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

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2 INTRODUCTION

The partial derivative of u with respect to t is defined in a similar way. Another way

of representing the partial of u with respect to x is

A partial differential equation is an equation involving one or more partial deriva-

tives of a dependent variable u. An example of such is the one-dimensional (ID)

diffusion equation

ut = ku

(1.2)

where u

represents the first partial of u with respect to t, k is a constant diffusion

coefficient, and u

represents the second partial of u with respect to x. That is,

(x + h,t) -u

(x,t)

= hm (1.3)

h-*0 h

Some applications of PDEs may include a mixed partial, such as u

, where

(x,y + h,t)-u

(x,y,t)

= hm (1.4)

h^o h

with u

—

u(x,y,t).

It is common for the dependent variable u to be a function of three spatial variables

x, y, and z, as well as time t. The general form of a PDE for u in this case may be

expressed as

" \%i y i

%·>

^"i

W")

^i) ^yi ^Ζΐ ^ti ^χχι ^xyi ^J/XÎ ^yy> ' ' '

^tti · · · ) — ^ v

*^/

1.2 CLASSIFICATION

The order of a PDE is the highest order derivative in Equation (1.5). The order of

the most common PDEs in science and engineering applications is two or less. In the

event that u = u(x, y), Equation (1.5) may be expressed as

+ Bu

4- Cuyy + Du

+ Euy + Fu = Q (1.6)

The PDE is said to be linear if each coefficient A

—

Q is at most a function of x

or y. Otherwise, the equation is nonlinear. A nonlinear equation is quasilinear if

it is linear in its highest order derivatives. Examples of PDEs with identification of

their order and linearity are given in Table 1.1. If the term Q on the right-hand side

of Equation (1.6) is zero, the PDE is said to be homogeneous. Otherwise the PDE is

classified as nonhomogeneous.

Suppose Equation (1.6) is linear. All equations of this form may be further

classified as parabolic, hyperbolic, or elliptic. Heat flow and diffusion problems

are typically described by parabolic forms of Equation (1.6). These equations have

coefficients A, B, and C satisfying the property B

—

AAC

—

0. Hyperbolic forms of

CANONICAL FORMS

Table 1.1 PDEs: Order and Linearity.

PDE

UyU

+ au

= 2

+ auuy = 0

+ au

= u

+ 1

+ OiUy — 1

-f Myy = C0S(X

+ y)

1¿1¿

OLUy

= 0

ÏX

-f uu

—

Order

1st

2nd

Linearity

nonlinear

quasilinear

linear

nonlinear

quasilinear

the equation are those for which B

—

4AC > 0. Common equations of this type are

associated with vibrating systems and wave motion. Finally, when B

—

4AC < 0,

the equation is elliptic. Equations of this type typically represent steady-state (time

independent) phenomena.

If any of the coefficients A, B, or C are functions of x or y, the characterization

of the equation as parabolic, hyperbolic, or elliptic may be a function of location in

the xy-plane. As an example, consider the PDE

y Μχχ + Λ/ΧΙ

-T~

Uyy ~Γ ¿U

(1.7)

The expression B

—

AAC is equal to x

—

for Equation (1.7). Consequently, the

equation is parabolic on the curve x = 4y

, hyperbolic for points (x, y) such that

x > Ay

, and elliptic for ordered pairs (x, y) that satisfy Ay

> x and x > 0. These

regions are depicted in Figure 1.1.

1.3 CANONICAL FORMS

For the case where the dependent variable u is a function of at most two independent

variables, as indicated in Equation 1.6, there are three common types of

PDEs.

The

Laplace equation is u

4- u

— 0, and is elliptic on the entire xy-plane. The heat

equation has the form u

—

= 0, where the independent variable t represents time

and x represents a spatial dimension. The heat equation is parabolic on the entire xy-

plane. The wave equation u

—

0 is hyperbolic on the entire xi/-plane. Here,

the dependent variable u represents the displacement or wave height as a function of

time t and location x.

It can be shown that with a smooth, nonsingular change of coordinates, the sign of

the discriminant B

—4AC will not

change.

Therefore, it

possible to make

change

of coordinate transformation in which an elliptic PDE is transformed to a Laplace

equation, a parabolic PDE is transformed to the heat equation, and a hyperbolic

equation is transformed into the wave equation form. Consequently, the Laplace,

4 INTRODUCTION

4 parabolic

Figure 1.1 Parabolic, hyperbolic, and elliptic regions for Equation (1.7).

heat, and wave equations are referred to as the canonical form in their respective

category.

1.4 COMMON PDES

The following list gives examples of PDEs that are common to physical applications.

One-dimensional heat equation : u

= ku

, where u(x,t) represents the

temperature of a solid body as a function of time t and spatial dimension x,

and k is the thermal diffusivity.

Two-dimensional (2D) heat equation : u

= k(u

+ u

), where i¿(x, y, t) is

the temperature of a solid body, and k is the thermal diffusivity.

One-dimensional convective-diffusion equation : 9

= — u6

+ k9

, where

the dependent variable 0(x, t) may be temperature or perhaps a concentration

of a given chemical in a fluid, u(t) is the velocity of the fluid, and k is a

diffusion coefficient.

Two-dimensional convective-diffusion equation :

= -u6

- v9

+ k{0

+ 9yy), where

<9(x,

y, t) may be temperature or per-

haps a concentration of a given chemical in a fluid, u(x, y, t) is the velocity

of the fluid in the x direction, v(x,y,t) is the velocity of the fluid in the y

direction, and

is a diffusion coefficient.

One-dimensional wave equation : u

—

, where u(x,t) represents the

displacement of a vibrating string from its equilibrium or initial position, and

c is the speed of the wave.

CAUCHY-KOWALEVSKI THEOREM 5

6. Two-dimensional wave equation : u

= c

+ u

), where u(x, y, t) rep-

resents the displacement of

vibrating membrane from its equilibrium or initial

position, and c is the speed of the wave.

Two-dimensional Laplace equation : V

u = u

+ u

= 0, where u(x,y)

may be the temperature of a solid plate.

8. Two-dimensional Poisson's equation : V

u = u

+ u

—

f(x,y),

where

u(x, y) may be an electrostatic field property.

9. Berger's equation in E

: u

+ uu

= 0, where u(x, t) is the velocity of a

stream of particles or fluid flow with zero viscosity.

10.

Schrödinger equation in

: u

= i \S7

u + Vu], where u(x, y, z, t) is veloc-

ity and V(x,y,z) denotes the potential, with application

quantum mechanics.

1.5 CAUCHY-KOWALEVSKI THEOREM

The Cauchy-Kowalevski theorem on existence and uniqueness of solutions to systems

of PDEs is stated in this section. The theorem establishes sufficient conditions on the

individual PDEs of the system, as well as the initial conditions, called Cauchy data.

No proof of the theorem is given in this text. The interested reader may consult [26]

for a proof for the case of

system of linear PDEs. A proof for the case of quasilinear

systems may be found in [17]. Note: Later in the text general nonlinear systems may

be reduced to quasilinear systems by differentiation. Yet another valuable reference

is Volume II of the classic works by Courant and Hubert [14].

Suppose i¿i, U2, ..., UM are dependent variables of time t and spatial variables

.xi, X2, ··, XN- The general expression for a second-order PDE of variable u

(1 < m < M) is

-Um „ Λ &xn \

at*™ \ optóte

X\...O3

)

(1.8)

where 1 < I < M and jo +

+ · · · 4- JN = j < k

and

< k

. The notation

reflects the requirement that the PDE for an arbitrary dependent variable u

be such

that the /c

th partial derivative of u

with respect to t may be isolated on the left-

hand side of the equation. The right-hand side of the equation is a function F

the independent variables t, xi,... XN, the dependent variables i¿i,...,

UM,

and the

partial derivatives of

the

dependent variable, with order less than or equal to the order

. The order of any partial derivative of u

with respect to any spatial variable x¿

is a non-negative integer

j¿.

Finally, the order

of the partial of u

with respect to

t must be an integer strictly less than the order km

Recall from experience with ODEs that a first-order equation

I-™

6 INTRODUCTION

requires an initial condition y (to) =

for determining the unique particular solution.

The second-order ODE

must have initial values for both y and ^ prescribed for the particular solution to be

found.

In a similar way, initial data is needed in the case of the system of PDEs for a

particular solution to be determined. Equation (1.8) is an expression for the /c

partial derivative of dependent variable u

with respect to time t. Consequently,

initial functions for all partial derivatives of order zero to k

—

1 of each dependent

variable must be specified. The general form of such data is

-^—(¿o,xi,.. ·

,XM)

<l>ij(xi,

· · · ,XM) (1.9)

where (j)i,j(x\, ..., XM) is a given function for 0 < j < k

—

1 and 1 < I < M.

Before the Cauchy-Kowalevski statement is given, the concept of an analytic

function is introduced. For that purpose, suppose function / is defined in terms of

variables xi,...,

XN-

The point x° = (x?, x®,..., x%) is said to be interior to the

domain D of / if there exists a positive real number e such that the n-dimensional

"ball" of radius

(often referred to as the e-neighborhood of x°) is contained entirely

within D. The function / is said to be analytic at x° if / can be expressed as a power

series expanded about x°, such as

/(*!,...,**)=£ ( Σ

ki,.:.,kA*i-4)

-'^XN-x%)

that is valid for all x in the e-neighborhood of x°. The powers ki are non-negative

integers.

It is now possible to state the Cauchy-Kowalevski theorem on existence and

uniqueness.

Theorem 1.1 (Cauchy-Kowalevski) Suppose the system of PDEs, with each PDE

of form as in Equation

(1.8),

is such that each F

is analytic in a neighborhood of

the point

£(b

2i · ' '

%? ·

· ·

dJm-J0

d^x

"-d^

1 ι

2">"">

Ν'

Further, suppose initial data, expressed as functions as shown in Equation

(1.9),

such that each φι^ is analytic in a neighborhood of(x®,..., x%). Then there exists a

unique analytic solution to the system of PDEs

a neighborhood of

(to,

x\,...,

x%).

Note: The theorem establishes the existence of a unique analytic solution.

INITIAL BOUNDARY VALUE PROBLEMS 7

1.6 INITIAL BOUNDARY VALUE PROBLEMS

Figure 1.2 depicts a thin, uniform rod of length L oriented along the x-axis so that

the left end of the rod corresponds with the origin. At time t

—

0 the rod has a known

temperature distribution as a function of x that is denoted by f(x), 0 < x < L.

This distribution represents the initial condition (IC) for the temperature of the rod.

Additionally, the temperature boundary condition (BC) at x — 0 and x — L is

known for all time t > 0. These conditions may be such that the temperature is

known, or the spatial partial derivative of temperature u

is prescribed.

x = L

Figure 1.2 Heat transfer in a rod of

length

Under the assumption that the rod is "narrow" so that the spatial dependence of

temperature is in terms of x only, the governing PDE for temperature is the ID heat

equation u

— ku

. (A careful derivation of this equation is provided in Chapter

4.) The objective is to determine the function, or functions, i¿(x, t) that satisfies the

PDE, as well as the initial and boundary conditions. This is an example of an initial

boundary value problem (IBVP). A well-posed problem is one in which the PDE

and properly defined ICs and BCs result in the existence of a unique solution u(x, t).

The IBVPs will be designated throughout the text using the following format:

IBVP<

' Ut =

u(x

«(0,

- kh

= T

0<:

v<L,t >0 (PDE)

(IC)

(BC1)

(BC2)

(1.10)

Note: BCs specified in Equation (1.10) imply the ends of the rod remain at constant

temperatures T

and ΤΊ, respectively. When the value of the dependent variable

itself is prescribed along a boundary as in this example, the condition is said to be

a Dirichlet condition. Dirichlet boundary conditions are sometimes referred to as

BCs of the first kind.

There are instances when boundary conditions prescribe the value of the dependent

variable u and its normal derivative

every point along the

boundary.

Such conditions

are referred to as Robin conditions. They are also known as BCs of the second kind.

An example of an IBVP with such BCs is given below:

Ut =

u(x

u(0,

— I\i Uu

0) =

t)-

,t) +

fix)

(0,t)

(L,t)

--0

= 0

0<x< L,t >0 (PDE)

(IC)

(BC1)

(BC2)

IBVP< -v-»-y-JV-; v-w

(U1)

8 INTRODUCTION

Yet, another prescription is the Neumann condition in which the value of the

normal (i.e., perpendicular to the boundary surface) derivative is given. An example

of an IBVP with Neumann BCs is given below:

= ku

0 < x < L, t > 0 (PDE)

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

(L12)

(0,t)=0 (BC1)

(L,t)=0 (BC2)

IBVP<

Neumann boundary conditions are sometimes call BCs of the third kind.

Some applications may prescribe BCs of different types on different segments

of the domain boundary. For example, one may encounter a heat transfer problem

for the rod of length L as depicted in Figure 1.2. The boundary condition at x =

0 is prescribed as Dirichlet condition, while the BC at x — L may be a Neumann

condition. Such cases will be referred to as mixed boundary conditions.

1.7 SOLUTION TECHNIQUES

Experience from solving ordinary differential equations (ODEs) suggests that solution

techniques for PDEs are varied, and depend on such equation characteristics as

linearity and order. It is reasonable, and correct, to assume methods for solving

PDEs are more numerous and equation specific due to the increased complexity of

multivariable functions.

In keeping with the similarities for solving ODEs, solutions techniques for PDEs

may be divided into two general groups: analytical and numerical. Analytical

techniques are those that strive to find "exact" formulas for the dependent variable

was a function of all independent variables. Values for u can be determined for

all times and all locations within the temporal and spatial domain of the problem.

Analytical methods include separations of variables, where a PDE in n variables is

reduced to n ODEs, integral transforms, where a PDE of n variables is transformed

to a PDE in n - 1 variables (a PDE in two variables would be transformed to an

ODE),

calculus of variations, where the solution to the PDE is the same function

that minimizes a companion energy function, and eigenfunction expansion, where

the solution to the PDE is given as an infinite sum of eigenfunctions that solve a

corresponding eigenvalue problem.

Numerical methods result in approximate values of the dependent variable u at

prescribed, discrete locations within a finite domain of the independent variables.

There are various numerical methods, and many are specific to PDE type. A unifying

feature is the transformation of the PDE into an alternate expression that commonly

includes a system of linear algebraic equations. Such techniques include the finite

difference method, the finite element method, the finite volume method, and the

finite analytic method. A somewhat more detailed description of these numerical

approaches is given in Chapter 8

This text will focus on the analytical methods of separation of variables and

eigenfunction expansion that are especially suitable for certain linear PDEs. The

SEPARATION OF VARIABLES 9

finite difference numerical method, which has a reasonable realm of application for

various types of PDEs, will be the primary numerical method. In Section 1.8, an

introductory example on the method of separation of variables is presented.

1.8 SEPARATION OF VARIABLES

The method of separation of variables is, perhaps, the most common analytical

scheme for solving common linear

PDEs.

It has a long history in science and applied

mathematics, dating back to the time of the French mathematician Jean Baptiste

Joseph Fourier (1768-1830). In fact, the method is commonly referred to as the

"Fourier method." The basic features will be presented in this section by working

through the simple ID heat transfer IBVP given in Equation (1.13), leaving greater

detail and general results to subsequent sections in the text.

= ku

0 < x < 1, t > 0 (PDE)

IBVP { u(x,0) = f(x) (IC) (1.13)

u(0,t)=0 (BC1)

t u{l,t) =0 (BC2)

The method of separation is applicable to linear, homogeneous PDEs. Addition-

ally, linear homogeneous BCs of the form

(0,t) + bu(0,t) = 0

(l,t) + du(l,t) = 0

must hold. The coefficients a,b,c, and d are constants.

The method begins by assuming the function u(x, t) may be represented by the

product of functions X(x), a function of x only, and T(t), a function oft only. That

u(x,t) = X(x)T(t) (1.14)

Substituting this form of u(x, t) into the PDE of Equation (1.13) results in

X(x)T'(t) = kX"{x)T{t) (1.15)

where T

{t) represents the ordinary first derivative of T with respect to t, and X"{x)

represents the ordinary second derivative of X with respect to x. Both sides of

Equation (1.15) are divided by kX{x)T(t) resulting in

T'(t)

Χ^ή

kT(t) X(x)

;

Note: The left-hand side of Equation (1.16) is a function of t only, while the right-

hand side of Equation (1.16) is a function of

only. Because variables x and t are

10 INTRODUCTION

independent, Equation (1.16) is true only if both sides are equal to a constant value.

For a matter of convenience, we will let this constant be represented by

—A.

That is,

T{t) _ X"{x)

WiT)~~

-~X{x)'

(M7)

X"{x)+\X{x)=Q (1.18)

T'(t) + k\T(t) = 0 (1.19)

The process transforms the original PDE in two variables (x and t) into two ODEs,

one for T as a function of t, and the other for X as a function of x.

The boundary conditions given in Equation (1.13), when applied to the separated

form u(x, t) = X(x)T(t), result in the following:

u(0,t)=0 => X(0)T(t)=0

=> X(0) = 0 (1.20)

and

u(l,t)=0 => X(l)T(t)=0

=> X(1) = 0 (1.21)

Combining Equations (1.18), (1.20), and (1.21) gives

X"(x)+\X(x)=0 X(0) = 0 X(1)=0 (1.22)

which is one form of the general Sturm-Liouville problem.

The solution process for Equation (1.22) is broken into cases corresponding to

possible values of

Case

= 0

The case for A = 0 implies X

{x) = 0, which means X(x) = A + Bx rep-

resents the general solution to the ODE. Applying the boundary conditions gives

X(0) = A + B'0 = A = 0and X(l) = 0 + B · 1 = 0 => B = 0, Consequently,

only the trivial solution X(x) =0 results when

= 0.

Case

> 0

> 0, then let

= a

(a > 0), and the ODE becomes

X"{x)

+ a

X{x) = 0

which, as a second-order, linear, homogeneous ODE, has the general solution

X{x) = A cos ax 4- B sin ax

SEPARATION OF VARIABLES

Invoking the boundary conditions give

X(0)

= 0

=> AcosO

£sinO

= 0

A = 0

and

X(l)

0=> J5sina

= 0

For nontrivial solutions (i.e.,

B φ

0), it must be that

sin

a = 0

a =

ηπ,

n =

1,2,3...

So,

any function with form

(x)

βίηηπχ,

n =

1,2,3...

is a solution to the Sturm-Liouville problem given in Equation (1.22). Such functions

are referred to as eigenfunctions, and the quantities

(ηπ)

are called eigen-

values for the associated Sturm-Liouville problem. The linearity and homogeneity

of the ODE and BCs

the the Sturm-Liouville problem (1.22) allow any linear

combination of eigenfunctions as a solution.

Case

< 0

For

λ <

0, let

λ = -a

> 0). The ODE of Equation (1.22) becomes

X"(x)

- a

X(x) = 0

which has a general solution of the form

X(x)=Ae

+Be-

Invoking the boundary conditions once again gives

X(0)

= 0

=Φ

œ0

0=>5

= -i

and

X(l)

= 0

- Ae~

= 0

=» 2ylsinha

A = 0

Consequently, only the trivial solution given by

A = B

—

0 exists.

In summary, the Sturm-Liouville problem yields eigenvalues

of a

= ηπ

and

eigenfunctions

(x) =

8ΐη(ηπχ) for

n =

1,2,3, The ODE in

has allowed

us to determine the eigenvalues and eigenfunctions. With the eigenvalues identified,

we turn our attention to the ODE for

The eigenvalues for

determined by the solutions for Equation (1.22) mean that

Equation (1.19) becomes

T'(t) + (nn)

kT(t) = 0

(1.23)

'sinhir

—

\{e

—

and sinha? is zero if and only if

= 0.