Velten K. Mathematical Modeling and Simulation: Introduction for Scientists and Engineers

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3.5 Some Theory You Should Know 145

position of ﬁrst-order ODEs is underlined by the fact that higher-order ODEs can

be reformulated as ﬁrst-order ODEs. For example, setting z = y



in Equations

3.44–3.46, we get the following equivalent ﬁrst-order formulation:



= z (3.51)



=−y (3.52)

y(0) = 1 (3.53)

z(0) = 0 (3.54)

3.5.2

First-order ODEs

To keep things simple, we will thus conﬁne ourselves here to a formal deﬁnition

of ﬁrst-order ODEs. This can be done as follows [98]:

Deﬁnition 3.5.1 (First-order ODE) Let  ⊂ R

, F :  → R a continuous func-

tion. Then,



(t) = F(t, y(t)) (3.55)

is a ﬁrst-order ODE in the unknown function y(t). A function y :[a, b] → R is called

a solution of the ODE (3.55) if this equation is satisﬁed for every t ∈ [a, b] ⊂ R.

Note that the deﬁnition implicitly assumes that (t, y(t)) ∈  holds for all t ∈ [a, b].

In many practical cases, we will have  = R

, that is, the ODE will be deﬁned for

any t ∈ R and for any y ∈ R. An exception is, for example, the equation



(3.56)

where the right-hand side is undeﬁned for y = 0, which means we have to set

 = R

\ R ×{0}. Applying Deﬁnition 3.5.1 to the last equation, we have

F(t, y) =

(3.57)

which is continuous on  = R

\ R ×{0} as required by the deﬁnition. Applied to

(3.40), on the other hand, we have

F(t, y) = y (3.58)

which is again a continuous function on  = R

as required. For Equation 3.42,

we have instead

F(t, T) = r ·

(

− T

)

(3.59)

146 3 Mechanistic Models I: ODEs

which is also obviously continuous on  = R

(remember that we can use any

variable as the unknown of the ODE, and hence as the second argument of ∓!).

We may, thus, conclude that Equations 3.40, 3.42 and 3.57 are indeed ﬁrst-order

ODEs in the sense of Deﬁnition 3.5.1. The last equations should have made it clear

what is meant by F(t, y(t)) on the right-hand side of Equation 3.55. This term just

serves as a placeholder for arbitrary expressions involving t and y(t), which may of

course be much more complex than those in the last examples.

Note that the solution concept used in the deﬁnition complies with our discussion

of ODE solutions in the previous sections. For example, as discussed in Section 3.3,

y(t) = e

,viewedasafunctiony : R → R, solves Equation 3.40 since this equation is

satisﬁed for all t ∈ R. In terms of Deﬁnition 3.5.1, this means we have used [a, b] =

[−∞, ∞] here. Similarly, as discussed in Section 3.4.1, T(t) = T

− (T

− T

) ·e

−r·t

again viewed as a function T : R → R, solves Equation 3.42 since this equation is

satisﬁed for all t ∈ R.

3.5.3

Autonomous, Implicit, and Explicit ODEs

Equations 3.40 and 3.42 are examples of so-called autonomous ODEs since their

right-hand side does not depend on t explicitly. Note that there is, of course, an

implicit t dependence in these equations via y(t)andT(t), respectively. In terms

of Deﬁnition 3.5.1, we can say that the right-hand side of autonomous ODEs can

be written as F(y). Equation 3.42 would turn into a nonautonomous ODE, for

example, if T

, the room temperature, would be time dependent. Note that in this

case T(t) = T

− (T

− T

) · e

−r·t

would no longer solve Equation 3.42. If T

is time

dependent, you can verify that the derivative of this function would be



(t) = T



(t)(1 − e

−rt

) + r(T

(t) −T

−rt

(3.60)

which obviously means that Equation 3.42 is not satisﬁed.

3.5.4

TheInitialValueProblem

As it was mentioned above, ODEs without extra conditions are not uniquely

solvable. For example, Equation 3.40 is solved by f (t) = ce

for arbitrary c ∈ R.We

have seen that this can be overcome using the initial conditions for the unknown

function, which leads to initial value problems. A formal deﬁnition can be given as

follows:

Deﬁnition 3.5.2 (Initial value problem) Let  ⊂ R

, F :  → R a continuous

function and y

∈ R . Then,



(t) = F(t, y(t)) (3.61)

3.5 Some Theory You Should Know 147

y(a) = y

(3.62)

is an initial value problem for the ﬁrst-order ODE, Equation 3.61. A function

y :[a, b] → R solves this problem if it satisﬁes Equation 3.62 as well as 3.61 for

every t ∈ [a, b] ⊂ R.

Theoretically, it can be shown that this initial value problem is uniquely solvable if

F is Lipschitz continuous [98–100]. Basically, this continuity concept limits F in how

fast it can change. We will not go into a discussion of this concept here since unique

solvability is no problem in the majority of applications, including the examples in

this book. Equations 3.40–3.43 are obvious examples of initial value problems in

the sense of Deﬁnition 3.5.2.

3.5.5

Boundary Value Problems

Characteristic of initial value problems is the fact that conditions on the unknown

function are imposed for only one value of the independent variable. Problems

where conditions are imposed at more than one point are called boundary value

problems. Problems of this kind provide another possible way to select one particular

solution among the many solutions of an ODE. As we will see in Chapter 4, boundary

value problems are particularly important in partial differential equation models.

This is related to the fact that many boundary value problems are formulated in

terms of spatial coordinates, while, on the other hand, most differential equation

models involving spatial coordinates are formulated using partial differential

equations. If models involving spatial coordinate are written in terms of an ODE,

they can frequently be generalized in terms of a PDE. For example, the ODE



(x) = 0 (3.63)

describes the temperature distribution in a one-dimensional, homogeneous body

that is obtained after a very long (in fact, inﬁnite) time when you do not change the

environment of that body. Since this temperature distribution is time independent,

it is usually called the stationary temperature distribution of that body (compare

Deﬁnition 1.6.2). You may imagine, for example, a metal rod, which is insulated

against heat ﬂow except for its ends where you keep it at constant temperature for

a long time. Equation 3.63 is a special case of the heat equation,whichprovidesa

general description of the variation of temperature in time and space [97, 101]. The

general form of this equation is

∂T

∂t

= k



∂

∂x

∂

∂y

∂

∂z



(3.64)

This equation is a PDE since it involves derivatives with respect to several vari-

ables, and it is discussed in more detail in Section 4.2. Assuming a one-dimensional

body means that temperature variations in the y and z directions can be neglected

148 3 Mechanistic Models I: ODEs

(see the discussion of spatial dimensionality in Section 4.3.3), which means

we have

∂

∂y

∂

∂z

= 0 (3.65)

in Equation 3.64. On the other hand, the stationarity of T leads to

∂T

∂t

= 0 (3.66)

Inserting Equations 3.65 and 3.66 into Equation 3.64, Equation 3.63 is obtained

as desired.

Now let us be more speciﬁc about that metal rod. Consider a metal rod of length

1 m with constant temperatures 20

◦

and 10

◦

C at its ends. Mathematically, this

means that Equation 3.63 turns into the problem



(x) = 0, x ∈ [0, 1] (3.67)

T(0) = 20 (3.68)

T(1) = 10 (3.69)

This is a boundary value problem in the sense explained above since the

conditions accompanying the ODE are imposed at two different points. The math-

ematical problem (Equations 3.67–3.69) is easily solved based on the observation

that Equation 3.67 is satisﬁed exactly by the straight lines. The boundary conditions

(3.68) and (3.69) ﬁx the endpoints of this straight line such that the solution

becomes

T(x) = 20 −10x (3.70)

This equation describes a linear transition between the endpoint temperatures,

and this is of course what one would naturally expect here since we have assumed

a homogeneous metal rod. Looking at our derivation of Equation 3.70, it is

obvious that this is the unique solution of the boundary value problem (Equations

3.67–3.69). Note that Equation 3.67 can be considered as the ‘‘second simplest’’

ODE. The only simpler ODE is this:



(x) = 0 (3.71)

Equation 3.71 describes a straight line with slope zero, parallel to the x axis. It

is again a nice demonstration of the fact that initial or boundary conditions are

needed to obtain unique solution of ODEs. Without such a condition, Equation

3.71 is solved by the entire family of straight lines parallel to the x axis, that is, by

all functions of the form

T(x) = c (3.72)

3.5 Some Theory You Should Know 149

for arbitrary c ∈ R. Only after imposing an initial condition such as T(0) = 4, the

c in Equation 3.72 is ﬁxed and a unique solution of Equation 3.71 is obtained

(T(x) = 4inthiscase).

3.5.6

Example of Nonuniqueness

Let us go back to our consideration of boundary value problems. Unfortunately,

in contrast to initial value problems, it cannot be proved in general that boundary

value problems would lead us to unique solutions of an ODE. For example, consider



(x) =−y(x) (3.73)

In Equations 3.44–3.46, we have looked at this equation as an initial value

problem. Now without any extra conditions, the general solution of Equation 3.73

canbewrittenas

y(x) = a · sin(x) +b · cos(x) (3.74)

This means: For any values of a, b ∈ R, Equation 3.74 is a solution of Equation 3.73

(see also the remarks on the solution of Equation 3.73 in Section 3.6). Now it

depends on the particular boundary conditions that we impose whether we get

a uniquely solvable boundary value problem or not. Let us ﬁrst consider the

problem



(x) =−y(x) (3.75)

y(0) = 1 (3.76)





= 0 (3.77)

Inserting the general solution, Equation 3.74, into Equations 3.76 and 3.77, you

can easily verify that a = 0andb = 1 in Equation 3.74, which means that we get

y(x ) = cos(x) as the unique solution of this boundary value problem. On the other

hand, you will ﬁnd that the boundary value problem



(x) =−y(x) (3.78)

y(0) = 0 (3.79)

y(π ) = 0 (3.80)

is solved by any function of the form y(x) = a · sin(x)(a ∈ R), which means there

is no unique solution of this problem. Finally, inserting the general solution,

Equation 3.74, into conditions (3.82) and (3.83) of the boundary value problem



(x) =−y(x) (3.81)

150 3 Mechanistic Models I: ODEs

y(0) = 0 (3.82)

y(2π ) = 1 (3.83)

you are led to the contradictory requirements b = 0andb = 1, which means

that there is no solution of this boundary value problem. This means that we

cannot expect any general statement assuring unique solvability of boundary value

problems, as it was possible in the case of initial value problems.

In the problems considered below as well as in our discussion of solution

methods, we will conﬁne ourselves to initial value problems. While it was indicated

in the above discussion of the metal rod problem that boundary value problems

can frequently be seen as special cases of problems that are naturally formulated

in a PDE context, initial value problems, on the other hand, may be considered as

the ‘‘natural’’ setting for ODEs, since time serves as the independent variable in

the majority of applications (note that ‘‘initial value’’ immediately appeals to time

as the independent variable). The fact that unique solvability can be proved only for

initial value problems may be considered as an additional aspect of the naturalness

of these problems. This is not to say, however, that boundary value problems over

ODEs are unimportant. From a numerical point of view, boundary value problems

over ODEs are particularly interesting: they involve numerical methods such as

the shooting method and the multiple shooting method for the solution of ODEs,

as well as global methods like ﬁnite differences or collocation methods [102]. All

these, however, are beyond the scope of a ﬁrst introduction into mathematical

modeling.

3.5.7

ODE Systems

Note that not all initial value problems discussed above are covered by Deﬁnitions

3.5.1 and 3.5.2. Equations 3.47–3.50 or 3.51–3.54 involve systems of two ODEs

and two state variables, while the above deﬁnitions refer only to one ODE and one

state variable. This is solved in the usual way based on a vectorial notation. Using

boldface symbols for all vectorial quantities, the vector versions of Deﬁnitions 3.5.1

and 3.5.2 read as follows:

Deﬁnition 3.5.3 (System of ﬁrst-order ODEs) Let  ⊂ R

n+1

, F :  → R

continuous function. Then,



(t) = F(t, y(t)) (3.84)

is an ODE of ﬁrst order in the unknown function y(t). A function y :[a, b] ⊂

R → R

is called a solution of the ODE (3.84) if this equation is satisﬁed for every

t ∈ [a, b].

3.5 Some Theory You Should Know 151

Deﬁnition 3.5.4 (Initial value problem for a system of ﬁrst-order ODEs) Let

 ⊂ R

n+1

, F :  → R

a continuous function and y

∈ R

. Then,



(t) = F(t, y(t)) (3.85)

y(a) = y

(3.86)

is an initial value problem for the ﬁrst-order ODE, Equation 3.85. A function

y :[a, b] → R

solves this problem if it satisﬁes Equation 3.86 as well as Equation

3.85 for every t ∈ [a, b] ⊂ R.

As before, unique solvability of the initial value problem can be proved if F is

Lipschitz continuous [99]. The interpretation of Equations 3.47–3.50 or 3.51–3.54

in terms of these deﬁnitions goes along the usual lines. For example, using the

identiﬁcations

= T

(3.87)

a = 0

(t, y(t)) = r

· (y

(t) −y

(t))

(t, y(t)) = r

· (T

− y

(t))

Equations 3.47–3.50 turn into



(t) = F

(t, y(t)) (3.88)



(t) = F

(t, y(t)) (3.89)

(a) = y

(3.90)

(a) = y

(3.91)

Using the vectors

y =





, y





, y









, F(t, y(t)) =



(t, y(t))



(3.92)

you ﬁnally arrive at the form of the initial value problem as in Deﬁnition 3.5.4

above. So, as usual with vectorial laws, after writing a lot of stuff you see that

(almost) everything remains the same in higher dimensions.

152 3 Mechanistic Models I: ODEs

3.5.8

Linear versus Nonlinear

Particularly when solving ODEs (Section 3.6), it is important to know whether you

are concerned with a linear or nonlinear ODE. As usual, linear problems can be

treated much easier, and nonlinearity is one of the main causes of trouble that

you may have. You remember that linearity can be phrased as ‘‘unknowns may be

added, subtracted, and multiplied with known quantities’’. Applying this to ODEs,

it is obvious that a linear ODE is this:



(x) = a(x) · y(x) + b(x) (3.93)

This can be generalized to a system of linear ODEs as follows:



(x) =



j=1

(x) · y

(x) + b

(x), i = 1, ..., n (3.94)

Using y(x) = (y

(x), y

(x), ..., y

(x)), b(x) = (b

(x), b

(x), ..., b

(x)) and

A(x) =

⎛

⎜

⎝

(x) a

(x) ... a

(x)

(x) a

(x) ... a

(x)

. ...

(x) a

(x) ... a

(x)

⎞

⎟

⎠

(3.95)

Equation3.94canbewritteninmatrixnotationasfollows:



(x) = A(x) · y(x) +b(x) (3.96)

Most ODEs considered so far have been linear. For example, the ODE of the

body temperature model (Equation 3.29 in Section 3.4.1.1),



(t) = r · (T

− T(t)) (3.97)

can be reformulated as



(t) =−rT(t) +r · T

(3.98)

which attains the form of Equation 3.93 if we identify y = T, x = t, a(x) =−r,and

b(x) = r · T

. As another example, consider the ODE system of the alarm clock

model (Equations 3.36 and 3.37 in Section 3.4.2.1)



(t) = r

· (T

(t) −T

(t)) (3.99)



(t) = r

· (T

− T

(t)) (3.100)

3.6 Solution of ODE’s: Overview 153

Let us write this as



(t) =−r

· T

(t) +r

· T

(t) (3.101)



(t) =−r

· T

(t) +r

· T

(3.102)

Then, using x = t, y = (y

, y

) = (T

, T

), b = (b

, b

) = (0, r

· T

)and

A(x) =



−r

0 −r



(3.103)

Equation 3.96 is obtained from Equations 3.99 and 3.100, which means that

Equations 3.99 and 3.100 are a linear system of ODEs.

A nonlinear ODE considered above is (compare Equation 3.57 in Section 3.5.2)



(3.104)

This cannot be brought in the form of Equation 3.93 since it involves a division

operation. As discussed above, we have of course to be careful regarding the domain

of deﬁnition of this equations right-hand side, in contrast to the unrestricted

domains of deﬁnition of the linear ODEs considered above, which illustrates our

above statement that ‘‘nonlinearity is one of the main causes of trouble that you

may have with ODEs’’. A number of nonlinear ODE models are discussed below,

such as the predator–prey model (Section 3.10.1) and the wine fermentation model

(Section 3.10.2).

3.6

Solution of ODE’s: Overview

3.6.1

Toward the Limits of Your Patience

As it was already mentioned above, solving differential equations is a big challenge

and a big topic in mathematical research since a long time, and it seems unlikely

that mathematicians working in this ﬁeld will loose their jobs in the near future.

It is the signiﬁcance of differential equations in the applications that drives this

research effort. While the example applications considered so far referred to simple

technical systems (see the body temperature, alarm clock, and steel rod examples),

the simplicity of these examples is, on the other hand, a nice demonstration of the

broad range of differential equation applications. They are needed not only for an

investigation of sophisticated and complex systems; rather, as it was demonstrated

in Section 3.3, differential equations are frequently a part of the true mathematical

structure of a model as soon as exponential or other transcendental functions

are used.

154 3 Mechanistic Models I: ODEs

In Section 3.3, a comparison of the solution of linear systems of equations versus

the solution of differential equations was used to give you an idea of what it is

that makes differential equations hard to solve. As it was discussed there, it is the

inﬁnite dimensionality of these equations, that is, the fact that these equations ask

for an inﬁnite-dimensional unknown:afunction.Agreatpartofwhatisdoneby

mathematicians working on the numerical solution of differential equations today

can be described as an effort to deal with the inﬁnite dimensionality of differential

equations in an effective way. This is even more important when we are dealing

with PDEs, since, as explained above, these equations ask for functions depending

on several variables, that is, which are inﬁnite dimensional in the sense explained

above not only with respect to one independent variable (time in many cases), but

also with respect to one or more other variables (such as spatial coordinates).

Particularly when solving PDEs, you will easily be able to explore the limits of

your computer, even if you consider models of systems that do not seem to be

too complex at a ﬁrst glance, and even if you read this book years after this text is

written, that is, at a time when computers will be much faster than today (since

more complex problems are solved on faster computers). PDE models involving

a complex coupling of ﬂuid ﬂow with several other phenomena, such as models

of casting and solidiﬁcation or climate phenomena, may require several hours of

computation time or more, even on today’s fastest computers (Section 4.6.8 and

[103, 104]). PDE’s may thus not only help us to explore the limits of our computers,

but also of our patience ...

3.6.2

Closed Form versus Numerical Solutions

There are two basic ways how differential equations can be solved, which correspond

to Section 3.7 on closed form solutions and Section 3.8 on numerical solutions.

A closed form solution – which is sometimes also called an analytical solution –isa

solution of an equation that can be expressed as a formula in terms of ‘‘well-known’’

functions such as e

and sin(x). All solutions of ODEs considered so far are closed

form solutions in this sense. In the body temperature example above (Section 3.4.1),

Equation 3.25 is a closed form solution of the initial value problem (Equations 3.29

and 3.30). In practice, however, most ODEs cannot be solved in terms of closed

form solutions, frequently due to nonlinear right-hand sides of the equations.

Approximate numerical solutions of such ODEs are obtained using appropriate

numerical algorithms on the computer.

The borderline between closed form and numerical solutions is not really sharp

in the sense that well-known functions such as e

and sin(x) are of course also

computed approximately using numerical algorithms, and also in the sense that it

is a matter of deﬁnition what we call ‘‘well-known functions’’. For example, people

working in probability and statistics frequently need the expression

√



−t

dt (3.105)