Maple在偏微分方程与边界值问题求解中的应用

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"Partial Differential Equations and Boundary Value Problems with Maple" 是一本第二版的教材，通过结合数学理论和Maple软件的应用，讲解如何解决偏微分方程（PDEs）和边界值问题。本书旨在教授读者理解和解决复杂的数学问题，特别是那些在物理、工程和其他科学领域常见的偏微分方程。作者George A. Articolo通过正式的数学概念阐述了PDE的基础知识，并展示了Maple软件的强大计算能力。Maple是一款强大的数学软件，广泛用于符号计算、数值计算以及可视化，它能帮助用户高效地处理复杂的数学运算和求解问题。在书中的内容可能包括以下几个方面： 1. 偏微分方程基础：介绍PDE的基本定义、分类（如线性和非线性，常微分和偏微分等）以及它们在物理现象中的应用，如热传导、波动和流体动力学等领域。 2. 边界条件和边界值问题：解释如何定义和处理边界条件，以及如何设定和求解边界值问题。这通常涉及到确定解在特定边界上的行为，以满足实际问题的要求。 3. 解的存在性和唯一性：探讨解的存在性和唯一性的定理，这对于理解PDE解决方案的性质至关重要。 4. 数值方法：介绍用Maple进行数值解法，如有限差分法、有限元法和格林函数法等，这些方法对于无法找到解析解的复杂问题特别有用。 5. Maple软件的应用：提供详细的Maple代码示例，展示如何利用Maple求解PDE，包括设置问题、运行计算和可视化结果。 6. 实例分析：书中可能包含各种实际问题的案例研究，比如声波传播、电磁场或化学反应扩散等，通过这些实例来巩固理论知识并提高实际应用技能。 7. 习题与练习：书后可能配有丰富的习题集，旨在帮助读者巩固所学，提升解决PDE问题的能力。通过阅读此书，读者不仅能够掌握偏微分方程的理论知识，还能学会使用Maple这一强大的工具，从而更有效地解决实际中的科学和工程问题。无论是初学者还是有经验的科研人员，都能从中受益，提升在PDE领域的计算能力。

Basic Review 3

The derivative of f(x) with respect to x:

> diff(f(x),x);

2 x (0.6)

The indeﬁnite integral of g(x) written symbolically:

> Int(g(x),x);



dx (0.7)

The evaluated indeﬁnite integral of g(x)

> int(g(x),x);

(0.8)

The deﬁnite integral of f(x) over the ﬁnite closed interval [1, 4] written symbolically:

> Int(f(x),x=1..4);



dx (0.9)

The evaluated deﬁnite integral of f(x) over the interval [1, 4]:

> int(f(x),x=1..4);

21 (0.10)

Factorization:

> factor(xˆ2−x−2);

(x +1)(x−2) (0.11)

Substitution:

> g(2):=subs(x=2,g(x));

g(2) := 8 (0.12)

Summation:

> S:=Sum(n*x,n=1..3);

S :=



n=1

nx (0.13)

4 Chapter 0

Evaluation of the previous command:

> S:=value(%);

S :=6x (0.14)

We must be aware of three other important items when using Maple worksheets. When we

want to declare new values of variables in an entire problem, we wipe out all the previous

declarations of these values by using the simple command

> restart:

When we want to use special computational packages that facilitate the use of Maple for

speciﬁc applications, we must bring these packages into the worksheet area by using a speciﬁc

command. For example, to implement the graphics capability of Maple, we bring the “plot”

package into the worksheet area by using the command

> with(plots):

To implement the integral transform commands into the worksheet, such as Fourier and

Laplace and their corresponding inverses, we bring the “transform” package into the worksheet

using the Maple command

> with(inttrans):

Generally, the commands to bring in special packages are made at the very beginning of the

Maple worksheet.

The preceding command-operations cover most of those in the text that use the Maple code.

There are other commands that are also valuble and will become apparent in the development

of the solutions of particular problems. Note the minimal number of operations and the

adherence to traditional style. Mastery of the preceding concepts, at the very beginning, will

set aside any problems we might have later with the code, allowing us to focus primarily on the

mathematics. Please be aware that different versions or releases of Maple have different

characteristics in entering commands and that the Maple help section should be read and used

to resolve any difﬁculties.

0.2 Preparation for Linear Algebra

A linear vector space consists of a set of vectors or functions and the standard operations of

addition, subtraction, and scalar multiplication. In solving ordinary and partial differential

equations, we assume the solution space to behave like an ordinary linear vector space.

A primary concern is whether or not we have enough of the correct vectors needed to span the

solution space completely. We now investigate these notions as they apply directly to

two-dimensional vector spaces and differential equations.

Basic Review 5

We use the simple example of the very familiar two-dimensional Euclidean vector space R2;

this is the familiar (x, y) plane. The two standard vectors in the (x, y) plane are traditionally

denoted as i and j. The vector i is a unit vector along the x-axis, and the vector j is a unit

vector along the y-axis. Any point in the (x, y) plane can be reached by some linear

combination, or superposition, of the two standard vectors i and j. We say the vectors “span”

the space. The fact that only two vectors are needed to span the two-dimensional space R2 is

not coincidental; three vectors would be redundant. One reason for this has to do with the fact

that the two vectors i and j are “linearly independent”—that is, one cannot be written as a

multiple of the other. The other reason has to do with the fact that in an n-dimensional

Euclidean space, the minimum number of vectors needed to span the space is n.

A more formal mathematical deﬁnition of linear independence between two vectors or

functions v1 and v2 reads as “The two vectors v1 and v2 are linearly independent if and only

if the only solution to the linear equation

c1 v1 +c2 v2 = 0

is that both c1 and c2 are zero.” Otherwise, the vectors are said to be linearly dependent.

In the simple case of the two-dimensional (x, y) space R2, linear independence can be

geometrically understood to mean that the two vectors do not lie along the same direction

(noncolinear). In fact, any set of two noncolinear vectors could also span the vector space of

the (x, y) plane. There are an inﬁnite number of sets of vectors that will do the job. One

common connection between all sets, however, is that all the sets can be shown to be linearly

dependent; that is, all the sets can be shown to be reducible to linear combinations of the

standard i and j vectors.

For example, the two vector sets

S1 ={i, j}

and

S2 ={i +j, 2 i −3 j}

are both linearly independent sets of vectors that span the two-dimensional (x, y) space. Note

that the vectors within each set are linearly independent, but the vectors between sets are

linearly dependent.

A set of vectors S ={v1,v2,v3,...,vn} that are linearly independent and that span the space is

called a set of “basis” vectors for that particular vector space. Thus, for the two-dimensional

Euclidean space R2, the vectors i and j form a basis, and for the three-dimensional Euclidean

space R3, vectors i, j, and k form a basis. The number of vectors in a basis is called the

“dimension” of the vector space.

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