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I have been referred by my friend to Ace_Tutor therefore this question is posted to Ace_Tutor only. I have a timed exam starting at 5:00 PM (CST) on 16th May, 2018 for the course “Numerical Methods for Engineering”. The exam covers Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs). I have attached all relevant chapters of my ebook which are covered in this exam therefore you please look at the attached chapters and the exam will be based on these chapters. Make sure that you have very good understanding and command on the attached chapters. These chapters are from the book “Numerical Methods for Engineers by Steven C. Chapra and Raymond P. Canale”As soon as i will get the exam I will send that to you and then you will tell me what all questions you can do 100% correct thereafter you need to deliver correct detailed solutions of those selected questions with in 1 hour and 30 minutes.
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Numerical Methods
for Engineers
PART SEVEN
ORDINARY DIFFERENTIAL
EQUATIONS
PT7.1
MOTIVATION
In the first chapter of this book, we derived the following equation based on Newtonâ??s
second law to compute the velocity y of a falling parachutist as a function of time t
[recall Eq. (1.9)]:
dy
c
5g2 y
m
dt
(PT7.1)
where g is the gravitational constant, m is the mass, and c is a drag coefficient. 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 parameters. 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 variable. 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 equation (or PDE) that involves two or more independent variables.
Differential equations are also classified as to their order. For example, Eq. (PT7.1)
is called a first-order equation because the highest derivative is a first 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
equation,
m
dx
d 2x
1 c 1 kx 5 0
dt
dt 2
(PT7.2)
where c is a damping coefficient and k is a spring constant. Similarly, an nth-order equation would include an nth derivative.
Higher-order equations can be reduced to a system of first-order equations. For Eq.
(PT7.2), this is done by defining a new variable y, where
y5
dx
dt
(PT7.3)
which itself can be differentiated to yield
dy
d 2x
5 2
dt
dt
(PT7.4)
699
700
ORDINARY DIFFERENTIAL EQUATIONS
Equations (PT7.3) and (PT7.4) can then be substituted into Eq. (PT7.2) to give
m
dy
1 cy 1 kx 5 0
dt
(PT7.5)
or
cy 1 kx
dy
52
m
dt
(PT7.6)
Thus, Eqs. (PT7.3) and (PT7.6) are a pair of first-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 first-order equations.
Some of the engineering applications in Chap. 28 deal with the solution of second-order
ODEs by reduction to a pair of first-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
y5
# ag 2 m yb dt
c
(PT7.7)
The right-hand side of this equation is called an indefinite integral because the limits of
integration are unspecified. This is in contrast to the definite 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 indefinite 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
( 1 2 e2(cym)t )
c
(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 fits 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 aâ??s and f â??s are specified
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
PT7.1
u
l
FIGURE PT7.1
The swinging pedulum.
MOTIVATION
701
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, Newtonâ??s second law can be used to develop the following differential equation
(see Sec. 28.4 for the complete derivation):
g
d 2u
1 sin u 5 0
2
l
dt
(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
g
d 2u
1 u50
l
dt 2
(PT7.11)
We have, therefore, transformed Eq. (PT7.9) into a linear form that is easy to solve
analytically.
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
engineers.
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 define mechanisms of change.
When combined with continuity laws for energy, mass, or momentum, differential equations 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 Newtonâ??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 velocity as a function of time (Fig. PT7.2). This equation could be utilized in a number of
different ways, including design purposes.
702
ORDINARY DIFFERENTIAL EQUATIONS
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
Newtonâ??s second law
of motion
dv
F
5
m
dt
Fourierâ??s heat law
q 5 2k¿
Fickâ??s law of diffusion
Faradayâ??s law
(voltage drop across
an inductor)
Variables and Parameters
Velocity (v), force (F), and
mass (m)
dT
dx
Heat flux (q), thermal conductivity (k9)
and temperature (T)
J 5 2D
dc
dx
Mass flux (J), diffusion coefficient (D),
and concentration (c)
¢VL 5 L
di
dt
Voltage drop (DV L ), inductance (L),
and current (i)
F = ma
Physical law
dv = g â?? c v
m
dt
ODE
Analytical Numerical
gm
v = c (1 â?? eâ?? (c/m)t)
vi 1 = vi (g â??
c
v) t
m i
Solution
FIGURE PT7.2
The sequence of events in the application of ODEs for engineering problem solving. The example shown is the velocity of a falling parachutist.
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 significance cannot be solved using the analytical methods of calculus. Thus, the methods discussed in the following chapters are
extremely important in all fields of engineering.
PT7.2
PT7.2
703
MATHEMATICAL BACKGROUND
MATHEMATICAL BACKGROUND
A solution of an ordinary differential equation is a specific function of the independent
variable and parameters that satisfies 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:
dy
5 22×3 1 12×2 2 20x 1 8.5
dx
(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.
y
4
3
x
(a)
dy/dx
8
3
â??8
(b)
x
704
ORDINARY DIFFERENTIAL EQUATIONS
how the zero values of the derivatives correspond to the point at which the original function is flatâ??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 differential 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
un11
1C
n11
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 equation 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 infinite number of possible functions (corresponding to an infinite 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.
y
C=3
C=2
C=1
C=0
C = â??1
C = â??2
x
PT7.3
ORIENTATION
705
Therefore, to specify the solution completely, a differential equation is usually accompanied by auxiliary conditions. For first-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 satisfies both the differential
equation and the specified 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 â??pinned downâ??â?? 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 reflective of the physical fact that at time zero the vertical velocity was zero. If the parachutist had already been in vertical motion at time zero, the
solution would have been modified 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 specified at the same value of the independent 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 specification 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
eigenvalues.
PT7.3
ORIENTATION
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 objectives 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 differential 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 Eulerâ??s method, which has a
very straightforward graphical interpretation. Then, we use visually oriented arguments
706
ORDINARY DIFFERENTIAL EQUATIONS
PT 7.1
Motivation
PT 7.2
Mathematical
background
PT 7.3
Orientation
25.1
Euler’s
method
PART 7
Ordinary
Differential
Equations
PT 7.6
Advanced
methods
25.2
Heun and
midpoint
methods
PT 7.5
Important
formulas
25.3
Runge-Kutta
CHAPTER 25
Runge-Kutta
Methods
EPILOGUE
PT 7.4
Trade-offs
25.4
Systems of
ODEs
25.5
Adaptive RK
methods
28.4
Mechanical
engineering
28.3
Electrical
engineering
CHAPTER 26
Stiffness/
Multistep
Methods
CHAPTER 28
Case Studies
CHAPTER 27
Boundary Value
and Eigenvalue
Problems
28.2
Civil
engineering
28.1
Chemical
engineering
26.2
Multistep
methods
27.1
Boundaryvalue problems
27.3
Software
packages
26.1
Stiffness
27.2
Eigenvalues
FIGURE PT7.5
Schematic representation of the organization of Part Seven: Ordinary Differential Equations.
PT7.3
ORIENTATION
707
to develop two improved versions of Eulerâ??s methodâ??the Heun and the midpoint techniques. After this introduction, we formally develop the concept of Runge-Kutta (or RK)
approaches and demonstrate how the foregoing techniques are actually first- and secondorder RK methods. This is followed by a discussion of the higher-order RK formulations
that are frequently used for engineering problem solving. In addition, we cover the application of one-step methods to systems of ODEs. Finally, the chapter ends with a
discussion of adaptive RK methods that automatically adjust the step size in response to
the truncation error of the computation.
Chapter 26 starts with a description of stiff ODEs. These are both individual and
systems of ODEs that have both fast and slow components to their solution. We introduce the idea of an implicit solution technique as one commonly used remedy for this
problem.
Next, we discuss multistep methods. These algorithms retain information of previous
steps to more effectively capture the trajectory of the solution. They also yield the truncation error estimates that can be used to implement step-size control. In this section, we
initially take a visual, intuitive approach by using a simple methodâ??the non-self-starting
Heunâ??to introduce all the essential features of the multistep approaches.
In Chap. 27 we turn to boundary-value and eigenvalue problems. For the former,
we introduce both shooting and finite-difference methods. For the latter, we discuss several approaches, including the polynomial and the power methods. Finally, the chapter
concludes with a description of the application of several software packages and libraries for solution of ODEs and eigenvalues.
Chapter 28 is devoted to applications from all the fields of engineering. Finally, a
short review section is included at the end of Part Seven. This epilogue summarizes and
compares the important formulas and concepts related to ODEs. The comparison includes
a discussion of trade-offs that are relevant to their implementation in engineering practice. The epilogue also summarizes important formulas and includes references for
advanced topics.
PT7.3.2 Goals and Objectives
Study Objectives. After completing Part Seven, you should have greatly enhanced
your capability to confront and solve ordinary differential equations and eigenvalue problems. General study goals should include mastering the techniques, having the capability
to assess the reliability of the answers, and being able to choose the â??bestâ??â?? method (or
methods) for any particular problem. In addition to these general objectives, the specific
study objectives in Table PT7.2 should be mastered.
Computer Objectives. Algorithms are provided for many of the methods in Part
Seven. This information will allow you to expand your software library. For example,
you may find it useful from a professional viewpoint to have software that employs the
fourth-order Runge-Kutta method for more than five equations and to solve ODEs with
an adaptive step-size approach.
In addition, one of your most important goals should be to master several of the
general-purpose software packages that are widely available. In particular, you should
become adept at using these tools to implement numerical methods for engineering
problem solving.
708
ORDINARY DIFFERENTIAL EQUATIONS
TABLE PT7.2 Specific study objectives for Part Seven.
1. Understand the visual representations of Eulerâ??s, Heunâ??s, and the midpoint methods
2. Know the relationship of Eulerâ??s method to the Taylor series expansion and the insight it provides
regarding the error of the method
3. Understand the difference between local and global truncation errors and how they relate to the
choice of a numerical method for a particular problem
4. Know the order and the step-size dependency of the global truncation errors for all the methods
described in Part Seven; understand how these errors bear on the accuracy of the techniques
5. Understand the basis of predictor-corrector methods; in particular, realize that the efficiency of the
corrector is highly dependent on the accuracy of the predictor
6. Know the general form of the Runge-Kutta methods; understand the derivation of the second-order
RK method and how it relates to the Taylor series expansion; realize that there are an infinite
number of possible versions for second- and higher-order RK methods
7. Know how to apply any of the RK methods to systems of equations; be able to reduce an nth-order
ODE to a system of n first-order ODEs
8. Recognize the type of problem context where step size adjustment is important …
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