使用MATLAB, OCTAVE和SCILAB模拟ODE-PDE模型

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"Simulation of ODE-PDE models with MATLAB, OCTAVE and SCILAB" 是一本由Alain Vande Wouwer, Philippe Saucez和Carlos Vilas合著的书籍，专注于利用MATLAB, OCTAVE和SCILAB进行常微分方程(ODE)和偏微分方程(PDE)模型的模拟。这本书由Springer出版社出版，以其高质量和清晰度而著称，适合对科学和工程应用感兴趣的读者。该书的核心内容涉及以下几个关键知识点： 1. **常微分方程(ODE)模拟**：ODE是描述系统动态行为的基本工具，广泛应用于物理、生物、化学、工程等多个领域。MATLAB提供了ode45等内置求解器，能够高效、精确地求解各种类型的ODE问题。OCTAVE和SCILAB作为开源替代品，也具有类似的求解功能，但可能在某些高级特性和用户界面方面有所不同。 2. **偏微分方程(PDE)模拟**：PDE用于描述空间和时间上的连续过程，如热传导、流体动力学和电磁学等。书中会讲解如何在MATLAB中使用pdepe等工具箱，以及在OCTAVE和SCILAB中实现PDE的数值解法。这包括有限差分、有限元或边界元方法等。 3. **MATLAB/Simulink**：Simulink是MATLAB的一个扩展，特别适用于系统仿真和动态系统建模。书中可能涵盖了如何使用Simulink构建和模拟复杂的ODE和PDE模型，包括多域建模和实时仿真。 4. **编程实践**：除了理论知识，作者可能还提供了实际案例和代码示例，帮助读者理解和应用这些工具解决实际问题。这将涵盖MATLAB, OCTAVE和SCILAB的语法和编程技巧，以提高读者的实践能力。 5. **跨平台比较**：由于涉及MATLAB、OCTAVE和SCILAB三种不同的软件环境，书中可能对比了它们在处理ODE和PDE问题时的优缺点，帮助读者选择最适合特定任务的工具。 6. **科学和工程应用**：书中的案例将集中在科学和工程领域的具体应用，比如控制系统设计、生物系统建模、材料科学或者环境工程等，这将使读者能将所学知识直接应用到实际工作场景。 7. **版权信息**：此书的版权由Springer International Publishing持有，所有权利受法律保护。虽然本书提供了丰富的教育资源，但未经许可，禁止翻译、复制、引用等行为。这本书是学习和掌握使用MATLAB, OCTAVE和SCILAB进行ODE和PDE模拟的宝贵资源，适合科研人员、工程师以及对此领域感兴趣的学者阅读。通过深入学习，读者可以提升在数值计算和仿真方面的技能，从而更好地理解和解决复杂系统的问题。

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2 1 An Introductory Tour

where X is the dependent variable, t the independent variable and µ

max

agiven

constant.

Equation (1.1) also requires an initial condition

X(t

) = X

(1.2)

where t

is the speciﬁed (initial) value of t and X

the speciﬁed (initial) value of X.

The solution to Eq. (1.1) is the dependent variable, X, as a function of the inde-

pendent variable, t. This function is

X(t) = X

(

max

(t−t

)

(1.3)

Equation (1.3) can be easily conﬁrmed as the solution of Eq. (1.1) subject to initial

condition (1.2).

We can note some interesting properties of the solution, Eq. (1.3):

• If µ

max

< 0, then X (t) decreases exponentially with t

• If µ

max

= 0, then X (t) is constant at X

• If µ

max

> 0, then X (t) increases exponentially with t

Thus, the value of µ

max

leads to markedly different solutions. For example, for the

case of bacterial growth, µ

max

> 0, while for radioactive decay, µ

max

< 0.

Another important property of Eq. (1.1) is that it is linear. In other words, X and

dX/dt are to the ﬁrst power. Generally, linear ODEs are relatively easy to solve in the

sense that analytical solutions such as Eq. (1.3) can be derived. Conversely, nonlinear

ODEs (for which X and/or dX/dt are not to the ﬁrst power) generally are not easily

solved analytically (these conclusions, of course, depend on the number of ODEs

and the particular form of their nonlinearities).

Because analytical solutions are generally difﬁcult, if not impossible, to derive,

we have to resort to numerical methods of solution which, in principle, can be used to

compute solutions to any system of ODEs, no matter how large (how many equations)

and how nonlinear the equations are. However, we will not obtain an analytical

solution (as a mathematical function, e.g., Eq. (1.3)). Rather, we will obtain the

dependent variables as a function of the independent one in numerical form. This

is the central topic of this book, i.e., the computation of numerical solutions. The

challenge then is to compute numerical solutions in reasonable time with acceptable

numerical accuracy.

As an illustration of how Eqs. (1.1) and (1.2) might be extended, we consider the

growth of bacteria (X) on a single substrate (S) in a culture (see for instance, [2])

S → X

is a pseudo-stoichiometric coefﬁcient (or yield coefﬁcient representing the mass

of substrate consumed/number of cells produced).

For the culture, which we consider as the contents of a reactor operated in batch

mode (no feed or product streams), the dynamic model is described by two coupled

ODEs derived from two mass balances for X and S

1.1 Some ODE Applications 3

Table 1.1 Bacterial growth laws

Case Growth rate model Law

1 µ = µ

max

Zero-order (constant) law

2 µ(S) = µ

max

Monod law: substrate limitation

3 µ(S) = µ

max

+S+S

Haldane law: substrate limitation and inhibition

4 µ(S, X) = µ

max

X+S

Contois law: substrate limitation and biomass inhibition

= ϕ(S, X) (1.4)

=−ν

ϕ(S, X) (1.5)

where S denotes the concentration of substrate in the reactor (mass of substrate/

volume reactor) and X the concentration of biomass (number of cells/volume

reactor). The solution to Eqs. (1.4) and (1.5) will be the dependent variables, X

and S, as a function of t in numerical form.

The kinetics (rate equation for the reaction) can be written as

ϕ(S, X) = µ(S, X)X (t) (1.6)

where µ(S, X) denotes the speciﬁc growth rate. Depending on the expression of

µ(S, X), the two mass balance equations for X and S can be linear or nonlinear. For

example, the speciﬁc growth rate models in Table 1.1 are used to model bacterial

growth. Case 1 in Table 1.1 corresponds to a constant speciﬁc growth rate, and

therefore to the linear Eq. (1.1). Cases 2 to 4 are nonlinear speciﬁc growth rates

whichifusedinEqs.(1.4) and (1.5) will preclude analytical solutions. Thus, we will

develop numerical solutions to Eqs. (1.4) and (1.5) when using cases 2–4.

The MATLAB function bacteria_odes and the MATLAB script

Main_bacteria can be used to compute numerical solutions to Eqs. (1.4) and

(1.5) for the four cases of Table 1.1. We can note the following points about the

function bacteria_odes:

1. The function named bacteria_odes is deﬁned by function xt =

bacteria_odes(t,x) The inputs to the function are t

, the independent vari-

able of Eqs. (1.4) and (1.5) (time), and x, the vector of dependent variables (X, S).

The function computes the derivatives from the RHS of Eqs. (1.4) and (1.5), and

returns these derivatives as a vector xt. Initially (at t = 0), X and S are available

from the initial conditions for Eqs. (1.4) and (1.5) (not yet speciﬁed). After the

integration of Eqs. (1.4) and (1.5) has progressed (by a numerical integration still

to be explained), the values of X and S are inputs to bacteria_odes from the

ODE integrator.

2. A set of global variables is deﬁned, which can be shared between functions (such

as between the function bacteria_odes and the main program that will be

4 1 An Introductory Tour

described next). Note, also, that a line beginning with a % is a comment (and

therefore has no effect on the executionof the MATLAB code; rather, it is included

only for informational purposes).

3. Since ODE integrators, such as the one called by the main program, generally

require that the dependent variables be stored as a vector, the two dependent vari-

ables of Eqs. (1.4) and (1.5), X and S, are extracted from the dependent variable

vector x (again, recall that x is an input to the function bacteria_odes).

4. One of the growth functions in Table 1.1 is then selected by the switch utility

of MATLAB, that is controlled by the value of the variable kinetics which is

set in the main program and passed to the function bacteria_odess as a

global variable. This coding illustrates a very important feature of numerical

ODE solutions: The ODE complexity (in this case, the nonlinearity of the growth

function) is not a limitation. In other words, ODEs of any complexity can be

handled numerically; this is certainly not the case if we are attempting to derive

an analytical solution.

5. The RHS of Eqs. (1.4) and (1.5) are then computed. Note that we adopted a con-

venient convention for naming derivatives: We add a character (such as t)tothe

dependent variable name, so that, for example, the derivative dX/dt becomes Xt.

6. Finally, the derivative vector consisting of the two elements Xt and St is returned

to the ODE integrator as a column vector. Note that the derivative vector [Xt

St] (two elements of the vector in a row delimited by square brackets) is inter-

preted by MATLAB as a row vector. To make it a column vector (as required by

the ODE integrator), we apply the transpose operator, which in MATLAB can be

denoted simply with an apostrophe (’). Other option would be to separate both

elements by a semicolon, i.e., [Xt; St].

function xt = bacteria_odes( t , x )

% Global variables

global nu mumax Km Ki Kc kinetics

% Transfer dependent variables

X = x (1);

S = x (2);

% Select growth function

switch kinetics

% Constant

case ('constant ')

mu = mumax ;

% Monod

case ('monod ')

mu = mumax

S /(Km+S );

% Haldane

case ('haldane ')

mu = mumax

S /(Km+S+(S ^2)/Ki );

% Contois

case ('contois ')

mu = mumax

S /(Kc

X+S );

end

6 1 An Introductory Tour

switch method

% ode45

case ('ode45 ')

options = odeset ('RelTol ',1e−3,'AbsTol ',1e−3);

options = odeset (options ,'Stats ','on ',...

'Events ',@events );

t =[t0 : Dtplot : tf ];

[tout , xout]=ode45 (@bacteria_odes, t , x0 , options );

% ode15s

case ('ode15s ')

options = odeset ('RelTol ',1e−3,'AbsTol ',1e−3);

options = odeset (options ,'Stats ','on ',...

'Events ',@events );

t =[t0 : Dtplot :

tf ];

[tout , xout]=ode15s (@bacteria_odes , t , x0 , options );

end

% Plot results

figure (1)

plot(tout , xout (: ,1));

xlabel ('t ');

ylabel ('X (t )');

title ('Biomass concentration ');

figure (2)

plot(tout , xout (: ,2));

xlabel ('t ');

ylabel ('S (t )');

title ('Substrate concentration ')

Script Main_bacteria Main program that calls integrator ode15s (or ode45)

This completes the programming of Eqs. (1.4) and (1.5). Now we consider the

main program Main_bacteria that calls the function bacteria_odes and

two integrators from the ODE SUITE:

1. A deﬁning statement is not required at the beginning.

2. The same global variables are again deﬁned so that they can be shared between the

main program Main_bacteria and the subordinate function bacteria_ode.

3. The model parameters used in Eqs. (1.4) and (1.5) and the growth functions are

then deﬁned numerically. The parameter values are taken from [3] and are given

in Table 1.2. The units are g (grams), l (liters), and h (hours). When these values

are substituted into the RHS of Eqs. (1.4) and (1.5) and the growth functions of

Table 1.1, the net units must be those of X and S per hour (the units of the LHS

derivatives of Eqs. (1.4) and (1.5)). Then, when these differential equations are

integrated, the resulting solution will have the units of X and S as a function of

time in hours.

4. The growth function is selected using the variable kinetics which is then passed

as a global variable to the function bacteria_odes to compute a particular

growth function. Note in this case the Contois growth function is selected since

it is not in a comment.

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