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

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1.1 Brief Historical Comments 11

With the rapid development of the theory and applications of diﬀeren-

tial equations, the closed form analytical solutions of many diﬀerent types

of equations were hardly possible. However, it is extremely important and

absolutely necessary to provide some insight into the qualitative and quan-

titative nature of solutions subject to initial and boundary conditions. This

insight usually takes the form of numerical and graphical representatives of

the solutions. It was E. Picard (1856–1941) wh o ﬁrst developed the method

of successive approximations for the solutions of diﬀerential equations in

most general form and later made it an essential part of his treatment of

diﬀerential equations in the second volume of his Trait´ed’Analysepublished

in 1896. During the last two centuries, the calculus of ﬁnite diﬀerences in

various forms played a signiﬁcant role in ﬁnding the numerical solutions of

diﬀerential equations. Historically, many well known integration formulas

and numerical methods including the Euler–Maclaurin formula, Gregory

integration formula, the Gregory–Newton formula, Simpson’s rule, Adam–

Bashforth’s method, the Jacobi iteration, the Gauss–Seidel method, and the

Runge–Kutta method have been developed and then generalized in various

forms.

With the development of modern calculators and high-speed electronic

computers, there has b een an increasing trend in research toward the numer-

ical solution of ordinary an d partial diﬀerential equations during the twen-

tieth century. Special attention has also given to in depth studies of conver-

gence, stability, error analysis, and accuracy of numerical solutions. Many

well-known numerical methods including the Crank–Nicolson methods, the

Lax–Wendroﬀ method, Richtmyer’s method, and Stone’s implicit iterative

technique have been develop ed in the second half of the twentieth century.

All ﬁnite diﬀerence methods reduce diﬀerential equations to discrete forms.

In recent years, more modern and powerful computational methods such

as the ﬁnite element method and the boundary element method have been

developed in order to handle curved or irregularly shaped domains. These

methods are distinguished by their more general character, which makes

them more capable of dealing with complex geometries, allows them to

use non-structured grid systems, and allows more natural imposition of the

boundary conditions.

During the second half of the nineteenth century, considerable attention

was given to problems concerning the existence, uniqueness, and stability

of solutions of partial diﬀerential equations. These studies involved not

only the Laplace equat ion, but the wave and diﬀusion equations as well,

and were eventually extended to partial diﬀerential equations with variable

coeﬃcients. Through the years, tremendous progress has been made on

the general theory of ordinary and partial diﬀerential equations. With the

advent of new ideas and methods, new results and applications, both an-

alytical and numerical studies are continually being added to this subject.

Partial diﬀerential equations have been the subject of vigorous mathemat-

ical research for over three centuries and remain so today. This is an active

12 1 Introduction

area of research for mathematicians and scientists. In part, this is moti-

vated by the large number of problems in partial diﬀerential equations that

mathematicians, scientists, and engineers are faced with that are seemingly

intractable. Many of these equations are nonlinear and come from such

areas of applications as ﬂuid mechanics, plasma physics, nonlinear optics,

solid mechanics, biomathematics, and quantum ﬁeld theory. Owing to the

ever increasing need in mathematics, science, and engineering to solve more

and more complicated real world problems, it seems quite likely that partial

diﬀerential equations will remain a major area of research for many years

to come.

1.2 Basic Concepts and Deﬁnitions

A diﬀerential equation that contains, in addition to the dependent variable

and the independent var iables, one or more partial derivatives of the de-

pendent variable is called a partial diﬀerential equation. In general, it may

bewrittenintheform

f (x,y,...,u,u

,...,u

,...)=0, (1.2.1)

involving several independent variables x, y, ..., an unknown function u of

these variables, and the partial derivatives u

, u

, ..., u

, u

, ...,ofthe

function. Subscripts on dependent variables denote di ﬀerentiations, e.g.,

= ∂u/∂x, u

= ∂

/∂y ∂x.

Here equation (1.2.1) is considered in a suitable domain D of the n-

dimensional space R

in th e independent variables x, y, .... We seek func-

tions u = u (x,y,...) which satisfy equation (1.2.1) identically in D. Such

functions, if they exist, are called solutions of equation (1.2.1). From these

many possible solutions we attempt to select a particular on e by introducing

suitable additional conditions.

For instance,

+ u

= y,

+2yu

+3xu

=4sinx, (1.2.2)

)

+(u

)

=1,

− u

=0,

are partial diﬀerential equations. The functions

u (x , y)=(x + y)

u (x , y)=sin(x − y) ,

are solutions of the last equation of (1.2.2), as can easily be veriﬁed.

1.2 Basic Concepts and Deﬁnitions 13

The order of a partial diﬀerential equation is the order of the highest-

ordered partial derivative app earing in the equation. For example

+2xu

+ u

= e

is a second-order partial diﬀerential equation, and

xxy

+ xu

+8u =7y

is a third-order partial diﬀerential equation.

A partial diﬀerential equation is said to be linear if it is linear in the

unknown function and all its derivatives with coeﬃcients depending only

on the independent variables; it is said to be quasi-linear if it is linear in

the highest-ordered derivative of the unknown function. For example, the

equation

+2xyu

+ u =1

is a second-order linear partial diﬀerential equation, whereas

+ xuu

=siny

is a second-order quasi-linear partial diﬀerential equation. The equation

which is not l inear is called a nonlinear equation.

We shall be primarily concerned with linear second-order partial dif-

ferential equations, which frequently arise in problems of mathematical

physics. The most general second-order linear partial diﬀerential equation

in n independent variables has th e form



i,j=1



i=1

+ Fu = G, (1.2.3)

where we assume without loss of generality that A

= A

. We also assume

that B

, F ,andG are functions of the n independent variables x

If G is identically zero, the equation is said to be homogeneous; otherwise

it is nonhomogeneous.

The general solution of a linear ordinary diﬀerential equation of nth or-

der is a family of functions depending on n independent arbi trary constants.

In the case of partial diﬀerential equations, the general solution dep ends on

arbitrary functions rather than on arbitrary constants. To illustrate this,

consider the equation

=0.

If we integrate this equation with respect to y, we obtain

(x, y)=f (x) .

14 1 Introduction

A second integrati on with respect to x yields

u (x , y)=g (x)+h (y) ,

where g (x)andh (y) are arbitrary functions.

Suppose u is a function of three variables, x, y,andz. Then, for the

equation

=2,

one ﬁnds the general solution

u (x , y, z)=y

+ yf (x, z)+g (x, z) ,

where f and g are arbitrary functions of two variables x and z.

We recall that in the case of ordinary diﬀerential equations, the ﬁrst task

is to ﬁnd the general solution, and then a particular solution is determined

by ﬁnding the values of arbitrary constants from the prescribed conditions.

But, for partial diﬀerential equations, selecting a particular solution satis-

fying the supplementary conditions from the general solution of a partial

diﬀerential equation may be as diﬃcult as, or even more diﬃcult than, the

problem of ﬁnding the general solution itself. This is so because the gen-

eral solution of a partial diﬀerential equation involves arbitrary functions;

the specialization of such a solution to the particular form which satis-

ﬁes supplementary conditions requires the determination of these arbitrary

functions, rather than merely the determination of constants.

For linear homogeneous ordinar y diﬀerential equations of order n,a

linear combination of n linearly independent solutions is a solution. Unfor-

tunately, this is not true, in general, in the case of partial diﬀerential equa-

tions. This is due to the fact that the solution space of every homogeneous

linear part ial diﬀerential equation is inﬁnite dimensional. For example, the

partial diﬀerential equation

− u

= 0 (1.2.4)

can be transformed into the equation

by the transformation of variables

ξ = x + y, η = x − y.

The general solution is

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

where f (x + y) is an arbitrary function. Thus, we see that each of the

functions

1.3 Mathematical Problems 15

(x + y)

sin n (x + y) ,

cos n (x + y) ,

exp n (x + y) ,n=1, 2, 3,...

is a solution of equation (1.2.4). The fact that a simple equation such as

(1.2.4) yields inﬁnitely many solutions is an indication of an added diﬃculty

which must be overcome in the study of partial diﬀerential equations. Thus,

we generally prefer to directly determine the particular solution of a partial

diﬀerential equation satisfying prescribed supplementary conditions.

1.3 Mathematical Problems

A problem consists of ﬁnding an unknown function of a partial diﬀerential

equation satisfying appropriate supplementary conditions. These conditions

may be initial conditions (I.C.) and/or boundary conditions (B.C.). For ex-

ample, the partial diﬀerential equation (PDE)

− u

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

with I.C. u (x, 0) = sin x,0≤ x ≤ l, t>0,

B.C. u (0,t)=0, t ≥ 0,

B.C. u (l, t)=0, t ≥ 0,

constitutes a problem which consists of a par tial diﬀerential equation and

three supplementary conditions. The equation describes the h eat conduc-

tion in a rod of length l. The last two conditions are called the boundary

conditions which describe the function at two prescribed boundary points.

The ﬁrst condition is known as the initial condition which prescribes the

unknown function u (x, t) throughout the given region at some initial time

t, in this case t = 0. This problem is known as the initial boundary-value

problem. Mathematically speaking, the time and the space coordinates are

regarded as indep endent variables. In this respect, the initial condition is

merely a point prescribed on the t-axis and the boundary conditions are

prescribed, in this case, as two points on the x-axis. Initial conditions are

usually prescribed at a certain time t = t

or t = 0, but it is not customary

to consider the other end point of a given time interval.

In many cases, in addition to prescribing the unknown function, other

conditions such as their derivatives are speciﬁed on the boundary and/or

at time t

In considering the problem of unbounded domain, the solution can be

determined uniquely by prescribing initial conditions only. The correspond-

ing problem is called the initial-value problem or the Cauchy problem.The

mathematical deﬁnition is given in Chapter 5. The solution of such a prob-

16 1 Introduction

lem may be interpreted physically as the solution unaﬀected by the bound-

ary conditions at inﬁni ty. For problems aﬀected by the boundary at inﬁnity,

boundedness conditions on the behavior of solutions at inﬁnity must be pre-

scribed.

A mathematical problem is said to be well-posed if it satisﬁes the fol-

lowing requirements:

1. Existence: There is at least one solution.

2. Uniqueness: There is at most one solution.

3. Continuity: The solution depends continuously on the data.

The ﬁrst requirement is an obvious logical condition, but we must keep in

mind that we cannot simply state that the mathematical problem has a

solution just because the physical problem has a solution. We may well be

erroneously developing a mathematical model, say, consisting of a partial

diﬀerential equation whose solution may not exist at all. The same can be

said about the uniqueness requirement. In order to really reﬂect the physical

problem that has a unique solution, the mathematical problem must have

a unique solution.

For physical problems, it is not suﬃcient to know that the problem has

a unique solution. Hence the last requirement is not only useful but also

essential. If the solution is to have physical signiﬁcance, a small change in

the initial data must produce a small change in the solution. The data in a

physical problem are normally obtained from experiment, and are approxi-

mated in order to solve the problem by numerical or approximate methods.

It is essential to know that the process of making an approximation to the

data produces only a small change in the solution.

1.4 Linear Operators

An operator is a mathematical rule which, when applied to a function,

produces an other function. For example, in the expressions

L [u]=

∂

∂x

∂

∂y

M [u]=

∂

∂x

−

∂u

∂x

+ x

∂u

∂y

L =



∂

/∂x

+ ∂

/∂y



and M =



∂

/∂x

− ∂/∂x



+ x (∂/∂y) are called

the diﬀerential operators.

An operator is said to be linear if it satisﬁes the following:

1. A constant c may be taken outside the operator:

L [cu]=cL [u] . (1.4.1)

1.4 Linear Operators 17

2. The op erator operating on the sum of two functions gives the sum of

the operator operating on th e individual functions:

L [u

+ u

]=L [u

]+L [u

] . (1.4.2)

We may combine (1.4.1) and (1.4.2) as

L [c

+ c

]=c

L [u

]+c

L [u

] , (1.4.3)

where c

and c

are any constants. This can be extended to a ﬁnite

number of functions. If u

, u

, ..., u

are k functions and c

, c

, ..., c

are k constants, then by repeated application of equation (1.4.3)

⎡

⎣



j=1

⎤

⎦



j=1

L [u

] . (1.4.4)

We may now deﬁne the sum of two linear diﬀerential operators f ormall y.

If L and M are two linear operators, then the sum of L and M is deﬁned

(L + M)[u]=L [u]+M [u] , (1.4.5)

where u is a suﬃciently diﬀerentiable function. It can be readily shown

that L + M is also a linear operator.

The product of two linear diﬀerential operators L and M is the operator

which produces the same result as is obtained by the successive operations

of the operators L and M on u,thatis,

LM [u]=L (M [u]) , (1.4.6)

in which we assume that M [u]andL (M [u]) are deﬁned. It can be readily

shown that LM is also a linear operator.

In general, linear diﬀerential operators satisfy the following:

1.L+ M = M + L (commutative) (1.4.7)

2. (L + M)+N = L +(M + N

) (associative) (1.4.8)

3. (LM) N = L (MN) (associative) (1.4.9)

4.L(c

M + c

N)=c

LM + c

LN (distributive) . (1.4.10)

For linear diﬀerential operators with constant coeﬃcients,

5.LM= ML (commutative) . (1.4.11)

Example 1.4.1. Let L =

∂

∂x

+ x

∂

∂y

and M =

∂

∂y

− y

∂

∂y

18 1 Introduction

LM [u]=



∂

∂x

+ x

∂

∂y



∂

∂y

− y

∂u

∂y



∂

∂x

∂y

− y

∂

∂x

∂y

+ x

∂

∂y

− xy

∂

∂y

ML[u]=



∂

∂y

− y

∂

∂y



∂

∂x

+ x

∂u

∂y



∂

∂y

∂x

+ x

∂

∂y

− y

∂

∂y∂x

− xy

∂

∂y

which shows that LM = ML.

Now let u s consider a linear second-order partial diﬀerential equation.

In the case of two independent variables, such an equation takes the form

A (x, y) u

+ B (x, y) u

+ C (x, y) u

+D (x, y) u

+ E (x, y) u

+ F (x, y) u = G (x, y) , (1.4.12)

where A, B, C, D, E,andF are the coeﬃcients, and G is the nonhomoge-

neous term.

If we denote

L = A

∂

∂x

+ B

∂

∂x∂y

+ C

∂

∂y

+ D

∂

∂x

+ E

∂

∂y

+ F,

then equation (1.4.12) may be written in the form

L [u]=G. (1.4.13)

Very often the square bracket is omitted and we simply write

Lu = G.

Let v

, v

, ..., v

be n functions which satisfy

L [v

]=G

,j=1, 2,...,n

and let w

, w

, ..., w

be n functions which satisfy

L [w

]=0,j=1, 2,...,n.

If we let

= v

+ w

then, the function

u =



j=1

1.4 Linear Operators 19

satisﬁes the equation

L [u]=



j=1

This is called the principle of linear superposition.

In particular, if v is a particular solution of equation (1.4.13), that is,

L [v]=G,andw is a solution of the associated homogeneous equation, that

is, L [w]=0,thenu = v + w is a solution of L [u]=G.

The principle of linear superposition is of fundamental importance in

the study of partial diﬀerential equations. This principle is used extensively

in solving linear partial diﬀerential equations by the method of separation

of variables.

Suppose that there are inﬁnitely many solutions u

(x, y), u

(x, y), ...

(x, y), ... of a linear homogeneous partial diﬀerential equation Lu =0.

Can we say that every inﬁnite linear combination c

+···+ c

··· of these solutions, where c

, c

, ..., c

, ... are any constants, is again

a solution of the equation? Of course, by an inﬁnite linear combination, we

mean an inﬁnite series and we must require that the inﬁnite series

∞



k=0

= lim

n→∞



k=0

(1.4.14)

must be convergent to u. In general, we state that the inﬁnite series is a

solution of the homogeneous equation.

There is another kind of inﬁnite linear combination which is also used

to ﬁnd the solution of a given linear equation. This is concerned with a

family of solutions u (x, y; k) of the linear equation, where k is any real

number, not just the values 1, 2, 3,....Ifc

= c (k) is any function of the

real parameter k such that



c (k) u (x, y; k) dk or



∞

−∞

c (k) u (x, y; k) dk (1.4.15)

is convergent, then, under suitable conditions, the integral (1.4.15), again,

is a solution. This may be called the linear integral superposition principle.

To illustrate these ideas, we consider the equation

Lu = u

+2u

=0. (1.4.16)

It is easy to verify that, for every real k, the function

u (x , y; k)=e

k(2x−y)

(1.4.17)

is a solution of (1.4.16).

Multiplying (1.4.17) by e

−k

and integrating with respect to k over −1 ≤

k ≤ 1gives

20 1 Introduction

u (x , y)=



−1

−k

k(2x−y)

dk =

2x−y −1

2x − y − 1

(1.4.18)

It is easy to verify that u (x, y) given by (1.4.18) is also a solution of (1.4.16).

It is also easy to verify that u (x, y; k)=e

−ky

cos (kx), k ∈ R is a

one-parameter family of solutions of the Laplace equation

∇

u ≡ u

+ u

= 0 (1.4.19)

It is also easy to check that

v (x, y; k)=

∂

∂k

u (x , y; k) (1.4.20)

is also a one-parameter family of solutions of (1.4.19), k ∈ R.Further,for

any (x, y) in the upper half-plane y>0, the integral

v (x, y) ≡



∞

u (x , y, k) dk =



∞

−ky

cos (kx) dk, (1.4.21)

is convergent, and v (x, y) is a solution of (1.4.19) for x ∈ R and y>0.

This follows from direct computation of v

and v

. The solution (1.4.21)

is another example of the linear integral superposition principle.

1.5 Superposition Principle

We may express supplementary conditions using the operator notation. For

instance, the initial boundary-value problem

− c

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

u (x , 0) = g

(x)0≤ x ≤ l,

(x, 0) = g

(x)0≤ x ≤ l, (1.5.1)

u (0,t)=g

(t) t ≥ 0,

u (l, t)=g

(t) t ≥ 0,

maybewrittenintheform

L [u]=G,

[u]=g

, (1.5.2)

[u]=g

where g

are the prescribed functions and the subscripts on operators are

assigned arbitrarily.