Numerical Methods for Engineers




In the  rst chapter of this book, we derived the following equation based on Newton’s

second law to compute the velocity y of a falling parachutist as a function of time t

[recall Eq. (1.9)]:


dt 5 g 2


m y (PT7.1)

where g is the gravitational constant, m is the mass, and c is a drag coef cient. Such

equations, which are composed of an unknown function and its derivatives, are called

differential equations. Equation (PT7.1) is sometimes referred to as a rate equation

because it expresses the rate of change of a variable as a function of variables and pa-

rameters. Such equations play a fundamental role in engineering because many physical

phenomena are best formulated mathematically in terms of their rate of change.

In Eq. (PT7.1), the quantity being differentiated, y, is called the dependent variable.

The quantity with respect to which y is differentiated, t, is called the independent vari-

able. When the function involves one independent variable, the equation is called an

ordinary differential equation (or ODE). This is in contrast to a partial differential equa-

tion (or PDE) that involves two or more independent variables.

Differential equations are also classi ed as to their order. For example, Eq. (PT7.1)

is called a ! rst-order equation because the highest derivative is a  rst derivative. A

second-order equation would include a second derivative. For example, the equation

describing the position x of a mass-spring system with damping is the second-order


m d 2x

dt 2 1 c


dt 1 kx 5 0 (PT7.2)

where c is a damping coef cient and k is a spring constant. Similarly, an nth-order equa-

tion would include an nth derivative.

Higher-order equations can be reduced to a system of  rst-order equations. For Eq.

(PT7.2), this is done by de ning a new variable y, where

y 5 dx

dt (PT7.3)

which itself can be differentiated to yield


dt 5

d 2x

dt2 (PT7.4)



Equations (PT7.3) and (PT7.4) can then be substituted into Eq. (PT7.2) to give

m dy

dt 1 cy 1 kx 5 0 (PT7.5)



dt 5 2

cy 1 kx

m (PT7.6)

Thus, Eqs. (PT7.3) and (PT7.6) are a pair of  rst-order equations that are equivalent to

the original second-order equation. Because other nth-order differential equations can be

similarly reduced, this part of our book focuses on the solution of  rst-order equations.

Some of the engineering applications in Chap. 28 deal with the solution of second-order

ODEs by reduction to a pair of  rst-order equations.

PT7.1.1 Noncomputer Methods for Solving ODEs

Without computers, ODEs are usually solved with analytical integration techniques. For

example, Eq. (PT7.1) could be multiplied by dt and integrated to yield

y 5 # ag 2 cm yb dt (PT7.7) The right-hand side of this equation is called an inde! nite integral because the limits of

integration are unspeci ed. This is in contrast to the de nite integrals discussed previously

in Part Six [compare Eq. (PT7.7) with Eq. (PT6.6)].

An analytical solution for Eq. (PT7.7) is obtained if the inde nite integral can be

evaluated exactly in equation form. For example, recall that for the falling parachutist

problem, Eq. (PT7.7) was solved analytically by Eq. (1.10) (assuming y 5 0 at t 5 0):

y(t) 5 gm

c (1 2 e2(cym)t) (1.10)

The mechanics of deriving such analytical solutions will be discussed in Sec. PT7.2. For

the time being, the important fact is that exact solutions for many ODEs of practical

importance are not available. As is true for most situations discussed in other parts of

this book, numerical methods offer the only viable alternative for these cases. Because

these numerical methods usually require computers, engineers in the precomputer era

were somewhat limited in the scope of their investigations.

One very important method that engineers and applied mathematicians developed to

overcome this dilemma was linearization. A linear ordinary differential equation is one

that  ts the general form

an(x)y (n)

1 p 1 a1(x)y¿ 1 a0(x)y 5 f(x) (PT7.8)

where y(n) is the nth derivative of y with respect to x and the a’s and f ’s are speci ed

functions of x. This equation is called linear because there are no products or nonlinear

functions of the dependent variable y and its derivatives. The practical importance of

linear ODEs is that they can be solved analytically. In contrast, most nonlinear equations


cannot be solved exactly. Thus, in the precomputer era, one tactic for solving nonlinear

equations was to linearize them.

A simple example is the application of ODEs to predict the motion of a swinging

pendulum (Fig. PT7.1). In a manner similar to the derivation of the falling parachutist

problem, Newton’s second law can be used to develop the following differential equation

(see Sec. 28.4 for the complete derivation):

d 2u

dt 2 1


l sin u 5 0 (PT7.9)

where u is the angle of displacement of the pendulum, g is the gravitational constant,

and l is the pendulum length. This equation is nonlinear because of the term sin u. One

way to obtain an analytical solution is to realize that for small displacements of the

pendulum from equilibrium (that is, for small values of u),

sin u > u (PT7.10)

Thus, if it is assumed that we are interested only in cases where u is small, Eq. (PT7.10)

can be substituted into Eq. (PT7.9) to give

d 2u

dt 2 1


l u 5 0 (PT7.11)

We have, therefore, transformed Eq. (PT7.9) into a linear form that is easy to solve


Although linearization remains a very valuable tool for engineering problem solving,

there are cases where it cannot be invoked. For example, suppose that we were interested

in studying the behavior of the pendulum for large displacements from equilibrium. In

such instances, numerical methods offer a viable option for obtaining solutions. Today,

the widespread availability of computers places this option within reach of all practicing


PT7.1.2 ODEs and Engineering Practice

The fundamental laws of physics, mechanics, electricity, and thermodynamics are usually

based on empirical observations that explain variations in physical properties and states

of systems. Rather than describing the state of physical systems directly, the laws are

usually couched in terms of spatial and temporal changes.

Several examples are listed in Table PT7.1. These laws de ne mechanisms of change.

When combined with continuity laws for energy, mass, or momentum, differential equa-

tions result. Subsequent integration of these differential equations results in mathematical

functions that describe the spatial and temporal state of a system in terms of energy,

mass, or velocity variations.

The falling parachutist problem introduced in Chap. 1 is an example of the derivation

of an ordinary differential equation from a fundamental law. Recall that Newton’s second

law was used to develop an ODE describing the rate of change of velocity of a falling

parachutist. By integrating this relationship, we obtained an equation to predict fall veloc-

ity as a function of time (Fig. PT7.2). This equation could be utilized in a number of

different ways, including design purposes.

FIGURE PT7.1 The swinging pedulum.




In fact, such mathematical relationships are the basis of the solution for a great

number of engineering problems. However, as described in the previous section, many

of the differential equations of practical signi cance cannot be solved using the analyti-

cal methods of calculus. Thus, the methods discussed in the following chapters are

extremely important in all  elds of engineering.

TABLE PT7.1 Examples of fundamental laws that are written in terms of the rate of change of variables (t 5 time and x 5 position).

Law Mathematical Expression Variables and Parameters

Newton’s second law Velocity (v), force (F), and of motion mass (m)

Fourier’s heat law Heat fl ux (q), thermal conductivity (k9) and temperature (T)

Fick’s law of diffusion Mass fl ux ( J), diffusion coeffi cient (D), and concentration (c)

Faraday’s law Voltage drop (DV L), inductance (L), (voltage drop across and current (i) an inductor)


dt 5



q 5 2k¿ dT


J 5 2D dc


¢VL 5 L di


F = ma

Analytical Numerical

v = (1 – e– (c/m)t) gm

c vi + 1 = vi + (g – vi) t



= g – vdv




Physical law



FIGURE PT7.2 The sequence of events in the application of ODEs for engineering problem solving. The exam- ple shown is the velocity of a falling parachutist.



A solution of an ordinary differential equation is a speci c function of the independent

variable and parameters that satis es the original differential equation. To illustrate this

concept, let us start with a given function

y 5 20.5×4 1 4×3 2 10×2 1 8.5x 1 1 (PT7.12)

which is a fourth-order polynomial (Fig. PT7.3a). Now, if we differentiate Eq. (PT7.12),

we obtain an ODE:


dx 5 22×3 1 12×2 2 20x 1 8.5 (PT7.13)

This equation also describes the behavior of the polynomial, but in a manner different

from Eq. (PT7.12). Rather than explicitly representing the values of y for each value of

x, Eq. (PT7.13) gives the rate of change of y with respect to x (that is, the slope) at every

value of x. Figure PT7.3 shows both the function and the derivative plotted versus x. Notice

FIGURE PT7.3 Plots of (a) y versus x and (b) dy/dx versus x for the function y 5 20.5×4 1 4×3 2 10×2 1 8.5x 1 1.









– 8



how the zero values of the derivatives correspond to the point at which the original func-

tion is ! at—that is, has a zero slope. Also, the maximum absolute values of the derivatives

are at the ends of the interval where the slopes of the function are greatest.

Although, as just demonstrated, we can determine a differential equation given the

original function, the object here is to determine the original function given the differ-

ential equation. The original function then represents the solution. For the present case,

we can determine this solution analytically by integrating Eq. (PT7.13):

y 5 # (22×3 1 12×2 2 20x 1 8.5) dx Applying the integration rule (recall Table PT6.2)

#un du 5 u n11

n 1 1 1 C   n ? 21

to each term of the equation gives the solution

y 5 20.5×4 1 4×3 2 10×2 1 8.5x 1 C (PT7.14)

which is identical to the original function with one notable exception. In the course of

differentiating and then integrating, we lost the constant value of 1 in the original equa-

tion and gained the value C. This C is called a constant of integration. The fact that such

an arbitrary constant appears indicates that the solution is not unique. In fact, it is but

one of an in nite number of possible functions (corresponding to an in nite number of

possible values of C) that satisfy the differential equation. For example, Fig. PT7.4 shows

six possible functions that satisfy Eq. (PT7.14).

FIGURE PT7.4 Six possible solutions for the integral of 22×3 1 12×2 2 20x 1 8.5. Each conforms to a different value of the constant of integration C.


x C = 0

C = – 1

C = – 2

C = 3

C = 2

C = 1


Therefore, to specify the solution completely, a differential equation is usually ac-

companied by auxiliary conditions. For  rst-order ODEs, a type of auxiliary condition

called an initial value is required to determine the constant and obtain a unique solution.

For example, Eq. (PT7.13) could be accompanied by the initial condition that at x 5 0,

y 5 1. These values could be substituted into Eq. (PT7.14):

1 5 20.5(0)4 1 4(0)3 2 10(0)2 1 8.5(0) 1 C (PT7.15)

to determine C 5 1. Therefore, the unique solution that satis es both the differential

equation and the speci ed initial condition is obtained by substituting C 5 1 into Eq.

(PT7.14) to yield

y 5 20.5×4 1 4×3 2 10×2 1 8.5x 1 1 (PT7.16)

Thus, we have “pinned down’’ Eq. (PT7.14) by forcing it to pass through the initial

condition, and in so doing, we have developed a unique solution to the ODE and have

come full circle to the original function [Eq. (PT7.12)].

Initial conditions usually have very tangible interpretations for differential equations

derived from physical problem settings. For example, in the falling parachutist problem,

the initial condition was re! ective of the physical fact that at time zero the vertical veloc-

ity was zero. If the parachutist had already been in vertical motion at time zero, the

solution would have been modi ed to account for this initial velocity.

When dealing with an nth-order differential equation, n conditions are required to

obtain a unique solution. If all conditions are speci ed at the same value of the indepen-

dent variable (for example, at x or t 5 0), then the problem is called an initial-value

problem. This is in contrast to boundary-value problems where speci cation of conditions

occurs at different values of the independent variable. Chapters 25 and 26 will focus on

initial-value problems. Boundary-value problems are covered in Chap. 27 along with



Before proceeding to numerical methods for solving ordinary differential equations, some

orientation might be helpful. The following material is intended to provide you with an

overview of the material discussed in Part Seven. In addition, we have formulated objec-

tives to focus your studies of the subject area.

PT7.3.1 Scope and Preview

Figure PT7.5 provides an overview of Part Seven. Two broad categories of numerical

methods for initial-value problems will be discussed in this part of this book. One-step

methods, which are covered in Chap. 25, permit the calculation of yi11, given the dif-

ferential equation and yi. Multistep methods, which are covered in Chap. 26, require

additional values of y other than at i.

With all but a minor exception, the one-step methods in Chap. 25 belong to what

are called Runge-Kutta techniques. Although the chapter might have been organized

around this theoretical notion, we have opted for a more graphical, intuitive approach to

introduce the methods. Thus, we begin the chapter with Euler’s method, which has a

very straightforward graphical interpretation. Then, we use visually oriented arguments


FIGURE PT7.5 Schematic representation of the organization of Part Seven: Ordinary Differential Equations.













Boundary Value

and Eigenvalue



Case Studies


26.2 Multistep methods

26.1 Stiffness

PT 7.2 Mathematical background

PT 7.6 Advanced methods

PT 7.5 Important formulas

28.4 Mechanical engineering

28.3 Electrical


28.2 Civil


28.1 Chemical


27.1 Boundary-

value problems

27.3 Software packages

27.2 Eigenvalues

PT 7.4 Trade-offs

PT 7.3 Orientation

PT 7.1 Motivation

25.2 Heun and midpoint methods

25.1 Euler’s method

25.3 Runge-Kutta

25.4 Systems of


25.5 Adaptive RK


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