =

2n − 1

,n=1, 2, 3,...,

and hence, the eigenvalues are λ

=(2n − 1)

/4, and the corresponding

eigenfunctions are

sin



2n − 1



x, n =1, 2, 3,....

Example 8.1.2. Consider the Euler equation

′′

+ xy

′

+ λu =0, 1 ≤ x ≤ e

with the end conditions

y (1) = 0,y(e)=0.

By using the transformation (8.1.2), the Euler equation can be put into

the Sturm–Liouville form:





λy=0.

The solution of the Euler equation is

y (x)=c

√

+ c

−i

√

Noting that x

= e

ia ln x

= cos (a ln x)+i sin (a ln x), the solution y (x)

becomes

y (x)=A cos



√

λ ln x



+ B sin



√

λ ln x



where A and B are constants related to c

and c

. The end condition

y (1) = 0 gives A = 0, and the end condition y (e)=0gives

276 8 Eigenvalue Problems and Special Functions

sin

√

λ =0,B=0,

which in turn yields the eigenvalues

= n

,n=1, 2, 3,...,

and the corresponding eigenfunctions

sin (nπ ln x) ,n=1, 2, 3,....

Another type of problem that often occurs in practice is the periodic

Sturm–Liouville system.

The Sturm–Liouville equation



p (x)



+[q (x)+λs (x)] y =0,a≤ x ≤ b,

in which p (a)=p (b), together with the periodic end conditions

y (a)=y (b) ,y

′

(a)=y

′

(b)

is called a periodic Sturm–Liouvill e system.

Example 8.1.3. Consider the p eriodic Sturm–Liouville system

′′

+ λy =0, −π ≤ x ≤ π,

y (−π)=y (π) ,y

′

(−π)=y

′

(π) .

Here we note that p (x) = 1, hence p (−π)=p (π). When λ>0, we see

that the solution of the Sturm–Liouville equation is

y (x)=A cos

√

λx+ B sin

√

λx.

Application of the periodic end conditions yields



2sin

√

λπ



B =0,



√

λ sin

√

λπ



A =0.

Thus, to obtain a nontrivial solution, we must have

sin



√



π =0,A=0,B=0.

Consequently,

= n

,n=1, 2, 3,....

Since sin

√

λπ = 0 is satisﬁed for arbitrary A and B, we obtain two li nearly

independent eigenfunctions cos nx,andsinnx corresponding to the same

eigenvalue n

It can be readily shown that if λ<0, the solution of the Sturm–Liouville

equation does not satisfy the periodic end conditions. However, when λ =0

the corresponding eigenfunction is 1. Thus, the eigenvalues of the periodic

Sturm–Liouville system are 0,

, and the corresponding eigenfunctions

are 1, {cos nx}, {sin nx},wheren is a positive integer.

8.2 Eigenvalues and Eigenfunctions 277

8.2 Eigenvalues and Eigenfunctions

In Examples 8.1.1 and 8.1.2 of the regular Sturm–Liouville systems in the

preceding section, we see that there exists only one lin early independent

eigenfunction corresponding to the eigenvalue λ, which is called an eigen-

value of multiplicity one (or a simple eigenvalue). An eigenvalue is said to

be of multiplicity k if there exist k linearly independent eigenfunctions cor-

responding to the same eigenvalue. In Example 8.1.3 of the periodic Sturm–

Liouville system, the eigenfunctions cos nx,sinnx correspond to the same

eigenvalue n

. Thus, this eigenvalue is of multiplicity two.

In the preceding examples, we see that the eigenfunctions are cos nx

and sin nx for n =1, 2, 3,.... It can be easily shown by using trigonometric

identities that



−π

cos mx cos nx dx =0,m= n,



−π

cos mx sin nx dx =0, for all integers m, n,



−π

sin mx sin nx dx =0,m= n.

We say that these functions ar e orthogonal to each other in the interval

[−π, π]. The orthogonality relation holds in general for the eigenfunctions

of Sturm–Liouville systems

Let φ (x)andψ (x) be any real-valued integrable functions on an interval

I.Thenφ and ψ are said to be orthogonal on I with respect to a weight

function ρ (x) > 0, if and only if,

φ, ψ =



φ (x) ψ (x) ρ (x) dx =0. (8.2.1)

The interval I may be of inﬁnite extent, or it may be either open or closed

at one or both ends of the ﬁnite interval.

When φ = ψ in (8.2.1) we deﬁne the norm of φ by

φ =





(x) ρ (x) dx



. (8.2.2)

Theorem 8.2.1. Let the coeﬃcients p, q,ands in the Sturm–Liouville sys-

tem be continuous in [a, b]. Let the eigenfunctions φ

and φ

, corresponding

to λ

and λ

, be continuously diﬀerentiable. Then φ

and φ

are orthogonal

with respect to the weight function s (x) in [a, b].

Proof.Sinceφ

corresponding to λ

satisﬁes the Sturm–Liouville equa-

tion, we have



pφ

′



+(q + λ

s) φ

= 0 (8.2.3)

278 8 Eigenvalue Problems and Special Functions

and for the same reason

(pφ

′

)+(q + λ

s) φ

=0. (8.2.4)

Multiplying equation (8.2.3) by φ

and equation (8.2.4) by φ

, and sub-

tracting, we obtain

(λ

− λ

) sφ

= φ

(pφ

′

) − φ



pφ

′



(pφ

′

) φ

−



pφ

′



and integrating yields

(λ

− λ

)



sφ

dx =



′

− φ

′

!

= p (b)

(b) φ

′

(b) − φ

′

(b) φ

(b)

−p (a)

(a) φ

′

(a) − φ

′

(a) φ

(a)

(8.2.5)

the right side of which is called the boundary term of the Sturm–Liouville

system. The end conditions for the eigenfunctions φ

and φ

are

(b)+b

′

(b)=0,

(b)+b

′

(b)=0.

If b

= 0, we multiply the ﬁrst condition by φ

(b) and the second condition

by φ

(b), and subtract to obtain

(b) φ

′

(b) − φ

′

(b) φ

(b)

=0. (8.2.6)

In a similar manner, if a

= 0, we obtain

(a) φ

′

(a) − φ

′

(a) φ

(a)

=0. (8.2.7)

We see by virtue of (8.2.6) and (8.2.7) that

(λ

− λ

)



sφ

dx =0. (8.2.8)

If λ

and λ

are distinct eigenvalues, then



sφ

dx =0. (8.2.9)

Theorem 8.2.2. The eigenfunctions of a per iodic Sturm–Liouville system

in [a, b] are orthogonal with respect to the weight function s (x) in [a, b].

8.2 Eigenvalues and Eigenfunctions 279

Proof. The periodic conditions for the eigenfunctions φ

and φ

are

(a)=φ

(b) ,φ

′

(a)=φ

′

(b) ,

(a)=φ

(b) ,φ

′

(a)=φ

′

(b) .

Substitution of these into equation (8.2.5) yields

(λ

− λ

)



sφ

dx =[p (b) − p (a)]

(a) φ

′

(a) − φ

′

(a) φ

(a)

Since p (a)=p (b), we have

(λ

− λ

)



sφ

dx =0. (8.2.10)

For distinct eigenvalues λ

= λ

,(λ

− λ

) =0andthus,



sφ

dx =0. (8.2.11)

Theorem 8.2.3. For any y, z ∈ D (L), we have the Lagrange identity

yL [z] − zL[y]=

[p (yz

′

− zy

′

)] . (8.2.12)

Proof. Using the deﬁn ition of the Sturm–Liouville operator, we have

yL [z] − zL[y]=y





+ qyz − z





− qyz

[p (yz

′

− zy

′

)] .

Theorem 8.2.4. The Sturm–Liouville operator L is self-adjoint. In other

words, for any y, z ∈ D (L), we have

L [y] ,z = y, L [z], (8.2.13)

where < , > is the inner product in L

([a, b]) deﬁned by

f,g =



f (x)

g (x) dx. (8.2.14)

Proof. Since all constants involved in the boundary conditions of Sturm–

Liouville system are real, if z ∈ D (L), then

z ∈ D (L).

Also since p, q and s are real-valued,

L [z]=L [z]. Consequently, we

have

L [y] ,z−y, L [z] =



(

zL[y] − yL[z]) dx

=[p (zy

′

− y z

′

)]

, by (8.2.12) . (8.2.15)

280 8 Eigenvalue Problems and Special Functions

We next show that the right hand side of this equality vanishes for a

regular RSL system. If p (a) = 0, the result follows immediately. If p (a) > 0,

then y and z satisfy the boundary conditions of the form (8.1.5) at x = a.

That is,

⎡

⎣

y (a) y

′

(a)

z (a) z

′

(a)

⎤

⎦

⎡

⎣

⎤

⎦

=0.

Since a

and a

are not both zero, we have

z (a) y

′

(a) − y (a) z

′

(a)=0.

A similar argument can be used to the other end point x = b, so that the

right-hand side of (8.2.15) vanishes. This proves the theorem.

Theorem 8.2.5. All the eigenvalues of a regular Sturm–Liouville system

with s (x) > 0 are real.

Proof.Letλ be an eigenvalue of a RSL system and let y (x)bethe

corresponding eigenfunction. This means that y = 0 and L [y]=−λsy.

Then

0=L [y] ,y−y, L [y] =



λ − λ





s (x) |y (x)|

dx.

Since s (x) > 0in[a, b]andy = 0, the integrand is a positive number. Thus,

λ =

λ, and hence, the eigenvalues are real. This completes the proof.

Theorem 8.2.6. If φ

(x) and φ

(x) are any two solutions of the equation

L [y]+λsy =0on [a, b], then p (x) W (x; φ

,φ

)=constant, where W is

the Wronskian.

Proof.Sinceφ

and φ

are solutions of L [y]+λsy =0,wehave



dφ



+(q + λs) φ

=0,



dφ



+(q + λs) φ

=0.

Multiplying the ﬁrst equation by φ

and the second equation by φ

,and

subtracting, we obtain



dφ



− φ



dφ



=0.

Integrating this equation from a to x, we obtain

p (x)[φ

(x) φ

′

(x) − φ

′

(x) φ

(x)] = p (a)[φ

(a) φ

′

(a) − φ

′

(a) φ

(a)]

p (x) W (x; φ, φ

) = constant. (8.2.16)

This is called Abel’s formula where W is the Wronskian.