Functions And Their Applications In Mathematics Pdf
Engineering Mathematics with Examples and Applications provides a compact and concise primer in the field, starting with the foundations, and then gradually developing to the advanced level of mathematics that is necessary for all engineering disciplines. Therefore, this book's aim is to help undergraduates rapidly develop the fundamental knowledge of engineering mathematics. The book can also be used by graduates to review and refresh their mathematical skills. Step-by-step worked examples will help the students gain more insights and build sufficient confidence in engineering mathematics and problem-solving. The main approach and style of this book is informal, theorem-free, and practical. By using an informal and theorem-free approach, all fundamental mathematics topics required for engineering are covered, and readers can gain such basic knowledge of all important topics without worrying about rigorous (often boring) proofs. Certain rigorous proof and derivatives are presented in an informal way by direct, straightforward mathematical operations and calculations, giving students the same level of fundamental knowledge without any tedious steps. In addition, this practical approach provides over 100 worked examples so that students can see how each step of mathematical problems can be derived without any gap or jump in steps. Thus, readers can build their understanding and mathematical confidence gradually and in a step-by-step manner. Covers fundamental engineering topics that are presented at the right level, without worry of rigorous proofs. Includes step-by-step worked examples (of which 100+ feature in the work). Provides an emphasis on numerical methods, such as root-finding algorithms, numerical integration, and numerical methods of differential equations. Balances theory and practice to aid in practical problem-solving in various contexts and applications.
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Engineering Mathematics
with Examples
and Applications
Xin-She Yang
Middlesex University
School of Science and Technology
London, United Kingdom
Academic Press is an imprint of Elsevier
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ISBN: 978-0-12-809730-4
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Publisher: Nikki Levy
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Typeset by VTeX
Contents
About the Author ix
Preface xi
Acknowledgment xiii
Part I
Fundamentals
1. Equations and Functions
1.1. Numbers and Real Numbers 3
1.1.1. Notes on Notations and
Conventions 3
1.1.2. Rounding Numbers and Significant
Digits 5
1.1.3. Concept of Sets 6
1.1.4. Special Sets 8
1.2. Equations 8
1.2.1. Modular Arithmetic 10
1.3. Functions 11
1.3.1. Domain and Range 12
1.3.2. Linear Function 12
1.3.3. Modulus Function 14
1.3.4. Power Function 15
1.4. Quadratic Equations 16
1.5. Simultaneous Equations 19
Exercises 20
2. Polynomials and Roots
2.1. Index Notation 21
2.2. Floating Point Numbers 23
2.3. Polynomials 24
2.4. Roots 25
Exercises 28
3. Binomial Theorem and Expansions
3.1. Binomial Expansions 31
3.2. Factorials 32
3.3. Binomial Theorem and Pascal's Triangle 33
Exercises 35
4. Sequences
4.1. Simple Sequences 37
4.1.1. Arithmetic Sequence 38
4.1.2. Geometric Sequence 39
4.2. Fibonacci Sequence 40
4.3. Sum of a Series 41
4.4. Infinite Series 44
Exercises 45
5. Exponentials and Logarithms
5.1. Exponential Function 47
5.2. Logarithm 48
5.3. Change of Base for Logarithm 53
Exercises 54
6. Trigonometry
6.1. Angle 55
6.2. Trigonometrical Functions 57
6.2.1. Identities 58
6.2.2. Inverse 59
6.2.3. Trigonometrical Functions of Two
Angles 59
6.3. Sine Rule 62
6.4. Cosine Rule 62
Exercises 63
Part II
Complex Numbers
7. Complex Numbers
7.1. Why Do Need Complex Numbers? 67
7.2. Complex Numbers 67
7.3. Complex Algebra 68
7.4. Euler's Formula 71
7.5. Hyperbolic Functions 72
7.5.1. Hyperbolic Sine and Cosine 72
7.5.2. Hyperbolic Identities 74
7.5.3. Inverse Hyperbolic Functions 74
Exercises 75
v
vi Contents
Part III
Vectors and Matrices
8. Vectors and Vector Algebra
8.1. Vectors 79
8.2. Vector Algebra 80
8.3. Vector Products 83
8.4. Triple Product of Vectors 85
Exercises 86
9. Matrices
9.1. Matrices 87
9.2. Matrix Addition and Multiplication 90
9.3. Transformation and Inverse 93
9.4. System of Linear Equations 98
9.5. Eigenvalues and Eigenvectors 99
9.5.1. Distribution of Eigenvalues 104
9.5.2. Definiteness of a Matrix 107
Exercises 108
Part IV
Calculus
10. Differentiation
10.1. Gradient and Derivative 111
10.2. Differentiation Rules 114
10.3. Series Expansions and Taylor Series 117
Exercises 120
11. Integration
11.1. Integration 121
11.2. Integration by Parts 125
11.3. Integration by Substitution 128
Exercises 130
12. Ordinary Differential Equations
12.1. Differential Equations 131
12.2. First-Order Equations 132
12.3. Second-Order Equations 136
12.4. Higher-Order ODEs 142
12.5. System of Linear ODEs 143
Exercises 144
13. Partial Differentiation
13.1. Partial Differentiation 145
13.2. Differentiation of Vectors 146
13.3. Polar Coordinates 147
13.4. Three Basic Operators 149
Exercises 152
14. Multiple Integrals and Special
Integrals
14.1. Line Integral 153
14.2. Multiple Integrals 153
14.3. Jacobian 155
14.4. Special Integrals 157
14.4.1. Asymptotic Series 157
14.4.2. Gaussian Integrals 158
14.4.3. Error Functions 159
Exercises 161
15. Complex Integrals
15.1. Analytic Functions 163
15.2. Complex Integrals 165
15.2.1. Cauchy's Integral Theorem 166
15.2.2. Residue Theorem 168
Exercises 169
Part V
Fourier and Laplace Transforms
16. Fourier Series and Transform
16.1. Fourier Series 173
16.1.1. Fourier Series 173
16.1.2. Orthogonality 175
16.1.3. Determining the Coefficients 176
16.2. Fourier Transforms 179
16.3. Solving Differential Equations Using
Fourier Transforms 182
16.4. Discrete and Fast Fourier Transforms 183
Exercises 185
17. Laplace Transforms
17.1. Laplace Transform 187
17.1.1. Laplace Transform Pairs 189
17.1.2. Scalings and Properties 189
17.1.3. Derivatives and Integrals 191
17.2. Transfer Function 192
17.3. Solving ODE via Laplace Transform 194
17.4. Z-Transform 196
17.5. Relationships between Fourier, Laplace
and Z-transforms 197
Exercises 197
Part VI
Statistics and Curve Fitting
18. Probability and Statistics
18.1. Random Variables 201
Contents vii
18.2. Mean and Variance 202
18.3. Binomial and Poisson Distributions 203
18.4. Gaussian Distribution 207
18.5. Other Distributions 209
18.6. The Central Limit Theorem 211
18.7. Weibull Distribution 212
Exercises 214
19. Regression and Curve Fitting
19.1. Sample Mean and Variance 215
19.2. Method of Least Squares 217
19.2.1. Maximum Likelihood 217
19.2.2. Linear Regression 217
19.3. Correlation Coefficient 219
19.4. Linearization 221
19.5. Generalized Linear Regression 222
19.6. Hypothesis Testing 225
19.6.1. Confidence Interval 225
19.6.2. Student's t -Distribution 226
19.6.3. Student's t -Test 227
Exercises 228
Part VII
Numerical Methods
20. Numerical Methods
20.1. Finding Roots 231
20.2. Bisection Method 232
20.3. Newton-Raphson Method 233
20.4. Numerical Integration 234
20.5. Numerical Solutions of ODEs 237
20.5.1. Euler Scheme 237
20.5.2. Runge-Kutta Method 237
Exercises 241
21. Computational Linear Algebra
21.1. System of Linear Equations 243
21.2. Gauss Elimination 244
21.3. LU Factorization 247
21.4. Iteration Methods 249
21.4.1. Jacobi Iteration Method 250
21.4.2. Gauss-Seidel Iteration 253
21.4.3. Relaxation Method 253
21.5. Newton-Raphson Method 254
21.6. Conjugate Gradient Method 254
Exercises 255
Part VIII
Optimization
22. Linear Programming
22.1. Linear Programming 259
22.2. Simplex Method 260
22.2.1. Basic Procedure 261
22.2.2. Augmented Form 262
22.3. A Worked Example 263
Exercises 265
23. Optimization
23.1. Optimization 267
23.2. Optimality Criteria 269
23.2.1. Feasible Solution 269
23.2.2. Optimality Criteria 269
23.3. Unconstrained Optimization 270
23.3.1. Univariate Functions 270
23.3.2. Multivariate Functions 271
23.4. Gradient-Based Methods 275
23.4.1. Newton's Method 275
23.4.2. Steepest Descent Method 276
23.5. Nonlinear Optimization 278
23.5.1. Penalty Method 278
23.5.2. Lagrange Multipliers 279
23.6. Karush-Kuhn-Tucker Conditions 280
23.7. Sequential Quadratic Programming 281
23.7.1. Quadratic Programming 281
23.7.2. Sequential Quadratic
Programming 282
Exercises 283
Part IX
Advanced Topics
24. Partial Differential Equations
24.1. Introduction 287
24.2. First-Order PDEs 288
24.3. Classification of Second-Order PDEs 290
24.4. Classic Mathematical Models: Some
Examples 291
24.4.1. Laplace's and Poisson's
Equation 291
24.4.2. Parabolic Equation 292
24.4.3. Hyperbolic Equation 293
24.5. Solution Techniques 294
24.5.1. Separation of Variables 294
24.5.2. Laplace Transform 296
24.5.3. Fourier Transform 296
24.5.4. Similarity Solution 297
24.5.5. Change of Variables 298
Exercises 299
viii Contents
25. Tensors
25.1. Summation Notations 301
25.2. Tensors 302
25.2.1. Rank of a Tensor 303
25.2.2. Contraction 303
25.2.3. Symmetric and Antisymmetric
Tensors 304
25.2.4. Tensor Differentiation 305
25.3. Hooke's Law and Elasticity 306
Exercises 307
26. Calculus of Variations
26.1. Euler-Lagrange Equation 309
26.1.1. Curvature 309
26.1.2. Euler-Lagrange Equation 311
26.2. Variations with Constraints 314
26.3. Variations for Multiple Variables 316
Exercises 317
27. Integral Equations
27.1. Integral Equations 319
27.1.1. Fredholm Integral Equations 319
27.1.2. Volterra Integral Equation 320
27.2. Solution of Integral Equations 321
27.2.1. Separable Kernels 321
27.2.2. Volterra Equation 322
Exercises 324
28. Mathematical Modeling
28.1. Mathematical Modeling 325
28.2. Model Formulation 326
28.3. Different Levels of Approximations 328
28.4. Parameter Estimation 330
28.5. Types of Mathematical Models 332
28.5.1. Algebraic Equations 332
28.5.2. Tensor Relationships 333
28.5.3. Differential Equations: ODE and
PDEs 333
28.5.4. Functional and Integral
Equations 335
28.5.5. Statistical Models 336
28.5.6. Fuzzy Models 337
28.5.7. Learned Models 337
28.5.8. Data-Driven Models 338
28.6. Brownian Motion and Diffusion:
AWorkedExample 338
Exercises 340
A. Mathematical Formulas
A.1. Differentiation and Integration 341
A.2. Complex Numbers 341
A.3. Vectors and Matrices 341
A.4. Fourier Series and Transform 342
A.5. Asymptotics 343
A.6. Special Integrals 343
B. Mathematical Software Packages
B.1. Matlab 345
B.1.1. Matlab 345
B.1.2. MuPAD 346
B.2. Software Packages Similar to Matlab 347
B.2.1. Octave 347
B.2.2. Scilab 348
B.3. Symbolic Computation Packages 348
B.3.1. Mathematica 348
B.3.2. Maple 348
B.3.3. Maxima 349
B.4. R and Python 350
B.4.1. R 350
B.4.2. Python 350
C. Answers to Exercises
Bibliography 381
Index 383
Chapter 28
Mathematical Modeling
Chapter Points
•Mathematical modeling is introduced with the basic modeling procedure, including mathematical model formulation based on
physical laws, parameter estimation and normalization.
•Different levels of approximations are explained to discuss the assumptions, abstractions and the balance of accuracy and model
complexity.
•Different types of models are explained with some examples relevant to science and engineering applications.
•A worked example is presented in detail to model Brownian motion and diffusion.
28.1 MATHEMATICAL MODELING
Mathematical modeling is the process of formulating an abstract model in terms of mathematical language to describe
the complex behavior of a real system. Mathematical models are quantitative models and often expressed in terms of
ordinary differential equations and partial differential equations. Mathematical models can also be statistical models, fuzzy
logic models and empirical relationships. In fact, any model description using mathematical language can be called a
mathematical model. Mathematical modeling is widely used in natural sciences, computing, engineering, meteorology,
economics and finance. For example, theoretical physics is essentially all about the modeling of real-world processes using
several basic principles (such as the conservation of energy and momentum) and a dozen important equations (such as the
wave equation, the Schrödinger equation, the Einstein equation). Most of these equations are partial differential equations.
An important feature of mathematical modeling is its interdisciplinary nature. It involves applied mathematics, computer
sciences, physics, chemistry, engineering, biology and other disciplines such as economics, depending on the problem of
interest. Mathematical modeling in combination with scientific computing is an emerging interdisciplinary technology.
Many international companies use it to model physical processes, to design new products, to find solutions to challenging
problems, and to increase their competitiveness in international markets.
Example 28.1
One of the simplest models we learned in school is probably Newton's second law that relates the force F acted on a body with a
mass m to its acceleration a .Thatis
F= ma,
which is one of the most accurate models in science. This is a linear relationship and thus a linear model, but a very well-tested
model.
Apart from a simple mathematical formula, as a mathematical model, all the quantities involved such as force, mass and
acceleration must have appropriate units. For example, the unit of F is Newton (N), the unit of mass is kilogram (kg), while the
acceleration has a derived unit (a combination of units) of m/s2 . Therefore, a person of 80 kg has a weight (the force acted upon the
person by the Earth) is W= mg where g=9. 8 m/s2 is the acceleration due to gravity. That is
W= mg = 80 ( kg )×9 .8( m/s2 )=784 N.
If the units are wrong, even a good model will give wrong values. This highlights the importance of units and the parameters (e.g.,
ghere) in mathematical modeling.
Engineering Mathematics with Examples and Applications
Copyright © 2017 Elsevier Inc. All rights reserved. 325
326 PART | IX Advanced Topics
FIGURE 28.1 Mathematical modeling.
Mathematical modeling is an iterative, multidisciplinary process with many steps from the abstraction of the processes
in nature to the construction of the full mathematical models. The basic steps of mathematical modeling can be summarized
as meta-steps shown in Fig. 28.1 . The process typically starts with the analysis of a real world problem so as to extract the
fundamental physical processes by idealization and various assumptions. Once an idealized physical model is formulated, it
can then be translated into the corresponding mathematical model in terms of partial differential equations (PDEs), integral
equations, and statistical models. Then, the mathematical model should be investigated in great detail by mathematical
analysis (if possible), numerical simulations and other tools so as to make predictions under appropriate conditions. Then,
these simulation results and predictions will be validated against the existing models, well-established benchmarks, and
experimental data. If the results are satisfactory (but they rarely are at first), then the mathematical model can be accepted.
If not, both the physical model and mathematical model will be modified based on the feedback, then the new simulations
and prediction will be validated again.
After a certain number of iterations of the whole process (often many), a good mathematical model can properly be
formulated, which will provide great insight into the real world problem and may also predict the behavior of the process
under study.
For any physical problem in physics and engineering, for example, there are traditionally two ways to deal with it by
either theoretical approaches or field observations and experiments.
The theoretical approach in terms of mathematical modeling is an idealization and simplification of the real problem
and the theoretical models often extract the essential or major characteristics of the problem. The mathematical equations
obtained even for such over-simplified systems are usually very difficult for mathematical analysis. On the other hand, the
field studies and experimental approach can be expensive if not impractical. Apart from financial and practical limitations,
other constraining factors include the inaccessibility of the locations, the range of physical parameters, and time for carrying
out various experiments. As the computing speed and power of computers have increased dramatically in the last few
decades, a practical third way or approach is emerging, which is computational modeling and numerical experimentation
based on mathematical models. It is now widely acknowledged that computational modeling and computer simulations
serve as a cost-effective alternative, bridging the gap between theory and practice as well as complementing the traditional
theoretical and experimental approaches to problem solving.
Mathematical modeling is essentially an abstract art of formulating the mathematical models from their corresponding
real-world problems. The mastery of this art requires practice and experience, and it is not easy to teach such skills as the
style of mathematical modeling largely depends on each person's own insight, abstraction, type of problems, and experience
of dealing with similar problems. Even for the same physical process, different models could be obtained, depending on
the emphasis of some part of the process, say, based on your interest in certain quantities in a particular problem, while the
same quantities could be viewed as unimportant in other processes and other problems.
28.2 MODEL FORMULATION
Mathematical modeling often starts with the analysis of the physical process and attempts to make an abstract physical
model by idealization and approximations. From this idealized physical model, we can use the various first principles such
as the conservation of mass, momentum, energy and Newton's laws to translate into mathematical equations. However,
such transformation from practice to theory can rarely be achieved in a single step, thus an iterative loop between theory
and practice is needed, as pointed out by the famous statistician George Box.
As an example, let us now look at the example of the diffusion process of sugar in a glass of water. We know that the
diffusion of sugar will occur if there is any spatial difference in the sugar concentration. The physical process is complicated
Mathematical Modeling Chapter | 28 327
FIGURE 28.2 Representative element volume (REV).
and many factors could affect the distribution of sugar concentration in water, including the temperature, stirring, mass of
sugar, type of sugar, how you add the sugar, even geometry of the container and others. We can idealize the process by
assuming that the temperature is constant (so as to neglect the effect of heat transfer), and that there is no stirring because
stirring will affect the effective diffusion coefficient and introduce the advection of water or even vertices in the (turbulent)
water flow.
We then choose a representative element volume (REV) whose size is very small compared with the size of the cup so
that we can use a single value of concentration to represent the sugar content inside this REV. If this REV is too large, there
is a considerable variation in sugar concentration inside this REV. We also assume that there is no chemical reaction between
sugar and water (otherwise, we are dealing with something else). If you drop the sugar into the cup from a considerable
height, the water inside the glass will splash and thus fluid volume will change, and this becomes a fluid dynamics problem.
So we are only interested in the process after the sugar is added and we are not interested in the initial impurity of the water
(to a certain degree).
With these assumptions, the whole process is now idealized as the physical model of the diffusion of sugar in still water
at a constant temperature. Now we have to translate this idealized model into a mathematical model, and in the present
case, a parabolic partial differential equation or diffusion equation.
Let c be the averaged concentration in a representative element volume with a volume dV inside the cup, and let
be an arbitrary, imaginary closed volume (much larger than our REV but smaller than the container, see Fig. 28.2 ). We
know that the rate of change of the mass of sugar per unit time inside is
δ1 = ∂
∂t %%%
cdV , (28.1)
where t is time. As the mass is conserved, this change of sugar content in must be supplied in or flow out over the surface
= ∂ enclosing the region .LetJ be the flux through the surface, thus the total mass flux through the whole surface
is
δ2 =%%
J·d S.
Thus the conservation of total mass in requires that
δ1 + δ2 = 0,
or
∂
∂t %%%
cdV +%%
J·d S=0. (28.2)
This is essentially the integral form of the mathematical model. Using the divergence theorem of Gauss
%%
J·d S=%%% ∇·JdV, (28.3)
328 PART | IX Advanced Topics
we can convert the surface integral into a volume integral. We thus have
∂
∂t %%%
cdV +%%% ∇· JdV = 0. (28.4)
Since the domain is fixed (independent of t ), we can interchange the differentiation and integration in the first term, we
now get
%%%
∂c
∂t dV +%%% ∇·JdV =%%% [∂c
∂t +∇· J]dV = 0. (28.5)
Since the enclosed domain is arbitrary, the above equation should be valid for any shape or size of ; therefore, the
integrand must be zero. We finally have
∂c
∂t +∇·J=0. (28.6)
This is the differential form of the mass conservation. It is a partial differential equation (PDE), and this mathematical
model is a PDE.
Example 28.2
As we know that diffusion occurs from the higher concentration to lower concentration, the rate of diffusion is proportional to the
gradient ∇c of the concentration. The flux J over a unit surface area is given by Fick's law
J=− D∇ c,
where D is the diffusion coefficient which depends on the temperature and the type of materials. The negative sign means the
diffusion is opposite to the gradient. Substituting this into the mass conservation (28.6) ,wehave
∂c
∂t −∇· (D ∇c) =0,
or
∂c
∂t =∇· (D∇c).
In the simplified case when D is constant, we have
∂c
∂t = D∇2 c, (28.7)
which is the well-known diffusion equation.
This equation can be applied to study many phenomena such as heat conduction, pollutant transport, groundwater flow
and concentrations if we replace D by their corresponding physical parameters.
28.3 DIFFERENT LEVELS OF APPROXIMATIONS
As we have just seen, we can formulate some mathematical models once we have made some appropriate assumptions such
as the process being the same inside the cup as well near the edge of the cup. Otherwise, we may have to deal with the
so-called a boundary-layer phenomenon which is not much relevant for the current model.
The assumptions essentially determine the level of consideration and thus the level of approximations. Let us use gravity
as an example to explain this issue.
Mathematical Modeling Chapter | 28 329
Example 28.3
Newton's law of universal gravitation can be applied to almost any objects. It can often be written as the following formula:
F= Gm 1 m2
r2 ,
which essentially states that the force F between two masses m1 and m2 is inversely proportional to the square of the distance r .
Here, G is the universal gravitational constant G=6. 674 ×10 −11 Nm
2/kg 2 . However, in deriving this formula, some assumptions
were made. One of the assumptions is that the two masses are both point masses; that is, their geometrical size does not matter.
This is true for a system of planets where the distance between a planet (say, the Earth) to the Sun is sufficiently large compared to
the sizes of the celestial bodies.
Imagine that we are trying to calculate the gravitational force between the Earth and its satellites, we have to be careful about
the distance r used. Obviously, we can say that ris the distance between the center of the Earth to the center of the satellite. Now
the question what about the distribution of mass or the density inside the Earth? Should we consider it as spherically symmetric
so that the whole mass of the Earth is essentially concentrated at the center of the Earth? Will the topological variations such as
mountainous terrains affect the motion of the satellite? In fact, mountains do affect the motion of a satellite and in this case, we
cannot assume that the Earth is a point mass. This means that we cannot directly apply the above formula to the calculations.
It is known that Newton's gravitation can be applied to many scales from small atomic scales to astronomical scales. However,
Einstein's general theory of relativity treats this gravity from a completely different perspective, linking energy (rather than mass) to
space-time curvature. In this case, Einstein's field equation can be written in the following tensor form:
Rμν − 1
2Rg μν +gμν = 8πG
c4 T μν ,
where Tμν is the so-called stress-energy tensor, Rμν is the Ricci curvature tensor, R is the curvature, and gμν is the metric tensor
in the four-dimensional (4D) space-time manifold. In addition, c is the speed of light, while G is Newton's gravitational constant as
mentioned earlier. The parameter is Einstein's cosmological constant.
Einstein's model can be considered as a generalization of Newton's model, though from a different framework. Their assumptions
are different, and the accuracies are also different. Einstein's model works at all scales with a higher accuracy, but it is more
challenging to do calculations. Even so, Einstein's model may not work well at Planck's length scale.
This example may be an extreme example where one model seems to be very easy to calculate the force and its calcula-
tions can be done using secondary school mathematics by multiplications and division. The other model requires complex
tensor theory and sophisticated techniques to compute any observable results. In many applications, different approxima-
tions can be linked to different accuracies and the choice of approximations may depend on the accuracy of the solution
and ease of calculations. Let us look at another example.
Example 28.4
There is a very simple formula to link the relative humidity (RH) to the dewpoint temperature Td proposed by M.G. Lawrence (The
relationship between relative humidity and the dewpoint temperature in moist air: A simple conversion and applications, Bulletins
of the American Meteorological Society, vol. 87, no. 2, 225–233, 2005, http://dx.doi.org/10.1175/BAMS-86-2-225 ), which states
that
Td =T − 100 − RH
5 , (28.8)
where T is the so-called dry-bulb temperature. Obviously, when RH = 100% ,wehaveTd =T. This is a simple linear relationship
and it implies that a 5% decrease in RH would lead to one degree reduction in the dewpoint temperature. It is worth pointing out
that the unit here for temperature is degree Celsius (°C); otherwise, the formula will be different.
Obviously, we can simply re-arrange the above equation and get
RH =100 −5 (T − Td ),
to calculate RH once we have Td .
The formula (28.8) is sufficiently accurate only when RH is greater than 50%, and the accuracy is within °C or about 5% for
RH in the range of 0 to 30°C. The ease of calculations means that the accuracy is limited, though it is good enough for daily use.
330 PART | IX Advanced Topics
However, for meteorological applications, much higher accuracy may be needed. In this case, we can use
Td = B[ln(RH / 100) +K ]
A−[ ln(R H/ 100)+ K] ,K =AT
B+ T,
where A=17 .625 ,B=243 .04 °C. This formula can be accurate within about 0. 4% in the range from −40 °C to 50 °C. However, its
calculations involve logarithm and its derivations are based on more realistic but more sophisticated assumptions.
From the above two examples, we can see that we have to balance the ease of calculations and accuracy, depending on
the level of approximations. If we insist on getting more accurate results, we may have to consider many factors and minute
details. For example, in the previous section, if we wish to know the level of concentration of sugar more accurately at a
particular point, we have to know the initial distribution of the water temperature and the sugar cube size, and the way it
drops and where it drops and some other details. Such details seem impossible to know. We may wonder if it worths the
effort? Will these details provide higher accuracy? As an extreme example, let us imagine that we know the locations of
every single atom in the body of a monkey, can we predict the motion and path where a monkey will go at a given moment
at a given location? Probably not.
Mathematical modeling is a complicated process, which is essentially an iteration loop between theory and practice. The
famous British statistician George Box once said: 'All models are wrong, but some are useful'. Providing more details in a
more sophisticated way does not guarantee a higher accuracy. Box's original observations provides us a guiding philosophy
for mathematical modeling:
'Since all models are wrong the scientist cannot obtain a "correct" one by excessive elaboration. On the contrary following
William of Occam he should seek an economical description of natural phenomena. Just as the ability to devise simple but evocative
models is the signature of the great scientist so overelaboration and overparameterization is often the mark of mediocrity.' (Box
GEP, Science and statistics, J. American Statistical Association, 791–799, 1976.)
George Box
The philosophy of Occam, often referred to as the Occam's razor, states that "Entities are not to be multiplied without
necessity". That is, the simplest answer is usually the correct answer.
Isaac Newton's rules of reasoning in philosophy outlined in his Principia also clearly stated: "We are to admit no more
causes of natural things than such as are both true and sufficient to explain their appearances." Later, Albert Einstein also
famously said: "Make everything as simple as possible, but not simpler."
Such advice suggests that proper mathematical modeling requires to identify the right level of approximations and
the right kind of accuracy we wish to achieve, which consequently requires the understanding of the mechanisms of the
physical, chemical and/or biological processes at different levels of details, and the abstraction of such processes into the
right level of mathematical equations. Therefore, mathematical modeling is an art, requiring practice, practice and more
practice.
28.4 PARAMETER ESTIMATION
Another important topic in mathematical modeling is the ability to estimate the orders (not the exact numbers) of certain
quantities. If we know the order of a quantity and its range of variations, we can choose the right scales to write the
mathematical model in the non-dimensional form so that the right mathematical methods can be used to tackle the problem.
It also helps us to choose more suitable numerical methods to find the solution over the correct scales. The estimations will
often give us greater insight into the physical process, resulting in more appropriate mathematical models.
Let us look at an example to estimate the rate of heat loss at the Earth's surface, and the temperature gradients in
the Earth's crust and the atmosphere. We can also show the importance of the sunlight in the heat energy balance of the
atmosphere.
Example 28.5
We know that the average temperature at the Earth's surface is about Ts =300 K, and the thickness of the continental crust varies
from d=35 km to 70 km. The temperature at the upper lithosphere is estimated about T0 =900 ∼1400 K (very crude estimation).
Thus the estimated temperature gradient is about
dT
dz = T 0 − Ts
d≈9∼31 K/km.
Mathematical Modeling Chapter | 28 331
The observed values of the temperature gradient around the globe are about 10 to 30 K/km. The estimated thermal conductivity k
of rocks is about 1. 5∼ 4. 5 W/m K (ignoring the temperature dependence), we can use k=3 W/m K as the estimate for the thermal
conductivity of the crust. Thus, the rate of heat loss obeys Fourier's law of conduction
q=− k∇ T=− kdT
dz ≈0. 027 ∼0. 093 W/m 2 ,
which is close to the measured average of about 0.07 W/m2 . For oceanic crust with a thickness of 6∼ 7 km, the temperature gradient
(and thus rate of heat loss) could be five times higher at the bottom of the ocean, and this heat loss provides a major part of the
energy to the ocean so as to keep it from being frozen.
If this heat loss goes through the atmosphere, then the energy conservation requires that
kdT
dz crust +ka
dT
dh air =0,
where h is the height above the Earth's surface and ka =0 .020 ∼ 0 .025 W/mK is the thermal conductivity of the air (again, ignoring
the variations with the temperature). Therefore, the temperature gradient in the air is
dT
dh =− k
ka
dT
dz ≈− 3. 6∼− 4. 5K/km ,
if we use dT/dz =30 W/km. The negative sign means the temperature decreases as the height increases. The true temperature
gradient in dry air is about 10 K/km in dry air, and 6∼ 7 K/km in moist air. As the thermal conductivity increases with the humidity,
so the gradient decreases with humidity.
Alternatively, we know the effective thickness of the atmosphere is about 50 km (if we define it as the thickness of layers
containing 99.9% of the air mass). We know there is no definite boundary between the atmosphere and outer space, and the
atmosphere can extend up to several hundreds of kilometers. In addition, we can also assume that the temperature in space vacuum
is about 4 Kelvin (K) and the temperature at the Earth's surface is 300 K, then the temperature gradient in the air is
dT
dh ≈ 4−300
50 ≈− 6K/km,
which is quite close to the true gradient. The higher rate of heat loss (due to higher temperature gradient) means that the heat
supplied from the crust is not enough to balance this higher rate. That is where the energy of sunlight comes into play. We can see
that estimates of this kind will provide a good insight in the whole process.
Sometimes, it can be extremely challenging to get a good estimate because the processes involved are too complicated
to do any calculations without using sophisticated computer simulations. In this case, an estimation can get the order of the
quantities right, but they can be quite different by a factor. As an example, let us estimate the quantity of jet fuel consumption
during a takeoff of a typical aircraft such as a jumbo Boeing 747 passenger jet. Obviously, the fuel consumption depends on
many factors such as aircraft type, loads, takeoff conditions, weather, altitude and cruise speed and others. Even the same
aircraft traveling in the same route can use a different amount of fuel due to the variation of loads, number of passengers
and wind direction. If you ask the Boeing company, they may produce a long document with various assumptions without
giving you an accurate answer. If you ask airlines or their pilots, they can only give you the average total fuel consumption
of a given flight route, though pilots may give the average reading changes of their fuel gauges during takeoffs and their
answers may be surprisingly very different.
If we read around and search the Internet, we may have the following data: A typical Boeing 747 (say, from London to
New York) can consume about 230,000 liters of jet fuel (and much more for flying from London to Hong Kong). A typical
such aircraft can weight 500 tons or m= 5× 105 kg and the fuel consumption during the takeoff (from the sea level to the
cruise altitude) can typically consume 2000 liters to 4000 liters of jet fuel. Now the questions are: do all these figures make
any sense? How can we provide an estimate without taking too many factors into consideration?
Example 28.6
Let us use the energy conservation to do a very crude estimation. We know that the cruise altitude is about h=10 km (or h=104 m)
and the cruise speed of a Boeing 747 is about 570 mph or 920 km/h (that is, v=255 m/s). Let us assume the that the aircraft's mass
is m=5 ×105 kg (or 500 tons) and the acceleration due to gravity is g=10 m/s2 .
332 PART | IX Advanced Topics
We know its kinetic energy is 1
2mv 2 at its cruise speed and the potential energy mg h. So the total energy is
E=1
2mv 2 +mg h = 1
2× 5×105 ×( 255 ) 2 + 5×105 ×104 ×10 ≈6. 6× 10 10 J.
On the other hand, we know that the energy density of jet fuel is about ej =4 .0× 107 J/kg and the efficiency of the jet engine is
about 98% to 100% . So the fuel consumption W needed during the takeoff to supply the same energy is
We
j=E,
or
W= E
ej = 6 .6× 1010
4. 0× 107 ≈ 1600 kg ≈ 2000 liters,
where we have used 0.8 kg/liter as the approximate value of fuel density. This means that it requires about 2000 liters of jet fuel to
get the aircraft to the standard cruise altitude from the ground. Obviously, this estimate is much lower than the real fuel consumption
because we have not considered the air resistance and many other factors. The actual fuel needed is typically around 3000 liters to
5000 liters (and more) for a typical Boeing 747, depending on the actual load and other conditions.
Even with this simple estimation, it can provide a lower bound for the fuel consumption or at least some sense of the
order of the fuel consumption. That is, the consumption is about thousands of liters, not of a few hundred liters for a jumbo
jet.
Of course the choice of typical values is important in order to get a valid estimation. Such choice will depend on the
physical process and the scales we are interested in. The right choice will be perfected by expertise and practice.
28.5 TYPES OF MATHEMATICAL MODELS
Mathematical models can take many different forms, and thus it is not an easy task to classify all the mathematical models.
However, based on the mathematical forms they can take, we can loosely classify them into eight categories: algebraic
models, tensor models, differential equation models, integral models, statistical models, fuzzy models, learned models and
data-driven models. We will briefly explain each of these model categories in this section.
28.5.1 Algebraic Equations
Many physical laws can be expressed as algebraic relationships. For example, Newton's second law F= ma is a good
example, though strictly speaking we should express it in the vector form:
F=m a,(28.9)
because both force F and acceleration a are vectors. Another example is Ohm's law.
Example 28.7
Ohm's law relates the current I through a conductor with a resistance R and the voltage V applied across the conductor. That is
I= V
R,
or
V= IR.
Here, the main assumption is that the resistance does not vary with V or I . Obviously, in case of R independent of I , we can still
write this as an equivalent relationship as an approximation. In fact, R usually depends on temperature T and the flow of current
will generate heat (and thus vary the temperature).
Almost all the physical and chemical laws we learned in school are expressed as algebraic equations.
Mathematical Modeling Chapter | 28 333
28.5.2 Tensor Relationships
Sometimes, the quantities of interest are tensors such as stress and strain, we have to use tensor relationships to express
them. For example, the generalized Hooke's law we discussed earlier in this book can be written as
εij =1 + ν
Eσ ij − ν
Eσ kk δ ij ,(28.10)
which is a tensor equation.
Example 28.8
The Ohm's law for a simple conductor is very simple; however, it becomes a tensor relations for a complex, anisotropic medium
under an electric field E :
J=σ E,
where J is the current density that varies with its location. Here, σ=[ σij ]is the conductivity tensor of anisotropic conducting
material such as crystals and nanomaterials. Using the summation convention for tensors, we have
Ji =σ ij Ej .
When the properties of a medium change with certain quantity such as directions, a tensor relationship can occur.
28.5.3 Differential Equations: ODE and PDEs
Though some laws in physics can be expressed in simpler forms, it is sometimes useful to write in a more generalized form
for theoretical analysis and this often involves derivatives or integrals.
As we have seen in the chapter about partial differential equations, most mathematical models in physics are expressed
in PDEs and the conservation laws such as the conservation of mass and energy are expressed in integral forms.
Example 28.9
Newton's second law F= ma is often written in a differential form
F=d p
dt = d(m v)
dt ,
where p=m v is the linear momentum and v is the velocity of the object. This form is specially useful in mechanics when discussing
variable-mass dynamics such as the trajectory of a rocket where its mass m(t ) is not a constant but is varying with time t .Inthis
case, we have
F=d(m v)
dt = dm
dt v+m dv
dt =˙ m(t)v+ m(t) ˙
v.
As we have seen earlier in this chapter, mathematical modeling often produces some mathematical equations, often
partial differential equations. The next step is to identify the detailed constraints such as the proper boundary conditions
and initial conditions so that we can obtain a unique set of solutions.
For the sugar diffusion problem discussed earlier, we cannot obtain the exact solution in the actual domain inside the
water-filled glass, because we need to know where the sugar cube or grains were initially added. The geometry of the glass
also needs to be specified. In fact, this problem needs numerical methods such as finite element methods or finite volume
methods. The only possible solution is the long-time behavior: when t→∞ , we know that the concentration should be
uniform c(z, t →∞ )→ c∞ (= mass of sugar added/volume of water).
You may say that we know this final state even without mathematical equations, so what is the use of the diffusion
equation? The main advantage is that you can calculate the concentration at any time using the mathematical equation with
appropriate boundary and initial conditions, either by numerical methods in most cases or by mathematical analysis in some
very simple cases. Once you know the initial and boundary conditions, the whole system history will be determined to a
certain degree.
334 PART | IX Advanced Topics
The beauty of mathematical models is that many seemingly diverse problems can be reduced to the same mathematical
equation. For example, we know that the diffusion problem is governed by the diffusion equation
∂c
∂t = D∇2 c. (28.11)
The heat conduction is governed by the heat conduction equation
∂T
∂t = κ∇2 T, κ = K
ρcp
,(28.12)
where T is temperature and κ is the thermal diffusivity. K is thermal conductivity, ρis the density and cp is the specific heat
capacity. Mathematically speaking, whether it is the concentration of a pollutant or temperature, it is the same dependent
variable u in terms of the same parabolic partial differential equation.
As we have seen earlier in this book, to obtain a solution for a partial differential equation is not an easy task. However,
it may be possible to get a good estimate without solving the PDE itself, depending on the purpose of estimation.
In the one-dimensional (1D) case, the above equation becomes
∂u
∂t = κ ∂ 2 u
∂x2 .(28.13)
For a typical length L and a typical timescale t∗ , we can define a new time t= t∗ τ= and a new (scaled) spatial coordinate
x= Lξ .Thus,wehave
dt = t∗ dτ, dx = Ldξ,
and
∂
∂t = 1
t∗
∂
∂τ , ∂
∂x = 1
L
∂
∂ξ , ∂ 2
∂x2 = 1
L2
∂2
∂ξ2 .(28.14)
Now the parabolic PDE (28.13) becomes the following non-dimensional form:
1
t∗
∂u
∂τ = κ
L2
∂2 u
∂ξ2 ,(28.15)
or
∂u
∂τ = λ ∂ 2 u
∂ξ2 ,λ =κt ∗
L2 ,(28.16)
where λ is a non-dimensional parameter. In a special cases λ= 1, it becomes a parameterless diffusion equation that can
describe many phenomena across scales.
Let us look at an example.
Example 28.10
To estimate the cooling time for a hot iron ball or a piece of hot rock, we can use the characteristic time t∗ defined by
λ= 1= κt ∗
L2 ,
or
t∗ = L 2
κ.
Mathematical Modeling Chapter | 28 335
We know that the thermal conductivity is about K≈35 W/mK for iron (though the actual values vary greatly with the type of iron
and purity of iron), its density is ρ≈7850 Kg/m3 and its specific heat capacity cp ≈450 J/kg K. Thus, the thermal diffusivity of iron is
κ= K
ρcp ≈ 35
7850 × 450 ≈ 9.9× 10 −6 m 2 /s.(28.17)
For a small iron ball of d=0. 1 m, we have
t∗ = L 2
κ= 0 .12
9. 9× 10 −6 ≈ 1000 seconds,
which is about 16 minutes.
For a larger iron ball of d=1 m, the time scale of cooling is
t∗ = L 2
κ≈ 1
9. 9× 10 −6 ≈ 1.0× 105 seconds ≈ 1day .(28.18)
For a larger hot body L=1000 m, then that time scale is t∗ =106 days or 3200 years. This estimate of the cooling time scale is based
on the assumption that no more heat generated inside is supplied.
With such simple assumptions, we can do many sensible estimations. It leaves as an exercise to estimate the time for a
cup of hot coffee to cool down to the room temperature.
28.5.4 Functional and Integral Equations
Though most mathematical models are written as partial different equations, however, sometimes it might be convenient to
write them in terms of integral equations, and these integral forms can be discretized to obtain various numerical methods.
Example 28.11
In the discussion about the conservation of mass earlier in Eq. (28.2) , we write it in the integral form
∂
∂t %%%
cdV +%%
J·d S=0,
which is equivalent to the partial differential equation as follows:
∂c
∂t +∇·J=0.
One of the advantages of writing in integral forms is that it has a clearer meaning in the representative volume, and such
integral form is often used in numerical simulations such as the finite volume methods.
The advantage of writing it in terms of a differential equation is that we can use all the available solution techniques
to solve the problem of interest. Obviously, these two forms are closely related and often can be converted from one to
another.
However, sometimes, the physical laws or properties can be intrinsically represented by integral equations. Examples
include inverse problems, non-local problems and visco-elasticity or creep problems.
Example 28.12
Viscoelasticity is an important phenomenon in engineering. For example, the behavior of chocolate flow, glasses, rubbers, plastic
bags and a bicycle pump can be described by viscoelastic behavior. Loosely speaking, linear viscoelastic constitutive relationships
or laws can be expressed as
σ(t)= σ0 +% t
0
E(t − η)˙ ε(η)dη,
336 PART | IX Advanced Topics
where σ(t) is the time-dependent stress and ˙ εis the strain rate that can also depend on time t .E is the so-called relaxation modulus,
while σ0 is a constant. This stress-strain relationship, unlike the generalized Hooke's law, is essentially a Volterra integral equation.
28.5.5 Statistical Models
Both differential equations and integral equations are the mathematical models for continuum systems. Other systems are
discrete and different mathematical models are needed, though they could reduce to certain forms of differential equations
if some averaging is carried out.
On the other hand, many systems have intrinsic randomness, thus the description and proper modeling require statistical
models as we have seen in the chapters on probability and statistics. The numbers of telephone calls received in a call center
can be described by a Poisson distribution. Statistical models are a very important class of models that are widely used in
economics, finance, risk management, data mining and machine intelligence.
Let use discuss two examples: one about the frequency of earthquakes and the other about the Six Sigma design method-
ology for manufacturing.
Example 28.13
The Gutenberg-Richter law is an empirical, statistical relationship between the magnitude (M ) and the total number (N )ofearth-
quakes for a given period in a given region. It can be written as
log10 N= a− bM,
or
N=10 a−bM ,
where a and b are constants. The b -value is of more scientific importance. The typical values are: a=0. 5 to 2 and b≈1 . In general,
bis relatively stable and does not vary much from region to region.
This model suggests that there will be a 10-fold decrease in seismic activity for a unit increase in magnitude. That is to say, there
are about 10 times more magnitude-5 earthquakes than magnitude-6 earthquakes.
Sometimes, a mathematical model can appear as a procedure or methodology, even though it is based on solid mathe-
matics. Let us use Six Sigma as an example.
Six Sigma (6σ ) is a quality control methodology, pioneered at Motorola in the 1980s, for eliminating defects in products.
Assuming that the normal distribution applies to such scenarios, we have
p(x, μ, σ) =1
σ√ 2πe − (x−μ) 2
2σ 2 ,
where μ is the mean and σ is the standard deviation. Clearly, this function is symmetric on both sides of x= μ .Its
cumulative distribution function is
(x) =1
2# 1+erf (x −μ)
σ√ 2$ ,
which is the area under the normal distribution curve from −∞ to x . Obviously, when x→∞ ,wehave(∞ )= 1 because
erf(∞ )= 1.
The exact area under the curve from μ− kσ to μ+ kσ (where k> 0) can be calculated by
Ak =(μ +kσ) −(μ −kσ ) = 1
2 erf (k/ √ 2)−erf (− k/ √ 2).
Since the error function
erf(x) = 2
√π % x
0
e−τ2 dτ,
Mathematical Modeling Chapter | 28 337
is an odd function erf(− x) =− erf(x) ,wehave
Ak =1
2[ erf (k/ √ 2)−erf (− k/ √ 2) ]= erf(k
√2 ).
Thus, for k= 1, the area within one standard deviation between μ− σ to μ+ σ is
A1 =erf (1/√ 2 )=0 .68269,
which is about 68. 3%. In addition, the area within 2σ (between μ− 2σ to μ+ 2σ )is
A2 =erf (2/√ 2 )=0 .9545,
or about 95%. The area within 3σ is
A3 =erf (3/√ 2 )=0 .9973,
which about 99. 7%. Similarly, we have
A4 =0 .99993665 ,A
5=0. 9999994267,A
6=0. 9999999980268.
Suppose the product is an electric motor with a no-load speed or the spin speed of ω= 3000. 0± 0. 2 round per minutes
(rpm). If we try to use 3σ , there would 99. 7% products without defects, and the defect rate is about 0. 27% is still too high
for a good product because there about 2700 defective products out of 1 million. If we use 6σ ,wehaveA 6 =0. 999999998,
which means that there are about 2 defective parts in a billion. This may be a higher quality assurance, but the technologies
and costs to ensure this may be extremely expensive.
Example 28.14
The beauty of the Six Sigma methodology is to use ±4.5σ allowing the mean to shift ±1.5σ (thus 6σ in essence). Since k=4. 5 ,we
have
A4 .5 =0 .999993204653751,
which gives 6. 8 defective parts per million if the mean is not allowed to shift. Since the mean shift ±1.5σ , the actual defective parts
is half of the above value, that is, 3. 4 parts per million, which is the essence of the Six Sigma Standards in manufacturing.
Coming back to the motor specification of ω=3000 .0± 0 .2 or 3000. 0− 0. 2= 2999. 8 to 3000. 0+ 0. 2= 3000. 2 , to ensure this
standard specification with only about 3.4 defective motors per million, we have to ensure that manufacturing standard deviation σ
to be even smaller. Since it requires that 6σ= 0. 2 ,wehave
σ=0 .2
6≈ 0. 033,
which is the standard deviation to be achieved in the actual manufacturing process to ensure the desired quality target.
28.5.6 Fuzzy Models
Sometimes, a model may need to work on situations that are not clear cut as simply yes or no (or 1 or 0) data. In this case,
models based on fuzzy logic can be advantageous. The main concepts of fuzzy logic and fuzzy set theory were pioneered
by L.A. Zadeh, and they are essentially a form of multi-valued logic with a membership function to measure the closeness.
In the standard binary logic, only there are true (1) and false (0), while fuzzy logic allows any real value from 0 to 1. In
fact, fuzzy mathematics is itself a broad subject, and interested readers can refer to more advanced literature.
28.5.7 Learned Models
In many applications such as data mining and machine intelligence, the relationship cannot be directly expressed as a
deterministic, explicit function. Instead, the model or relationship is an implicit, dynamically varying model that learned
338 PART | IX Advanced Topics
from the data or available information. As the data or information changes, the learned relationship needs to be updated by
training or learning to reflect the new information. Such models can be called learned models.
One classical way to construct such model is to use regression analysis as we discussed in the context of statistics.
However, there are other, even better, ways to build learned models. Two good examples are artificial neural networks
(ANN) and support vector machines. A typical ANN uses a number of neurons arranged into different layers. Each neuron
can convert an input signal into a binary output (0 or 1), activated at a threshold, and each neuron in a layer can be connected
with other neurons in a layer in front of its layer. Typically, the network contains an input layer, an output layer and one or
more hidden layer in the middle. ANNs, especially the so-called convolution deep learning neural networks, have become
a powerful tool in machine learning and artificial intelligence.
Support vector machines (SVM) are a class of techniques for classification and regression analysis, they often use the
so-called kernel tricks to map data in one space to a higher-dimensional space so that their structures can be identified and
different groups or classes can be separated relatively easily by constructing some hyperplanes. Again, SVMs have also
become a powerful tool for analyzing data and extracting features in many applications, including engineering, finance and
machine intelligence.
28.5.8 Data-Driven Models
As the data volumes have increased dramatically in the two last decades, various techniques are needed to deal with such
big data, which have formed a new subject, called the big data science. As these data sets are usually unstructured with
high complexity, there is no simple mathematical model that can be constructed to explain the whole data. Even if it may
be possible to construct some mathematical models, such models need to be modified as more data flow in. In addition,
uncertainties and noise often present in the data, which makes the already challenging data mining tasks even more difficult.
In such applications, models become data-driven, indeterministic and dynamic, and thus it may impossible or mean-
ingless to try to seek a deterministic and simple mathematical model. In this case, we have to live with models that truly
data-driven and noisy. This is a relatively new area, and interested readers can refer to more advanced literature on data
science.
In this chapter, we have summarized the basic procedures and steps of mathematical modeling. We have also provided
a few examples to illustrate the types of models and different levels of approximation. Now let us end this chapter by
providing a detailed worked example as a case study.
28.6 BROWNIAN MOTION AND DIFFUSION: A WORKED EXAMPLE
Diffusion is relevant to many phenomena in physics, chemistry and biology. Thus, its importance has attracted much
attention in the last two centuries. In essence, diffusion can be modeled by Brown motion as a statistical process, and thus
the mathematical models are probabilistic.
Einstein provided in 1905 the first theory of Brownian motion of spherical particles suspended in a liquid, which can be
written, after some lengthy calculations, as
2 = k B
T
t
3πμa ,(28.19)
where a is the radius of the spherical particle, and μ is the viscosity of the liquid. T is the absolute temperature, and t is
time. kB =R/N Ais Boltzmann's constant where R is the universal gas constant and N Ais Avogadro's constant. 2 is the
mean square of the displacement.
Langevin in 1908 presented a very instructive but much simpler version of the theory. If u is the displacement, and
ξ= du
dt is the speed at a given instant, then the kinetic energy of the motion should be equal to the average kinetic energy
1
2k BT. (Here Langevin considered only one direction. For three directions, the kinetic energy becomes 3
2k BT, thus there is
a factor 3. This means that equation (28.28) should be u 2 =6Dt .) That is
1
2mξ 2 = 1
2kB T,
Mathematical Modeling Chapter | 28 339
where m is the mass of the particle. The spherical particle moving at a velocity of ξ will experience a viscous resistance
equal to
−6πμaξ =− 6πμa du
dt
according to Stokes' law. If F(t) is the complementary force acting on the particle at the instant so as to maintain the
agitation of the particle, we have, according to Newton's second law
md 2 u
dt2 =− 6πμadu
dt + F. (28.20)
Multiplying both sides by u ,wehave
mu d 2 u
dt2 =− 6πμaud 2 u
dt2 + Fu. (28.21)
Since (u 2 )/2= uu + u2 or
ud 2 u
dt2 = 1
2
d2 (u2 )
dt2 − ( du
dt ) 2 = 1
2
d
dt d(u2 )
dt − ξ 2 ,
we have
m
2
d
dt d(u2 )
dt − mξ 2 =− 3πμa d(u 2)
dt + Fu, (28.22)
wherewehaveuseduu =(u 2 )/2. If we consider a large number of identical particles and take the average, we have
m
2
d
dt d(u2 )
dt − m ξ 2 =− 3πμa d(u 2)
dt + Fu, (28.23)
where () means the average. Since the average of Fu = 0 due to the fact that the force is random, taking any signs and
directions. Let Z= d(u 2)/dt,wehave
m
2
dZ
dt +3πμaZ = kB T, (28.24)
wherewehaveusedmξ 2 =kB Tdiscussed earlier. This equation is now known as Langevin's equation. It is a linear
first-order differential equation. Its general solution can easily be found using the method discussed in the chapter on
differential equations
Zc =Ae −t/τ +kB T 1
3πμa ,τ =m
6πμa ,(28.25)
where A is the constant to be determined. For the Brownian motion to be observable, t must be reasonably larger than the
characteristic time τ≈ 10−8 seconds, which is often the case. So the first term will decrease exponentially and becomes
negligible when t>τ in the standard timescale of our interest. Now the solution becomes
Z= d(u2 )
dt = k B T 1
3πμa .(28.26)
Integrating it with respect to t ,wehave
u2 = kB T t
3πμa ,(28.27)
340 PART | IX Advanced Topics
which is Einstein's formula for Brownian motion. Furthermore, the Brownian motion is essentially a diffusion process with
u2 =2 Dt, D = k B T
6πμa ,(28.28)
where D is the equivalent diffusion coefficient. This suggests that large particles diffuse more slowly than smaller particles
in the same medium. Mathematically, it is very similar to the heat conduction process discussed in earlier chapters.
We know that the diffusion coefficient of sugar in water at room temperature is D≈ 0. 5× 10−9 m2 /s. In 1905, Einstein
was the first to estimate the size of sugar molecules using experimental data in his doctoral dissertation on Brownian motion.
Let us now estimate the size of the sugar molecules. Since kB =1. 38 ×10 −23 J/K, T =300 K, and μ =10 −3 Pa s, we
have
a= k B T
6πμD = 1 .38 × 10−23 × 300
6π× 10 −3 ×0. 5× 10 −9 ≈ 4.4× 10 − 10 m,(28.29)
which means that the diameter is about 8. 8× 10−10 m= 0. 88 nm. In fact, Einstein estimated for the first time the diameter
of a sugar molecule was 9. 9× 10−10 m even though the other quantities were not so accurately measured at the time.
From u 2 =2Dt , we can either estimate the diffusion distance for a given time or estimate the timescale for a given
length. For a sugar cube in a cup of water to dissolve completely to form a solution of uniform concentration (without
stirring), the time taken will be t= d 2 /2 D, where d= u 2 is the size of the cup. Using d= 5cm =0. 05 m and D =
0. 5× 10 −9 m2 /s, we have t= d 2 / 2D= 0. 052 /( 2× 0. 5× 10 −9 )≈ 2. 5× 106 seconds, which is about one month. This is
too slow; that is why we always try to stir a cup of tea or coffee.
EXERCISES
28.1. Discuss how the Lorenz equations for chaos were obtained.
28.2. Build a simple mathematical model to simulate the bungee jump process and discuss any assumptions made.
28.3. In mathematical modeling of epidemic spread of flu, the so-called SIR model is often used. Discuss the assumptions
used to derive the SIR model.
28.4. Try to write a simulator to model the pattern-formation equation
∂u
∂t = D( ∂ 2 u
∂x2 + ∂ 2 u
∂y2 )+γu( 1−u).
Discuss the possible boundary and initial conditions.
28.5. If your aim is to automatically identify any hand-written digits and letters such as post codes, discuss the models and
methods for completing the task.
... The solution of the four-dimensional spatiotemporal vibration equation is the other problem. In this part, we proposed a numerical analysis of space and time discretization to get approximate solutions, in which the core concept is based on the combination of the central difference method and the Runge-Kutta fourth-order (RK4) method (Yang, 2016). All the above-discussed equations of the solution algorithm have complied with Matlab software Valentine, 2020a, 2020b). ...
... Then the RK4 is used to measure multiple "slope" and a given time step to extrapolate the solution to the future time step (Yang, 2016). Combining Eq. (8) and Eq. ...
... The RK4 general iterate form is introduced (Yang, 2016). Combined with appropriate step size and iterative length, the motion governing equations are given by: ...
The pitting corrosion of pipelines subjected to the spatio-temporal earthquake was evaluated to illustrate the failure probability of multiple loading conditions. The pipeline vibration was modeled as a coupled 3D vibration equation considering the uncertain soil parameters. Using a combination of the Gamma process and Gaussian copula function implemented time-dependent corroded growth of single defect as well as the spatial-dependent between corroded depth growths of different defects. All modeling was embedded in a frame of Bayesian inference and Markov chain Monte Carlo (MCMC) simulation techniques to predict the future corroded growth by reported in-line inspections (ILIs). Moreover, the Monte Carlo simulation (MC) technique was used to evaluate the reliability of the pipeline system, including multiple corroded defects. A case study was employed to prove the application of all proposed models. The results indicate that the reliability analysis of pipelines under the earthquake considering pitting corrosion growth is of considerable significance to the accuracy of the evaluation, which can inform operators from process industry to mitigate the risk of pipeline failure.
... SBFEM is semi-analytical, and the TPAA takes advantage of accurate and stable solution accuracy when the step size varies. Subsequently, the membership functions of the viscoelastic constitutive parameters can be determined by a fuzzy decomposition theorem [13,43] with the interval bounds acquired at each of α-sublevels. The parallelization is realized for the construction of surrogate and the implementation of the particle swarm method for a further reduction of computational expense. ...
... The minimization of Equation (52) is realized using the particle swarm method [35][36][37], providing ϕ (j) for all α-sublevels. By virtue of the fuzzy decomposition theorem [13,43], the membership functions of fuzzy viscoelastic constitutive parameters can be constructed using ϕ (j) ...
When there exists fuzzy uncertainty in experimentally determined information, viscoelastic constitutive parameters to be identified are treated as fuzzy variables, and a two-stage strategy cooperating with particle swarm method is presented to identify membership functions of fuzzy parameters. At each stage, inverse fuzzy problem is formulated as a series of α-level strategy-based inverse interval problems, which are described by optimization problems and are solved utilizing particle swarm method. Forward interval analysis required in inverse interval analysis is conducted by solving two optimization problems via modified coordinate search algorithm. To alleviate heavy computational burden, dimension-adaptive sparse grid (DSG) surrogate is embedded in optimization process. The surrogate is constructed on a solid platform of high fidelity deterministic solutions, which is provided by scaled boundary finite element method and temporally piecewise adaptive algorithm. Eventually, membership functions of fuzzy parameters can be obtained by fuzzy decomposition theorem with interval bounds acquired at each of α-sublevels. Parallelization is realized for construction of DSG surrogate and implementation of particle swarm method for a further computation reduction. Numerical examples are provided to illustrate effectiveness of proposed approach, where regional inhomogeneity and impact of measurement points selection on identification results are explored.
... A study by See and Jamaian [19] successfully solved the state equations of optimal control problem for microalgae growth by the first order Runge-Kutta method, which also called the Euler method. The higher the order of Runge-Kutta method, the more accurate the numerical solutions obtained when using fixed step size for most cases [20]. To increase the accuracy of the numerical solution, the fourth-order Runge-Kutta (RK4) method is applied to solve the system of the state equations in this research. ...
... The Newton method is known to be one of the most popular iterative methods used to find zeros of a nonlinear objective function [35]. It is based on finding the root of ...
-
![Fatih Tosun]()
In order to certificate a helicopter, aviation safety agencies must know that all designed helicopter configurations can withstand all loads resulting from maneuvers defined in certification standards. In other words, the maneuvers defined in regulations must be performed for each appropriate combination of weight and center of gravity. Then, designers have to prove that the helicopter can fly safely across the entire flight spectrum. Therefore, load engineers perform all maneuvers defined in the helicopter usage spectrum in order to analyze all possible load values. In traditional methods, a trial and error approach is used to reach the maneuver. However, performing these maneuvers with a trial and error approach not only causes expensive computing but also requires more engineering effort. Additionally, it may cause some defects in the maneuvers. Therefore, this thesis study aims to achieve the desired helicopter maneuvers using optimization methods in order to reduce the calculation cost and engineering efforts and perform the maneuvers accurately. For this purpose, only selected maneuvers are performed in this thesis. In addition, various optimization methods in different configurations have been applied to solve these maneuvers. Thus, the most useful one among them has been decided by making the necessary comparisons. Finally, the most useful optimization method has been applied to maneuvers frequently performed by helicopters throughout its lifetime.
... The main problem of computer diagnostic is the separation of some groups of subjects from others: PD vs. control subjects, ET vs. control subjects, PD vs. ET. There are several standard methods for solving this problem, such as the support vector machine, the brute-force method, the branch and bound method, genetic algorithms, etc [19]- [23]. In our case, the difficulty is in that we have a very big quantity of parameters of the separation function, and the actual number of patients is quite small. ...
... Mathematical modelling is defined as the process of describing complex behaviours of a real system and formulating these complex behaviours mathematically to achieve a solution (Yang, 2017). Thus in this study, all parameters, decision variables, objective function, constraints should be determined while creating a mathematical model. ...
-
![Umit Unver]()
- Ozlem Kara
Energy efficiency can be considered as one of the key components of sustainability. This study introduces a decision support tool that aims to provide increased energy efficiency in a steel forging facility. Seven different products were selected as a sample product group in order to simulate overall production of the facility. The material, operations, parameters, decision variables, objective function and constraints are identified according to the facility. Energy consumptions for both the current production process and the proposed production process were evaluated and compared. The AMPL Software was used to obtain the lowest energy consuming production route. It is shown that the correct regulation of the production process would result in a 65% energy saving in the unit production of the products in the chosen sample group. The calculated savings correspond to about 6.57 TOE/yr. The decision support tool developed in this study provided cost savings without any investment in the production process and is therefore considered to be the most economic and most practical way to achieve the increased energy efficiency for a cleaner production.
-
![Antigoni Avramouli]()
In recent years, a highly sophisticated array of modeling and simulation tools in all areas of biological and biomedical research has been developed. These tools have the potential to provide new insights into biological mechanisms integrating subcellular, cellular, tissue, organ, and potentially whole organism levels. Current research is focused on how to use these methods for translational medical research, such as for disease diagnosis and understanding, as well as drug discovery. In addition, these approaches enhance the ability to use human-derived data and to contribute to the refinement of high-cost experimental-based research. Additionally, the conflicting conceptual frameworks and conceptions of modeling and simulation methods from the broad public of users could have a significant impact on the successful implementation of aforementioned applications. This in turn could result in successful collaborations across academic, clinical, and industrial sectors. To that end, this study provides an overview of the frameworks and disciplines used for validation of computational methodologies in biomedical sciences.
-
![Xin-She Yang]()
- Xing-Shi He
Before we proceed to analyse any nature-inspired algorithms from at least ten different perspectives, let us review the mathematical fundamentals concerning convergence, stability and probability distributions.
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![Xin-She Yang]()
- Xing-Shi He
Nature-inspired algorithms have become powerful and popular for solving problems in optimization, computational intelligence, data mining, machine learning, transport and vehicle routing. After the theoretical analyses in earlier chapters, it would be useful to provide examples and case studies to show that these algorithms can indeed work well in practice.
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![Xin-She Yang]()
This chapter first reviews the most relevant fundamentals of functions, vectors, differentiation, and integration. Then, it introduces some useful concepts such as eigenvalues, complexity, convexity, probability distributions, and optimality conditions. A function is a quantity which varies with another independent quantity or variable in a deterministic way. One of the simplest optimization problems is to find the minimum of a function in the real domain. One can extend the optimization procedure for univariate functions to multivariate functions using partial derivatives and relevant conditions. A very important concept related to optimization and convergence analysis is the Lipschitz continuity of a function. Whatever the real‐world applications may be, it is usually possible to formulate an optimization problem in a general mathematical form. All optimization problems with an explicit objective can be expressed as a nonlinearly constrained optimization problem.
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Functions And Their Applications In Mathematics Pdf
Source: https://www.researchgate.net/publication/312220301_Engineering_Mathematics_with_Examples_and_Applications
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