高等工程数学：模型应用经典教程

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"Advanced-Engineering-Mathematics" 是一本专为高级工程专业人士设计的权威教材，它涵盖了高等工程数学的广泛内容，特别强调了建模应用。该书由 CRC Press 出版，隶属于 Taylor & Francis Group，这是一家全球知名的学术出版社，总部位于美国佛罗里达州博卡拉顿。此版本为2009年的最新修订版，日期为2011年7月15日，电子书的国际标准书号为 13:978-1-4200-0944-6 (eBook-PDF)。本书内容深入且实用，主要涉及诸如微积分、线性代数、复变函数、微分方程、概率统计、矩阵论、泛函分析等高级数学理论，这些都是工程领域不可或缺的基础工具。作者 S. Graham Kelly 以其丰富的教学经验和专业知识，将复杂的数学概念通过实际案例和模型解析，使之易于理解和应用到实际工程问题中。书中特别注重将理论与实践相结合，通过具体的工程模型构建，帮助读者掌握如何在设计、分析和优化工程系统时运用高级数学技巧。同时，由于版权问题，作者和出版社已尽力追踪并承认所有复制材料的版权所有者，并在必要时寻求许可，确保尊重知识产权。尽管出版商努力确保信息的准确性和可靠性，但读者在使用本书中的数据和信息时仍需自行判断其适用性，并承担可能产生的后果。这本著作是工程师、科研人员以及研究生进行深入学习和研究的重要参考资料，对于提升解决复杂工程问题的能力具有显著价值。

6 Advanced engineering mathematics with modeling applications

varies over the range of the dependent variable. Heat transfer problems can

be formulated for bars, cylinders, spheres, and other common volumes. In

all problems, the differential equations are second-order in spatial variables

and  rst-order in time. Such problems and their variations are considered

throughout the study.

Vibrations of a beam are governed by a partial differential equation

which is fourth-order in a spatial coordinate and second-order in time. A

normal-mode solution is assumed, to determine the free response leading

to a fourth-order ordinary differential equation. A nontrivial solution of the

equation occurs only for certain values of the normal-mode parameter, the

natural frequency. The problem is equivalent to an eigenvalue problem in

which the natural frequencies are the square roots of the eigenvalues and

the mode shapes are the corresponding eigenvectors. The mode-shape vec-

tors satisfy an orthogonality condition. Variations of the basic beam problem

studied include stretched beams, beams with transverse loading, beams on

an elastic foundation, and sets of elastically connected beams.

Heat transfer problems and vibration problems describe different physi-

cal phenomena, but the underlying mathematics used in solving the prob-

lems is similar. The normal-mode solution used in the vibration problem

is a form of separation of variables, which is the method used to solve the

heat transfer problem. Product solutions in the heat transfer problem satisfy

orthogonality conditions similar to those satis ed by the mode shapes in

the vibration problem. In both cases, nonhomogeneous problems are solved

using expansions in terms of homogeneous solutions. Such expansions are

called eigenvector expansions or Fourier series expansions.

The remainder of this chapter focuses on mathematical modeling, includ-

ing nondimensionalization and scaling. The theory of vector spaces and lin-

ear operators, presented in Chapter 2, provides a mathematical foundation

on which solution methods can be based. The general theory of ordinary dif-

ferential equations and methods of solution are presented in Chapter 3. Vari-

ational methods, approximate solution techniques based on energy methods,

including the fundamentals of the  nite-element method, are developed

in Chapter 4. A comprehensive theory of eigenvalue problems for discrete

and continuous systems is presented in Chapter 5. The study concludes in

Chapter 6 with a study of partial differential equations.

1.2 Conservation laws and mathematical modeling

A  nite number of basic conservation laws are used in the mathematical

modeling of physical problems. Listed by name only, they include:

Conservation of mass

Conservation of linear momentum

Conservation of angular momentum

Conservation of energy

•

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8 Advanced engineering mathematics with modeling applications

Using the Eulerian viewpoint, you  x yourself to a point in space and

observe the time-dependent properties of the material surrounding you as

particles of mass move past you. The Eulerian or  eld viewpoint is used in

modeling systems in which a transport process occurs, such as the  ow of a

 uid. As you observe from a  xed point in space, particles of matter move by

you with a velocity v. You observe changes in properties such as temperature

caused by changes occurring locally in the  eld, as well as changes occurring

because of changes in temperature of the  uid particles that pass by you.

The total rate of change of temperature is the sum of the local rate of change,

∂T/∂t, and the convective rate of change, v ⋅ ∇T. This form of the rate of change

is called a material derivative or a substantial derivative and is written as:

∂

+⋅∇v (1.1)

A sys

tem is a mac

roscopic version of the Lagrangian viewpoint in which

conservation laws are applied to a  xed set of matter. A control volume is the

macroscopic version of the Eulerian viewpoint in which conservation laws

are applied to a de ned region in space.

There are several techniques used for mathematical modeling. One

method is to begin with a set of equations which have been formulated from

the basic conservation laws and, using appropriate assumptions, to apply the

conservation laws to the given system. This is often the modeling technique

used in transport problems, especially the transport of momentum. The

Navier-Stokes equations are a set of partial differential equations governing

the  ow of a Newtonian  uid and are formulated using the Eulerian view-

point. Written in vector form, they are

∂

+⋅+⋅=

∇ρ ρ∇vv0 (1.2)

ρρ∇ ∇μ∇

∂

+⋅

()

=− +

vvF v

(1.3)

w(x,t)

Figure 1.4 The Lagrangian viewpoint is used for modeling the vibrations of a beam.

The vibrations are described using time and x, the initial distance of a particle along

the neutral axis of the beam from its left end, as independent variables.

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Chapter 1: Foundations of mathematical modeling 9

Equation 1.2 is called the continuity equation and is a statement of conserva-

tion of mass. Equation 1.3 is the vector form of conservation of linear momen-

tum. The dependent variables in Equation 1.2 and Equation 1.3 are ρ, the

mass density of the  uid, v, the three-dimensional velocity vector, and p, the

pressure. The body forces are represented by F

, and μ is the dynamic coef-

 cient of viscosity. There are  ve dependent variables, but Equation 1.2 and

Equation 1.3 provide only four scalar equations. If the  ow is incompressible,

then conservation of mass implies that ∇ ⋅ v = 0, and Equation 1.2 implies that

the density is constant. In other cases, an equation of state relating pressure

and density is used along with Equation 1.2 and Equation 1.3. Equations of

state often involve temperature as a variable. If temperature varies, then an

energy equation is used in conjunction with Equation 1.2 and Equation 1.3.

A second method of modeling is to apply basic forms of the conservation

laws directly to the system. If time is an independent variable in the problem

formulation, the conservation laws are applied at an arbitrary instant. In all

cases, the conservation laws are applied for arbitrary values of the depen-

dent variables. Two of the basic laws are conservation of momentum and

conservation of energy.

Conservation of momentum is often applied to a free-body diagram of an

in nitesimal particle drawn at an arbitrary instant. Conservation of momen-

tum, Newton’s second law as applied to a particle, is formulated as:

∑

= m (1.4)

where ∑ F represents the resultant force acting on a free-body diagram of the

particle drawn at an arbitrary instant and a is the acceleration vector. Conser-

vation of angular momentum is an appropriate form of the moment equation.

Conservation of energy as applied to a control volume takes the form that

the rate of energy accumulation within the control volume is equal to the rate at

which energy is transferred into the control volume through its boundaries.

A complete mathematical formulation often requires that the basic conser-

vation laws be supplemented by laws speci c to the system being modeled.

For example, conservation of energy can be supplemented by Newton’s law

of cooling and Fourier’s conduction law. The Navier-Stokes equations are

written for a viscous  uid which satis es Newton’s viscosity law. Constitutive

equations are used to represent stress-strain relations for many solids.

The method of application of mathematical modeling is different for differ-

ent types of systems, yet the concepts are the same for all systems. The prob-

lem is  rst identi ed. Basic questions are answered. Why is a mathematical

model necessary? What level of abstraction is required? What assumptions

are necessary for successful modeling? What are the independent and depen-

dent variables? What parameters are used in the modeling? Are numerical

values of these parameters necessary? Based on answers to these questions,

the problem is re ned. Basic laws of physics, in some form, are applied to the

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10 Advanced engineering mathematics with modeling applications

abstracted system. Empirical laws speci c to the system are applied as well

as geometrical constraints.

Finally, a set of equations is developed. Before a solution method is applied,

an assessment is necessary. Will solving the equations enable the objectives

of the modeling to be met? Can additional physical insights be gained from

examining the equations? Can these physical insights be used to aid in the

solution of the equations? Answers to the previous two questions require

that the mathematical equations be written in a nondimensional form.

A solution method is chosen and applied, resulting in mathematical

equations de ning the state of the system in terms of the independent vari-

ables. Another assessment is then required. Can these equations be used to

meet the objectives of the modeling? Does the solution contradict any of the

assumptions? If all appears to be well with the solution, it is then used to

satisfy the objectives of the modeling.

1.3 Problem formulation

This section provides examples of mathematical modeling of physical sys-

tems. The differential equations are derived, but the methods of solution are

found in later chapters.

Example 1.1 A  n or ex te nded s urf ace, a s i llu st rate d i n Fig ur e 1.5, i s at tac he d

to a body to enhance the rate of heat transfer to or from the body. The base

of the  n is maintained at the surface temperature, while its tip is insulated.

The surface of the  n is surrounded by an ambient medium which is at a

constant temperature T

∞

. The convective heat transfer coef cient between

the  n and the ambient is h. The  n is made of a material of thermal con-

ductivity k. The thickness of the  n is small enough that conduction can be

assumed negligible across the thickness of the  n and a one-dimensional

model can be used.

A differential approach is used to derive a differential equation governing

the steady-state temperature distribution in the  n. Apply the model to (a)

a straight  n of constant thickness; (b) an annular  n of constant thickness;

and (c) a straight  n of triangular pro le. In each case, show how to deter-

mine the total heat transfer at the base of the  n.

Solution The approach used is to apply conservation of energy to a control

volume of a differential element drawn at an arbitrary instant. The same

basic model can be used for a straight  n and an annular  n. First, consider

a straight  n. Let x represent the perpendicular distance from the base of the

 n to a point on the  n. For a straight  n, assume that heat conduction occurs

only in the x-direction. Let b(x) represent the thickness of the  n, and let w be

the width of the  n at its base.

Consider a differential slice of the  n as illustrated in Figure 1.6. Heat is

transferred into the slice at a distance x from the wall by conduction at a rate

of qA, where q is the rate of conductive heat transfer per unit area and A is the

area of the face of the slice. Heat is transferred from the slice by conduction at

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