C++, Java, Maple与MATLAB解常微分方程全攻略

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《常微分方程在C++、Java、Maple及MATLAB中的程序设计》是一本由H.J. Lee和W.E. Schiesser编著的专业书籍，版权归属于Chapman & Hall/CRC出版社，于2004年出版。该书深入探讨了常微分方程（ODE）和偏微分方程（PDE）的求解方法，并提供了详细的编程示例，涵盖了C、C++、Fortran、Java、Maple以及MATLAB等广泛使用的编程语言。书中首先介绍了基础理论，帮助读者理解微分方程在物理学、工程学等领域的实际应用。然后，作者通过实战案例展示了如何利用这些编程语言来解决实际问题，包括数值求解算法，如欧拉法、龙格-库塔法等，以及如何利用各自语言提供的数学库函数或工具包来处理微分方程系统。 C++部分着重于高效的算法实现和内存管理，展示了如何编写可扩展的代码，而Java部分则强调其面向对象的设计原则，以及如何将计算密集型任务与Java的并发特性相结合。Maple作为一个专业的数学软件，展示了其强大的符号计算能力在微分方程求解中的应用，包括求解解析解、绘制图形和分析解的稳定性。MATLAB作为科学计算领域的常用工具，书中讲解了如何利用其内置函数如ode45和pdepe进行高效的数值模拟。书中还包含了版权声明和致谢，以及如何获取更多产品信息的联系方式。作者对引用材料的尊重体现在合理引用和列出广泛的参考文献，确保了信息的权威性和准确性。《常微分方程在C++、Java、Maple及MATLAB中的程序设计》不仅是一本技术手册，也是一本实用的教程，适合那些希望在实际工作中应用微分方程解决复杂问题的工程师、科学家以及计算机专业学生。它不仅教授编程技能，还提供了理解和应用微分方程理论的桥梁，对于提升跨学科研究的能力具有很高的价值。

Taylor series (Equation 1.18), in this case, to Equation 1.19. In other words, ε

is the truncation error for Euler’s method, Equation 1.19.

We could logically argue that the truncation error could be reduced (for a

given h)byincluding more terms in the Taylor, e.g., the second derivative

term (d

/dt

)(h

/2!). Although this is technically true, there is a practical

problem. For the general ODE, Equation 1.1, we have only the ﬁrst derivative

available

= f (y

)

The question then in using the second derivative term of the Taylor series

is “How do we obtain the second derivative, d

/dt

?”. One answer would

be to differentiate the ODE, i.e.,





df(y, t)

∂ f

∂y

∂ f

∂t

∂ f

∂y

f +

∂ f

∂t

(1.20)

Then we can substitute Equation 1.20 in Equation 1.18:

i+1

= y

+ f

h +



∂ f

∂y

f +

∂ f

∂t



(1.21)

where again subscript “i” means evaluated at point i.

As an example of the application of Equation 1.21, consider the model ODE

= f (y, t) = λy (1.22)

where λ is a constant. Then

= λy



∂ f

∂y

f +

∂ f

∂t



= λ

(

λy

)

(note: ∂ f /∂t =0 since f =λy does not depend on t) and substitution in

Equation 1.21 gives

i+1

= y

+ λy

h + λ

(

λy

)

= y

(1 + λh + (λh)

/2!)

(1+λh+(λh)

/2!) is the Taylor series of y

λh

up to and including the h

term,

but y

λh

is the analytical solution to Equation 1.22 with the initial condition

y(t

) = y

for the integration step, h = t

i+1

− t

. Thus, as stated previously,

Equation 1.21 ﬁts the Taylor series of the analytical solution to Equation 1.22

up to and including the (d

/dt

)(h

/2!) term.

Of course, we could, in principle, continue this process of including ad-

ditional terms in the Taylor series, e.g., using the derivative of the second

derivative to arrive at the third derivative, etc. Clearly, however, the method

quickly becomes cumbersome (and this is for only one ODE, Equation 1.1).

Application of this Taylor series method to systems of ODEs involves a lot of

differentiation. (Would we want to apply it to a system of 100 or 1000 ODEs?

We think not.)

Ideally, we would like to have a higher-order ODE integration method

(higher than the ﬁrst-order Euler method) without having to take derivatives

of the ODEs. Although this may seem like an impossibility, it can in fact be

done by the Runge Kutta (RK) method.Inother words, the RK method can be

used to ﬁt the numerical ODE solution exactly to an arbitrary number of terms

in the underlying Taylor series without having to differentiate the ODE.Wewill

investigate the RK method, which is the basis for the ODE integration routines

described in this book.

The other important characteristic of a numerical integration algorithm (in

addition to not having to differentiate the ODE) is a way of estimating the

truncation error, ε,sothat the integration step, h, can be adjusted to achieve a

solution with a prescribed accuracy. This may also seem like an impossibility

since it would appear that in order to compute ε we need to know the exact

(analytical) solution. But if the exact solution is known, there would be no need

to calculate the numerical solution. The answer to this apparent contradiction

is the fact that we will calculate an estimate of the truncation error (and not the

exact truncation error which would imply that we know the exact solution). To

see how this might be done, consider computing a truncation error estimate

for the Euler method. Again, we return to the Taylor series (which is the

mathematical tool for most of the numerical analysis of ODE integration).

Now we will expand the ﬁrst derivative dy/dt

i+1

+ ··· (1.23)

/dt

is the second derivative we require in Equation 1.18. If the Taylor

series in Equation 1.23 is truncated after the h term, we can solve for this

second derivative

i+1

−

(1.24)

Equation 1.24 seems logical, i.e., the second derivative is a ﬁnite difference

(FD) approximation of the ﬁrst derivative. Note that Equation 1.24 has the im-

portant property that we can compute the second derivative without having

to differentiate the ODE; rather, all we have to do is use the ODE twice, at

grid points i and i + 1. Thus, the previous differentiation of Equation 1.20 is

avoided. However, note also that Equation 1.24 gives only an approximation

for the second derivative since it results from truncating the Taylor series of

Equation 1.23. Fortunately, the approximation of Equation 1.24 will generally

is too large and causes the large overshoot of the numerical solution above

the exact solution (and thus, a relatively large value of ε

), the averaging of

the derivatives at i and i + 1inFigure 1.2 reduces this overshoot (and the

truncation error is reduced from ε

to ε

Equation 1.25 can be rearranged into a more useful form. If we assume

that the truncation error of Euler’s method, ε

,isdue mainly to the second

derivative term (d

/dt

)(h

/2!) of Equation 1.18 (which will be the case if

the higher-order terms in Equation 1.18 are negligibly small), then

i+1

−



i+1

−



and Equation 1.25 can be written as a two-step algorithm:

i+1

= y

h (1.26a)



i+1

−



(1.26b)

i+1

= y

h + ε

= y

i+1

+ ε

(1.26c)

With a little algebra, we can easily show that y

i+1

of Equation 1.25 and y

i+1

of Equation 1.26c are the same. While Equation 1.25 and Equations 1.26c are

mathematically equivalent, Equations 1.26 have an advantage when used in

a computer program. Speciﬁcally, an algorithm that automatically adjusts h

to achieve a prescribed accuracy, tol, can be programmed in the following

steps:

1. Compute y

i+1

by the Euler method, Equation 1.26a. The superscript p

in this case denotes a predicted value.

2. Compute the estimated error, ε

,from Equation 1.26b. Note that

i+1

/dt = f (y

i+1

), where t

i+1

= t

+ h.

3. Pose the question is ε

< tol?Ifno, reduce h and return to 1. If yes,

continue to 4.

4. Add ε

from 3 to y

i+1

to obtain y

i+1

according to Equation 1.26c. The

superscript c denotes a corrected value.

5. Increment i, advance t

to t

i+1

by adding h,goto1.totake the next step

along the solution.

The algorithm of Equations 1.26 is termed a predictor-corrector method, which

we will subsequently discuss in terms of a computer program.

To conclude this introductory discussion of integration algorithms, we in-

troduce the RK notation.Ifwedeﬁne k

and k

(termed Runge Kutta constants

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