数学物理与工程中的Mathematica应用

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"Mathematical Methods In Physics And Engineering With Mathematica" 本书《Mathematical Methods In Physics And Engineering With Mathematica》是Ferdinand F. Cap所著，属于CRC Press出版的Applied Mathematics and Nonlinear Sciences Series系列。该系列由Goong Chen和Thomas J. Bridges担任主编。该书于2003年由CRC Press LLC发行，并提供了相关的图书信息和ISBN号。书中内容可能涵盖了物理和工程中的数学方法，特别强调了如何利用Mathematica这一强大的计算软件来解决这些问题。Mathematica是一种广泛使用的计算机程序语言，特别适用于数值和符号计算，以及复杂的图形和数据可视化。通过使用Mathematica，读者能够更深入地理解物理和工程领域的数学模型，并能高效地求解相关问题。这本书可能会涵盖以下关键知识点： 1. **数学物理**：书中可能会详细介绍数学在物理学中的应用，包括经典力学、电磁学、热力学、量子力学等领域的数学模型和解析技巧。 2. **工程数据处理**：作者可能讨论了如何利用Mathematica进行数据分析和处理，这对于理解和优化工程系统至关重要，例如信号处理、控制系统设计、结构分析等。 3. **Mathematica编程**：读者可以学习到如何编写Mathematica代码，解决物理和工程问题，包括求解微积分方程、偏微分方程、线性代数问题，以及数值模拟等。 4. **数值方法**：书中可能涉及各种数值方法，如欧拉法、龙格-库塔方法、有限差分和有限元方法，这些都是在物理和工程领域中处理复杂问题的常用工具。 5. **可视化技术**：利用Mathematica的可视化功能，读者可以直观地理解物理现象和工程系统的动态行为，例如创建二维和三维图形，动画展示动态过程。 6. **符号计算**：Mathematica在符号计算方面的强大能力可能被用来解决理论问题，例如简化表达式、求解方程组和积分等。 7. **索引和参考文献**：书中提供的索引和参考文献有助于读者进一步研究相关主题，拓宽知识面。通过阅读本书，工程和物理专业的学生以及研究人员将能够掌握利用Mathematica解决实际问题的技能，提高他们在这个领域的专业能力。这本书对于那些希望将高级数学应用于实际问题的人来说，无疑是一本宝贵的资源。

What is a boundary problem?

The separation constants are

here

determined by the dimensions

a b

of the

membrane. From (1.1.41) wethenhave





(1.1.49)

Apparently our boundary value problem (1.1.35) with (1.1.36) has solutions

only for sp ecial discrete values of

. If a dierential equation like (1.1.35)

or (1.1.39) has solutions only for sp ecial values of a parameter, the prob-

lem is called an

eigenvalue problem

is the

eigenvalue

and the solutions

(

) =

cos(



) are called

eigenfunctions

. In contrast to this, solu-

tions of (1.1.42), (1.1.43) will be called

modes

particular solutions

Let us nowconsider (1.1.34). One possible solution is

(

)=cos

!t:

(1.1.50)

Due to (1.1.49) and the denition used for

in (1.1.35) the constant

(

angular

frequency

) has the dimension sec

;

] and assumes the discrete values





;



(1.1.51)

The

are called

eigenfrequencies

partial dierential equation

is called

linear

, if it is of the rst degree in

the dep endentvariable

(

x y

)

u

(

x y  t

)andtheir partial derivatives

u

i.e., if eachtermeither consists of the pro duct of a known function of the inde-

pendentvariables, and the dependentvariable or one of its partial derivatives.

The equation may also contain a known function of the independentvariables

only (

inhomogeneous equation

). If this term do es not app ear, the

equation

called

homogeneous

. It is a general prop ertyofall linear dierential equations

that twoparticular solutions can b e sup erp osed, which means that the sum of

twosolutions is again a (new) solution (

superposition principle

). So the gen-

eral solution is a sup erp osition of all particular solutions. Hence the general

symmetric solution of (1.1.30) maybewritten as

(

x y  t

mn

cos



x

2

;



cos



y

2

;



cos

t

(1.1.52)

where the

are given by(1.1.51) and the partial amplitudes

maybe

determined from an

initial condition

(

x y 

0) =

(

x y

)



(1.1.53)

where

(

x y

)isagiven function, i.e., the deection of the membrane at the

time

=0.

Using nowthesolutions (1.1.42), wehave

(

x y

cos



;





cos

y:

(1.1.54)

12 Introduction

1.2 Classication of partial dierential equations

Ordinary dierential equations containing only one indep endentvariable (and

two point boundary conditions

)are of minor imp ortance for problems of physics

and engineering. Hence, we mainly concentrate on partial dierential equa-

tions.

Partial dierential equations possess a large manifold of solutions. Instead

of integration constants, arbitrary functions app ear in the solution. As an

example, we consider (1.1.15) in a sp ecialization of two indep endentvariables

(

x t

(

x t

)

(1.2.1)

By inserting into (1.2.1) it is easy to prove that

(

x t

(

;

) (1.2.2)

is a solution of (1.2.1). Here

and

are arbitrary functions. Such a solu-

tion of a partial dierential equation of order

with

arbitrary functions

is called a

general solution

. If a partial dierential equation contains

inde-

pendentvariables

x y :::

, one can nd a

complete solution

,which contains

integration constants. If afunction satises the partial dierential equation

and the accompanying b oundary conditions and has no arbitrary function or

constants, it is called a

particular solution

. If a solution is not obtainable by

assigning particular values to the parameters in the general (complete) solu-

tion, it is called a

singular solution

. It describ es an envelop e of the family of

curves represented by the general solution. But the expression is also used for

a solution containing a

singular point

, a

singularity

. We will later consider

such solutions containing singular p oints where the solution tends to innity.

There are other essential prop erties characterizing various types of partial

dierential equations. Wegivesome examples below:

(

x y

(

x y

)=0

(1.2.3)

This

Laplace

equation

is linear, homogeneous and has constantcoe"cients.

The equation

(

x t

)

;

(

x t

)=0 (1.2.4)

is of the rst order, linear, homogeneous and has variable coe"cients. (For a

rst-order partial dierential equation a b oundary problem cannot b e formu-

lated.) An example of a nonlinear homogeneous partial dierential equation

is given by

(

x y

(

x y

)=0 (1.2.5)

and the equation

(

x t

(

x t

)

(

x t

)=0 (1.2.6)

@~n

(

x y

)

(

x y

(

x y

)(1.2.12)

on the boundary. This condition is ofimp ortance in heat ow and uid

mechanics. It is p ossible to prove1.2]the

solvability

boundary problems

see Table 1.1.

Classication of partial dierential equations

is called

quasilinear

. (The derivatives are linear). An inhomogeneous example

is given bythe linear

Poisson

equation

(

x y

(

x y



(

x y

)



(1.2.7)

where



is a given function. The most

general linear partial dierential equa-

tion

of twoindep endentvariables has the form

(

x y

)

(

x y

)+2

(

x y

)

(

x y

(

x y

)

(

x y

(

x y

)

(

x y

)

(

x y

)

(

x y

(

x y

)

(

x y

(

x y

)

(1.2.8)

Now equations satisfying

(

x y

)

(

x y

)

(

x y

) (1.2.9)

in a certain domain in the

x y

plane are called

hyperbolic equations

(

x y

)

(

x y

)

(

x y

) (1.2.10)

characterizes

el liptic equations

and if

(

x y

(

x y

)

(

x y

) (1.2.11)

the equation is called

parabolic

. Thus, an equation is hyperb olic, elliptic or

parabolic within a certain domain in the

x y

plane. This is an imp ortant

distinctive mark determining the

solvability

of a

boundary problem

Boundary curves

or surfaces may be

open

closed

. A closed boundary

surface is one that surrounds the domain everywhere, conning it to a nite

surface or volume. Asimple closed smo oth curveis called

Jordan

curve

. An

open surface is one that do es not completely enclose the domain but lets it

extend to innity in at least one direction. Then the

Dirichlet

boundary

conditions

(

rst boundary value problem

)xthe value

(

x y

)onthebound-

ary,

Neumann

boundary conditions

(

second boundary value problem

)xthe

value of the

normal derivative

@u=@~n

on the boundary and a

Cauchy

condi-

tion

xes both value and normal derivativeatthe

same

place. The

Cauchy

condition actually represents an

initial condition

. The normal derivative is

the directional derivativeofafunction

(

x y

)inthe direction of the normal

at the point of the b oundary where the derivative is taken. A generalized

Neumann

boundary condition

(

thirdboundary value problem

)xes

(

x y

)

(

x y

)

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