使用Matlab解决非线性、微分及偏微分方程的高级数值方法

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"Advanced Numerical Methods with Matlab vol 2" 是一本专注于利用Matlab进行高级数值方法应用的书籍，特别是解决非线性、微分和偏微分方程的策略。该书由Bouchaib Radi和Abdelkhalak El Hami共同撰写，旨在帮助数学和机械工程领域的学者与从业者提升在Matlab环境下处理复杂计算问题的能力。书中涵盖了从基本概念到高级技巧的广泛内容，旨在帮助读者深入理解并应用数值方法来解决实际工程和科学问题。部分内容提到了“Resolution of Nonlinear, Differential and Partial Differential Equations”，这表明书籍将详细探讨如何利用Matlab工具和算法来求解这类方程。非线性方程的求解通常涉及迭代法，如牛顿-拉弗森方法或二分法；微分方程的数值解可能涵盖欧拉方法、龙格-库塔方法等；而偏微分方程（PDEs）的数值解可能包括有限差分法、有限元法或谱方法。书中可能还会讨论矩阵运算、数值稳定性分析、误差估计、收敛性测试等关键主题。此外，可能会介绍Matlab中的相关函数和工具箱，如ode45用于常微分方程（ODEs）的求解，pdepe用于偏微分方程的求解，以及fsolve用于非线性方程组的求解。通过实例和练习，读者可以学习如何编写Matlab代码，实现这些算法，并理解它们在实际问题中的应用。此外，对于那些对研究或私人研究、批评或评论目的的合理使用，根据版权、设计和专利法案1988年的规定，书籍允许在一定范围内复制、存储或传输。然而，任何形式的复制、存储或传播都需事先获得出版商的书面许可，除非是按照复制权许可协会（CLA）颁发的条款和许可证进行的。 "Advanced Numerical Methods with Matlab vol 2" 是一本面向高级用户的指南，它将帮助读者精通使用Matlab进行数值计算，特别是在处理非线性、微分和偏微分方程方面的技能。通过这本书，读者不仅可以提升编程技能，还能深化对数值方法理论基础的理解。

Solving Nonlinear Equations 7

The idea of the method is to choose a function f for the original function F , such

that:

– F (x)=0→ f(x)=x.

– The sequence (x

) deﬁned by x

∈ [a, b]; x

= f(x

n−1

) for n ≥ 1 converges

The problem of approximating ¯x up to  is therefore reduced to that of computing

, such that |x

− ¯x|≤.

EMARK.– For any given function F (x)=0, there are an inﬁnite number of choices

of f such that F (x)=0is equivalent to f(x)=x (e.g. f(x)=x + λF (x); λ ∈ R

∗

The challenge of the ﬁxed-point method lies in choosing f in such a way that the

sequence (x

) converges, and does so as quickly as possible. There are two natural

criteria for selecting the best function: convergence and rate of convergence. The

choice of the initial point x

also inﬂuences both of these criteria.

XAMPLE.– Suppose that we wish to compute the value of

√

2. This is equivalent to

ﬁnding the positive root α =

√

2 of the function

f(x)=x

− 2,

that is, solving a nonlinear equation.

It is easy to check that α =

√

2 is a ﬁxed point of the function φ(x)=−

+x+

simply by noting that φ(

√

2) =

√

2. Moreover, ∀x ∈ [1, 2], |φ



(x)|≤

= C.We

can therefore deﬁne the sequence x

∈ [1, 2] and x

n+1

= φ(x

), and apply the mean

value theorem to deduce that ∃η ∈ [1, 2], such that

n+1

− α| = |φ(x

) − φ(α)| = |φ



(η)||x

− α|.

Therefore,

− α|≤C

− α|, ∀n ≥ 0.

This shows that the sequence x

converges to the root α. We need to perform 34

iterations of the ﬁxed-point method to compute an approximate value for

√

2 that is

accurate to ten digits after the decimal point.

1.3.3. First convergence criterion

HEOREM.– Let x ∈ [a, b] be an isolated root of F (x), and suppose that f(x) is a

continuous and differentiable function on [a, b] satisfying f(

x)=x.

Solving Nonlinear Equations 9

We can write |x

− ¯x| as a function of |x

− x

n−1

| as follows:

ASE 1.– −1 <f



(x) < 0 on [a, b].

In this case, the sequences (x

) and (x

2n+1

) are adjacent and their limit ¯x is

always located between any two consecutive terms of the sequence (x

). Therefore, in

this speciﬁc case, |x

−x|≤|x

−x

n−1

|, and the stopping condition |x

−x

n−1

|≤

is sufﬁcient to guarantee that x

and x

n−1

approximate ¯x up to , one from above and

the other from below. Therefore, we can approximate ¯x up to  without needing to ﬁnd

ASE 2.– 0 <f



(x) < 1 (or if the sign of f



is not known on [a, b]).

In this case, the sequence (x

) converges monotonically, and we can show that

− ¯x|≤

1 − k

n+1

− x

Therefore, to guarantee a precision of  when approximating ¯x by x

, we can

simply apply the stopping condition |x

n+1

− x

|≤(1 − k).

1.3.5. Second convergence criterion (local criterion)

HEOREM.– Let ¯x ∈ [ a, b] be an isolated root of F (x), and let f(x) be a C

function,

such that f(¯x)=¯x.

1) If |f



(¯x)| < 1, then there exists a neighborhood V of ¯x such that the recursive

sequence (x

) associated with f converges to ¯x, ∀x

∈ V.

Furthermore, (x

) converges monotonically if 0 ≤ f



(¯x) < 1, and converges

while oscillating around

x if −1 <f



(¯x) < 0.

The set V is called the domain of attraction of ¯x.

2) If |f



(¯x)| > 1, then, ∀x

, the sequence x

= f(x

n−1

) does not converge to ¯x.

3) If |f



(¯x)| =1, then we cannot draw any conclusions about the convergence

behavior of the sequence (x

), which is unstable (either convergent or divergent) and

very sensitive to rounding errors.

EMARK.–

1) The convergence criterion |f



(¯x)| < 1 only holds in the neighborhood of ¯x (we

might not have convergence for every x

in [a, b]). We therefore need to choose the

right x

(in the domain of attraction) to apply this criterion.

2) The smaller the value of |f



(¯x)|, the faster the convergence (|f



(¯x)| is known

as the rate of convergence).

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