C2999_PDF_Intro.pdf

(1005 KB) Pobierz

Introduction

Some Denitions,

Formulas, Methods, and Solutions

0.1. Classication of Second-Order Partial Differential

Equations

0.1.1. Equations with Two Independent Variables

0.1.1-1. Examples of equations encountered in applications.

Three basic types of partial differential equations are distinguished—parabolic,

hyperbolic,

and

elliptic.

The solutions of the equations pertaining to each of the types have their own characteristic

qualitative differences.

The simplest example of a

parabolic

equation is the heat equation

−

where the variables and play the role of time and the spatial coordinate, respectively. Note that

equation (1) contains only one highest derivative term.

The simplest example of a

hyperbolic

equation is the wave equation

−

where the variables and play the role of time and the spatial coordinate, respectively. Note that

the highest derivative terms in equation (2) differ in sign.

The simplest example of an

elliptic

equation is the Laplace equation

where and play the role of the spatial coordinates. Note that the highest derivative terms in

equation (3) have like signs.

Any linear partial differential equation of the second-order with two independent variables can

be reduced, by appropriate manipulations, to a simpler equation which has one of the three highest

derivative combinations specied above in examples (1), (2), and (3).

0.1.1-2. Types of equations. Characteristic equations.

Consider a second-order partial differential equation with two independent variables which has the

general form

¤ £

( , )

, , ,

( , )

= 0,

(1)

= 0,

(2)

= 0,

(3)

(4)

where , , are some functions of and that have continuous derivatives up to the second-order

inclusive.*

Given a point ( , ), equation (4) is said to be

parabolic

hyperbolic

elliptic

at this point.

In order to reduce equation (4) to a canonical form, one should rst write out the characteristic

equation

−2

= 0,

which splits into two equations

−

and

and nd their general integrals.

In this case, equations (5) and (6) coincide and have a common general integral,

( , )

in accordance with the relations

By passing from ,

to new independent variables ,

( , ),

where

( , ) is any twice differentiable function that satises the condition of nondegeneracy

(

of the Jacobian

( ,, ))

in the given domain, we reduce equation (4) to the canonical form

As , one can take

It is apparent that, just as the heat equation (1), the transformed equation (7) has only one

highest-derivative term.

In the degenerate case where the function

does not depend on the derivative

equation (7) is an ordinary differential equation for the variable , in which serves as a parameter.

The general integrals

of equations (5) and (6) are real and different. These integrals determine two different families of

real characteristics.

* The right-hand side of equation (4) may be nonlinear. The classication and the procedure of reducing such equations

to a canonical form are only determined by the left-hand side of the equation.

( , )

0.1.1-4. Canonical form of hyperbolic equations (case

−

0).

, , ,

0.1.1-3. Canonical form of parabolic equations (case

−

= 0,

(5)

(6)

$ #

" !

3 120 )

= 0).

(7)

This is the so-called rst canonical form of a hyperbolic equation.

The transformation

= +

= −

brings the above equation to another canonical form,

where

= 4

. This is the so-called second canonical form of a hyperbolic equation. Apart from

notation, the left-hand side of the last equation coincides with that of the wave equation (2).

In this case the general integrals of equations (5) and (6) are complex conjugate; these determine

two families of complex characteristics.

Let the general integral of equation (5) have the form

( , )

we reduce equation (4) to the canonical form

Apart from notation, the left-hand side of the last equation coincides with that of the Laplace

equation (3).

0.1.2. Equations with Many Independent Variables

where

are some functions that have continuous derivatives with respect to all variables to the

second-order inclusive, and

= {

, ,

[The right-hand side of equation (8) may be nonlinear.

The left-hand side only is required for the classication of this equation.]

At a point

, the following quadratic form is assigned to equation (8):

B ¥

)

@ £

998

8 8

@ £

9898

(x)

8 8

998

Consider a second-order partial differential equation with

has the form

B ¥

independent variables

@ £

, , ,

( , ),

where ( , ) and ( , ) are real-valued functions.

By passing from , to new independent variables ,

0.1.1-5. Canonical form of elliptic equations (case

−

0).

= −1,

in accordance with the relations

, , ,

−

, , ,

we reduce equation (4) to

( , ),

By passing from ,

to new independent variables ,

in accordance with the relations

B ¥

that

(8)

(9)

TABLE 1

Classication of equations with many independent variables

Type of equation (8) at a point

Parabolic (in the broad sense)

Hyperbolic (in the broad sense)

Elliptic

Coefcients of the canonical form (11)

B §

At least one coefcient of the

B §

is zero

All

are nonzero and some

B §

differ in sign

All

are nonzero and have like signs

By an appropriate linear nondegenerate transformation

8 8

998

B §

the quadratic form (9) can be reduced to the canonical form

B B §

where the coefcients assume the values

1, −1,

and

The number of negative and zero coefcients

in (11) does not depend on the way in which the quadratic form is reduced to the canonical form.

Table 1

presents the basic criteria according to which the equations with many independent

variables are classied.

Suppose all coefcients of the highest derivatives in (8) are constant,

const. By introducing

the

form

are the coefcients of the linear transformation (10), we reduce equation (8) to the canonical

@ ¤

8 8

998

Here, the coefcients are the same as in the quadratic form (11), and

= {

, ,

Among the parabolic equations, it is conventional to distinguish the parabolic

equations in the narrow sense, i.e., the equations for which only one of the coefcients, , is zero,

while the other is the same, and in this case the right-hand side of equation (12) must contain the

rst-order partial derivative with respect to .

In turn, the hyperbolic equations are divided into normal hyperbolic equations—

for which all but one have like signs—and ultrahyperbolic equations—for which there are two or

more positive and two or more negative .

Specic equations of parabolic, elliptic, and hyperbolic types will be discussed further in

Subsection 0.2.

References for Section

0.1: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), S. J. Farlow (1982), D. Colton

(1988), E. Zauderer (1989), A. N. Tikhonov and A. A. Samarskii (1990), I. G. Petrovsky (1991), W. A. Strauss (1992),

R. B. Guenther and J. W. Lee (1996), D. Zwillinger (1998).

B §

@ ¤

9898

B §

3 I

3 P

120 )

0.2. Basic Problems of Mathematical Physics

0.2.1. Initial and Boundary Conditions. Cauchy Problem.

Boundary Value Problems

Every equation of mathematical physics governs innitely many qualitatively similar phenomena

or processes. This follows from the fact that differential equations have innitely many particular

8 8

998

the new independent variables

in accordance with the formulas

B ¥

@ ¤

FEA

(

= 1,

, )

(10)

¤ B §

B §

(11)

, where

(12)

solutions. The specic solution that describes the physical phenomenon under study is separated

from the set of particular solutions of the given differential equation by means of the initial and

boundary conditions.

Throughout this section, we consider linear equations in the -dimensional Euclidean space

or in an open domain

(exclusive of the boundary) with a sufciently smooth boundary

0.2.1-1. Parabolic equations. Initial and boundary conditions.

In general, a linear second-order partial differential equation of the parabolic type with independent

variables can be written as

−

[ ]

(x, ),

(1)

Parabolic equations govern unsteady thermal, diffusion, and other phenomena dependent on time .

Equation (1) is called homogeneous if (x, )

≡ 0.

Cauchy problem

(

≥ 0,

). Find a function that satises equation (1) for >

and the

initial condition

(x) at

= 0.

(3)

Boundary value problem*

(

≥ 0,

). Find a function

the initial condition (3), and the boundary condition

that satises equation (1) for >

( >

0).

In general,

is a rst-order linear differential operator in the space variables

with coefcient de-

pendent on

and . The basic types of boundary conditions are described below in Subsection 0.2.2.

The initial condition (3) is called homogeneous if (x)

≡ 0.

The boundary condition (4) is called

homogeneous if (x, )

≡ 0.

0.2.1-2. Hyperbolic equations. Initial and boundary conditions.

Consider a second-order linear partial differential equation of the hyperbolic type with independent

variables of the general form

where the linear differential operator

is dened by (2). Hyperbolic equations govern unsteady

wave processes, which depend on time .

Equation (5) is said to be homogeneous if (x, )

≡ 0.

). Find a function that satises equation (5) for >

and the

Cauchy problem

(

≥ 0,

initial conditions

(x) at

= 0,

(6)

(x) at

= 0.

Boundary value problems

for parabolic and hyperbolic equations are sometimes called

mixed

initial-boundary value

problems.

Y ¢

(x, )

−

[ ]

(x, ),

[ ]

(x, ) at

(4)

(5)

B ¥

(x, )

@ £

8998

= {

≥

B ¦

(x, )

¡ ¢

B ¥

[ ]

≡

where

(x, ) ,

(2)

Plik z chomika:

malgosia68

Inne pliki z tego folderu:

Encyclopedic Dictionary of Mathematics4.pdf (38140 KB)
Encyclopedic Dictionary of Mathematics1.pdf (28991 KB)
C2999_PDF_C04.pdf (1827 KB)
History in mathematics education.pdf (7906 KB)
Mathematics and culture 2.pdf (5982 KB)

C2999_PDF_Intro.pdf

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika: