Bernatz R. Fourier Series and Numerical Methods for Partial Differential Equations

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12 INTRODUCTION

which is a linear first-order ODE whose general solution for a given n is

(t) =

e~^

2kt

(1.24)

where A

is an arbitrary constant.

Using the respective solutions for X(x) and T(t), the set of functions

{x,t)

= X

(x)T

(t)

= A

{n7r)2kt

sm(nnx), n = 1,2,3,... (1.25)

represent all possible functions that satisfy the PDE and the BCs in Equation (1.13).

Also,

the linear and homogeneous PDE, in combination with the linear and homo-

geneous BCs of Problem (1.13), allow any linear combination of functions shown in

Equation (1.25) to satisfy the PDE and BCs. Only the initial condition of Problem

(

1.13) is left to be satisfied.

The IC will be satisfied if it is possible to determine values for the arbitrary

constants A

, such that

CO CO

u(x,0) = ]P

e~(

nn)2k0

sin(nnx) = ^ A

sin(mrx) = f(x)

n=l n—1

The problem is rather easy if the function / were, for example,

f(x) = 2sin(7HE)

—

4sin(37nr) + - sin(77rx)

In this case, it is obvious that A\ = 2, As =

—4,

Aj = |, and A

= 0 for all n not

equal to 1,3, and 7.

It is less obvious if there exist A

, such that

co , ^2

2_"

δίη(ηπχ) = 50x(l

—

x) I x

—

- I (1.26)

n=l V

for all x on the interval [0, 1]. More generally, for what initial temperature distribu-

tions f(x) is it possible to find a sine series expansion, such that

y^ A

sin(ri7rx) = f(x)

n=l

for all x on the interval [0, 1]? This is, in fact, one of the many problems Fourier

investigated. It will be shown in subsequent sections of the text that such a Fourier

series expansion is possible for any / provided / meets certain continuity criteria.

Additionally, it will be shown that the coefficients A

may be determined using a

formula similar to

= 2 / f(x)sm(rnrx)dx (1.27)

SEPARATION OF VARIABLES 13

As a way of completing this separation of variables example, suppose the initial

temperature distribution is that given by the formula

on the

right-hand side of Equation

(1.26).

Equation (1.27) is used to determine a formula for the Fourier coefficients so

that the initial condition is satisfied. Therefore, the solution for i¿(x, t) is given by

u(x,t)

n=l V

f(x) s'm(nnx)dx I e

-(mr)

/ct

sin

(

n7rx

)

(1.28)

This expression gives the value of u(x,t) for any x in the interval

[0,1],

and

any t > 0 as an infinite series. An approximate value for u at a given x and t

is determined by a partial sum of the infinite series. A visual indication of how

accurately a partial sum will approximate the infinite series can be accomplished by

comparing the partial sum of the series at time t = 0 and the initial temperature

distribution /(#). Such is the case in Figure 1.3, where the partial sum of the first

five terms in the series and f(x) are plotted on the [0, 1] x-interval.

0.4

Figure 1.3 The initial temperature distribution f(x) (solid line) and the

five-term

Fourier

series partial sum approximation u(x, 0) (broken line).

Figure 1.4 shows a plot of

iM-

f(x) s'm(nnx)dx ) e

(ηπ) kt

sin(nnx)

u(x,t)

The graph of the approximation is a surface above the xt-plane. Note that as time

t increases, the temperature at all x locations decreases to zero as one would expect

when the ends of

the

rod are held at a constant zero temperature and there are sources

of heat for the rod. The rate at which the temperature declines to zero is determined

largely by the magnitude of the thermal diffusivity k. The value of the diffusivity k

in this case is 0.1.

14 INTRODUCTION

Figure 1.4 The surface u(x, t) using the

first

five

terms of

the

series.

This initial separation of variables example demonstrates, among other things, the

need to understand more about the process of representing initial or boundary value

functions as Fourier series expansions. This need will be addressed in Chapter 2.

EXERCISES

1.1

Determine

the given function

is a

solution

to the

PDE.

du du _ / \ x£t \

du du ,

„, o

-^ + c— = 0

M(X,

¿)

= f(x

- ct)

du du .

i./ 9 2\

"¿T

= c2

(

'*)

<*)

)

1.2

Classify

the

following PDEs

hyperbolic, parabolic,

elliptic.

^χχ

^Uu

óU

~Γ

¿Uyy

Uyy

= 0, a φ

¿O'U'xy

\ U'yy

- YlUxt

ÖWj;x ZU

y óUyy

^ΧΧ

+ ¿U

OUyy

α^

1.3

Determine

the

sets

order pairs

(x, y) on

which each

the following partial

differential equations is hyperbolic, parabolic,

elliptic. Make

sketch, such as that

shown

Figure

1.1,

identifying

the

region(s)

in the

xy-p\me corresponding

each

type.

^χχ

XUyy

— U

2XU

yUyy

= 0

(1 +

4- (1

= 0

2m¿

X2/

4- (1 -

—

4~ %U

y 4~

yUyy

= U

f) cos

í/n^

—

sin

xt^ = 0

1.4

a) Show that

the PDE u

X2/

4- 4?/^ = 0 is

parabolic

on the

entire

x?/-plane.

b) Use the change

variables £(#,

y) = x and

ry(x, ?/)

= i/

—

2x to

transform

the equation

part

(a) to the

canonical form uçç

= 0.

1.5

Following the example presented

Section

1.8,

rewrite the following PDEs

set of

ODEs

in X and Y, or X, Y, and T.

^XX 4" ^2/7/ I Wy = U

Ό)

Uyy

C U

= k(U

+Uyy)

V*xx X^yy

(14-

= 0

16 INTRODUCTION

1.6 Consider the B VP below with the four prescribed boundary conditions.

BVP{ "i„

3« + 7 (BCZ) (1-29)

%i ~T~ 2/2/

(x,0)

Ml,

2/)

u(x, 1) =

w(0,2/) =

= 3x-1

= 32/+ 7

-2y

- y

-1

4-2

(FDb)

(BC1)

(BC2)

(BC3)

(BC4)

We assume a solution of the form u(x, y) = Ax

+ 2?:n/ + C?/

+ Dx

-\-

Ey + F,

where constants A

—

F are to be determined.

a) What conditions on A

—

F are required so that the proposed form of u

satisfies the PDE.

b) If possible, determine values for A

—

F so that u satisfies each of the four

BCs.

1.7 The ID homogeneous wave equation is

Utt =

where the constant c represents the intrinsic wave speed. Let u(x,t) = X(x)T(t)

and use the method of separation of variables with separation constant — λ to find the

two resulting ODEs (one in X and one in T) for the case of the ID wave equation.

1.8 Find the general solution for the following homogeneous ODEs

a) y"{t)-Ay\t)+Ay{t) =

b) y"{t) - 2y'(t) + 2y(t) = 0

c) 2/"(t)+42/(i) = 0

1.9 Solve the following first-order linear IVP using an appropriate integrating

factor.

a) y'(x) + y(x) = cosx, y(0) — 1

b) y'{x) + 2xy(x) = xe~

\ y(0) = -1

1.10 The purpose of this exercise is to introduce, or review, the method of variation

of parameters

for finding a particular solution to a second-order, linear, non-

homogeneous ODE, such as

y"(t)

+ p(t)y'(t) + q(t)y(t) = f(t) (1.30)

Suppose y i and y^ are linearly independent solutions to homogeneous form of Equa-

tion (1.30). We seek a particular solution of Equation (1.30) of the form

Vv =

m +u

y2 (1.31)

where u\ and u

are, generally, functions oft determined by requiring

u[yi +

= 0

The French mathematician J.L. Lagrange is known to have used this method in 1774. Consequently, this

method is sometimes called the "Lagrange method."

EXERCISES 17

in addition to y

solving Equation (1.30). Under these requirements, show that

-V2Í ,

yif

)

W(y

)

where W(y\, y

) = 2/12/2

2/Í2/2 is the Wronskian of y\ and y

. [Hint: Under the

given requirements, show that the following system of equations:

^Í2/l+^2l/2 = 0

u'iy'i+u'^ = f

results.]

1.11 Find the general solution to the following nonhomogeneous ODEs by the

method of

(i)

undetermined coefficients, and (ii) variation of parameters (see Exercise

1.10).

a) y"(t)-3y'(t) + 2y(t)=8uit

b) y"{t) - 2y'(t) - 3y(t) =t + l

c) y"(t) + 4y(t) = te

1.12 Determine the solution to IBVP (1.13) for the various initial conditions f(x)

and thermal conductivity k given below. In each case, generate a 2D plot (e.g., Figure

1.3) of u(x, 0) and a 3D plot of u(x, t) for 0 < t < 2, such as that shown in Figure

1.4. Use a 50-term partial sum.

a) f(x) = cos πχ and k = 0.05

b) f(x) = e~

and k = 0.01

r 1 0.4<x<0.6

c) f(x) = I and k = 0.01

[ 0 otherwise

1.13 The function u(x, y) defined as

11 °°

u(x, y)

—

-x

—

-y

—

2x + V^

nV2 cos(nny) ΰθ8η(ηπ(2

—

x))

n=l

with

1 r

-—-- / (4(1 - y) - 2)V2cos(nny)dy n = 1, 2, 3,...

ΠΈ

smh{2nn)

is part of the solution to a steady-state problem that includes the required boundary

condition

-u

(0,y) = 4(l-j/)

Show that u(x, y) so defined satisfies the required boundary condition. Follow these

steps:

(i) Construct a 10-term approximation to u(x, y). (ii) Differentiate the result

in (i). (iii) Plot the result of part (ii) on the interval 0 < y < 1.

18 INTRODUCTION

1.14 Consider the IBVP

( u

= ku

+ q{x, t) 0 < x < 1, t > 0 (PDE)

IBVP ^ w(x, 0) = f(x)

u(0,t) = h

i w(i,t) = Λ

(ί;

(IC)

(BCl)

(BC2)

(1.32)

Assume u(x, t) = £/(x,

£)

4· i?(x, t), where i?(x, t) = A(t)x

JB(Í),

and substitute

for u(x, t), in terms of U and i?, in the IBVP given in 1.32. In doing so, derive an

IBVP for U that has a nonhomogeneous PDE, with nonhomogeneous term q*(x, t).

Determine an expression for q*(x, t). Next, determine the initial condition /*(£) for

the IBVP for U. Then, determine conditions on A(t) and B(t) that give homogeneous

BCs for the IBVP in U.

1.15 Repeat the process described in Exercise 1.14, but for the nonhomogeneous

IBVP

( u

= ku

+ q(x,t) 0 < x < 1, í > 0 (PDE)

IBVP<

u(x,0) = f(x)

(l,t)

= h

(t)

(IC)

(BCl)

(BC2)

(1.33)

1.16 Repeat the process described in Exercise 1.14, but for the nonhomogeneous

IBVP

= ku

+q(x,t)

0<x<l,¿>0 (PDE)

IBVP

u(x, 0) = f(x)

u(0,t) -u

(0,t

u(l,t) +u

(l,t

hi(t

(IC)

(BCl)

(BC2)

(1.34)

CHAPTER 2

FOURIER SERIES

The initial example of separation of variables presented in Section 1.8 demonstrated

the need to explore the Fourier series representation of functions that define initial

or boundary condition for IBVPs. That is the broad objective of this chapter. More

specifically, the primary question is what are necessary or sufficient characteristics of

a function / so that a Fourier sine, Fourier cosine, or general Fourier series expansion

accurately represents / on a given interval. To this end, the chapter begins with a

review of vector spaces, inner products, and orthogonality.

2.1 VECTOR SPACES

Recall that a vector space V is a nonempty set of objects (vectors) on which the

operations of vector addition and scalar multiplication are defined

that the following

10 axioms hold for all vectors u, v, and w in V, and for all scalars c and d in R.

The sum of u and v, denoted by u + v, is in V.

u + v = v-fu

(u + v) + w = u + (v + w)

Fourier Series and Numerical Methods for Partial Differential Equations, 19

First Edition. By Richard Bernatz

20 FOURIER SERIES

There exist a zero vector 0 in V , such that u + 0 = u.

For each u in V, there is a vector —u in V , such that u + (—u) = 0.

6. The scalar multiple of u by c, denoted by cu, is in V.

c(u + v) = c + cv

8. (c + d)u = eu -f du

9. c(du) = (cd)u

10.

lu = u

A set V is said to be "closed under vector addition" when Axiom 1 is satisfied.

Likewise, a set V is said to be "closed under scalar multiplication" when Axiom 6

holds.

Some important properties that apply to any vector space resulting from these

axioms include the uniqueness of the zero vector, as well as the uniqueness of the

—u element, sometimes referred to as the additive inverse of u.

The prototypical vector space is the Euclidean space R, where a vector is simply

a real number, vector addition is real number addition, and scalar multiplication is

simply real number multiplication. Consequently, R is closed under vector addition,

because the sum of any two real numbers is a real number, and closed under scalar

multiplication, because the product of any two real numbers is a real number. Other

typical vector spaces include R

and R

. In R

, for example, an arbitrary vector u is

given by the ordered triple (x, y, z) or {x\,X2, xs).

There are vector spaces that look significantly different than the Euclidean space

mentioned in the previous paragraph. The vector space most pertinent to the subject

at hand is the set of real-valued functions on the interval (a, b), which will be denoted

by F(o, b). For two arbitrary vectors f and g from F(a, 6), the vector sum is defined

f + g = (f + g)(x)

= f(x) + g(x) (2.1)

for all x G (a, b). Note: This vector addition is, in fact, defined in terms of real

number addition, as indicated in Equation (2.1). Because the set of real numbers is

closed under real number addition, the set F (a,

is closed under vector addition.

Multiplication of any f in F(a,

by any scalar c is defined as

cî = (cf)(x)

= cf(x) (2.2)

for all x in (a, b). The set F(a,

is closed under scalar multiplication defined in this

way because scalar-vector multiplication reduces to real number multiplication, and

the set of real numbers is closed under this operation.

The zero vector of F(a,

is defined as 0(x)

—

0 for all x G (a, b).

VECTOR SPACES 21

2.1.1 Subspaces

A subspace

of vector space V is a subset of V that satisfies the following properties

The zero vector of V is an element of

The set H is closed under vector addition.

The set El is closed under scalar multiplication.

Common subspaces of the vector space R

include the set {0} (referred to as the

zero vector space), any line or plane that includes the origin (the zero vector), and R

itself.

Common subspaces of F(a,

include, the zero vector space, all polynomials

of degree n or less, all continuous functions, all differentiable functions, and all

integrable functions [a function f is said to be integrable on (a,

if and only if the

definite integral / î{x)dx exists].

2.1.2 Basis and Dimension

Let B = {bi,

,...,

} be a subset of vectors from vector space V. The set B is

a spanning set (or the set B is said to span V) if every vector in V can be written as

a linear combination of the vectors in B. That is, there exist coefficients, or weights,

such that

v = cibi + c

h c

for every v

The set B is linearly independent if and only if

cibi + c

h c

= 0

implies each coefficient

c¿

is zero.

The set B is a basis for the vector space V if B is a spanning, linearly independent

subset of ¥. A vector space V can have more than one basis. However, it can be

shown that every basis for V has the same number of vectors.

The dimension of V is the number of vectors in any basis of V. The dimension

of the zero vector space {0} is defined to be zero. A vector space V with a finite

basis is said to be finite-dimensional. A vector space V that cannot be spanned by

a finite set is said to be infinite-dimensional. The dimension of a subspace El of

vector space V is always less than or equal to the dimension of

That is,

dim H < dim V

The set F(a,

is an example of an infinite-dimensional vector space. To under-

stand why this is true, let P(a,

be the set of polynomials of

any

order on the interval

(a, b). It should be obvious that F(a,

is a subspace of F(o, b)(see Exercise 2.1).

Assume P(o,

has finite dimension n. If so, then any basis B of P(a,

is finite

in number. Consequently, the exists an element bj of P(a, 6) such that the order of