Solving applied mathematical problems with MATLAB

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This textbook presents a variety of applied mathematics topics in science and engineering with an emphasis on problem solving techniques using MATLAB®. The authors provide a general overview of the MATLAB language and its graphics abilities before delving into problem solving, making the book useful for readers without prior MATLAB experience. They explain how to generate code suitable for various applications so that readers can apply the techniques to problems not covered in the book. Examples, figures, and MATLAB scripts enable readers with basic mathematics knowledge to solve various applied math problems in their fields while avoiding unnecessary technical details.

Preface	p. xi
1 Computer Mathematics Languages - An Overview	p. 1
1.1 Computer Solutions to Mathematics Problems	p. 1
1.1.1 Why should we study computer mathematics language?	p. 1
1.1.2 Analytical solutions versus numerical solutions	p. 4
1.1.3 Mathematics software packages: an overview	p. 5
1.2 Summary of Computer Mathematics Languages	p. 6
1.2.1 A brief historic review of MATLAB	p. 6
1.2.2 Three widely used computer mathematics languages	p. 7
1.2.3 Introduction to free scientific open-source softwares	p. 7
1.3 Outline of the Book	p. 8
Exercises	p. 9
2 Fundamentals of MATLAB Programming	p. 11
2.1 Fundamentals of MATLAB Programming	p. 12
2.1.1 Variables and constants in MATLAB	p. 12
2.1.2 Data structure	p. 13
2.1.3 Basic structure of MATLAB	p. 14
2.1.4 Colon expressions and sub-matrices extraction	p. 15
2.2 Fundamental Mathematical Calculations	p. 16
2.2.1 Algebraic operations of matrices	p. 16
2.2.2 Logic operations of matrices	p. 18
2.2.3 Relationship operations of matrices	p. 19
2.2.4 Simplifications and presentations of analytical results	p. 19
2.2.5 Basic number theory computations	p. 21
2.3 Flow Control Structures of MATLAB Language	p. 22
2.3.1 Loop control structures	p. 22
2.3.2 Conditional control structures	p. 24
2.3.3 Switch structure	p. 25
2.3.4 Trial structure	p. 26
2.4 Writing and Debugging MATLAB Functions	p. 27
2.4.1 Basic structure of MATLAB functions	p. 27
2.4.2 Programming of functions with variable inputs/outputs	p. 30
2.4.3 Inline functions and anonymous functions	p. 31
2.5 Two-Dimensional Graphics	p. 31
2.5.1 Basic statements of two-dimensional plotting	p. 32
2.5.2 Other two-dimensional plotting statements	p. 34
2.5.3 Implicit function plotting and applications	p. 36
2.5.4 Graphics decorations	p. 36
2.6 Three-Dimensional Graphics	p. 39
2.6.1 Plotting of three-dimensional curves	p. 39
2.6.2 Plotting of three-dimensional surfaces	p. 40
2.6.3 Viewpoint setting in 3D graphs	p. 43
Exercises	p. 44
3 Calculus Problems	p. 47
3.1 Analytical Solutions to Calculus Problems	p. 47
3.1.1 Analytical solutions to limit problems	p. 48
3.1.2 Analytical solutions to derivative problems	p. 50
3.1.3 Analytical solutions to integral problems	p. 55
3.2 Series Expansions and Series Evaluations	p. 58
3.2.1 Taylor series expansion	p. 59
3.2.2 Fourier series expansion	p. 62
3.2.3 Series	p. 65
3.2.4 Sequence product	p. 67
3.3 Numerical Differentiation	p. 67
3.3.1 Numerical differentiation algorithms	p. 68
3.3.2 Central-point difference algorithm	p. 69
3.3.3 Gradient computations of functions with two variables	p. 71
3.4 Numerical Integration Problems	p. 72
3.4.1 Numerical integration from given data using trapezoidal method	p. 72
3.4.2 Numerical integration of single variable functions	p. 74
3.4.3 Numerical solutions to double integrals	p. 77
3.4.4 Numerical solutions to triple integrals	p. 79
3.5 Path Integrals and Line Integrals	p. 80
3.5.1 Path integrals	p. 80
3.5.2 Line integrals	p. 81
3.6 Surface Integrals	p. 83
3.6.1 Scalar surface integrals	p. 83
3.6.2 Vector surface integrals	p. 84
Exercises	p. 85
4 Linear Algebra Problems	p. 89
4.1 Inputting Special Matrices	p. 90
4.1.1 Numerical matrix input	p. 90
4.1.2 Defining symbolic matrices	p. 94
4.2 Fundamental Matrix Operations	p. 95
4.2.1 Basic concepts and properties of matrices	p. 95
4.2.2 Matrix inversion and generalized inverse of a matrix	p. 102
4.2.3 Matrix eigenvalue problems	p. 106
4.3 Fundamental Matrix Transformations	p. 109
4.3.1 Similarity transformations and orthogonal matrices	p. 109
4.3.2 Triangular and Cholesky decompositions	p. 111
4.3.3 Jordan transformations	p. 114
4.3.4 Singular value decompositions	p. 116
4.4 Solving Matrix Equations	p. 118
4.4.1 Solutions to linear algebraic equations	p. 118
4.4.2 Solutions to Lyapunov equations	p. 121
4.4.3 Solutions to Sylvester equations	p. 124
4.4.4 Solutions to Riccati equations	p. 125
4.5 Nonlinear Functions and Matrix Function Evaluations	p. 126
4.5.1 Element-by-element computations	p. 126
4.5.2 Matrix function evaluations	p. 127
Exercises	p. 133
5 Integral Transforms and Complex Variable Functions	p. 137
5.1 Laplace Transforms and Their Inverses	p. 137
5.1.1 Definitions and properties	p. 138
5.1.2 Computer solution to Laplace transform problems	p. 139
5.2 Fourier Transforms and Their Inverses	p. 142
5.2.1 Definitions and properties	p. 142
5.2.2 Solving Fourier transform problems	p. 142
5.2.3 Fourier sine and cosine transforms	p. 144
5.2.4 Discrete Fourier sine, cosine transforms	p. 147
5.3 Other Integral Transforms	p. 147
5.3.1 Mellin transform	p. 148
5.3.2 Hankel transform solutions	p. 149
5.4 Z Transforms and Their Inverses	p. 150
5.4.1 Definitions and properties of Z transforms and inverses	p. 150
5.4.2 Computations of Z transform	p. 151
5.5 Solving Complex Variable Function Problems	p. 152
5.5.1 Complex variable functions and mapping visualization	p. 152
5.5.2 Concept and computation of residues	p. 152
5.5.3 Partial fraction expansion for rational functions	p. 155
5.5.4 Inverse Laplace transform using PFEs	p. 159
5.5.5 Computing closed-path integrals	p. 160
Exercises	p. 162
6 Nonlinear Equations and Numerical Optimization Problems	p. 165
6.1 Nonlinear Algebraic Equations	p. 166
6.1.1 Graphical method for solving nonlinear equations	p. 166
6.1.2 Quasi-analytical solutions to polynomial-type equations	p. 168
6.1.3 Numerical solutions to general nonlinear equations	p. 172
6.1.4 Nonlinear matrix equations	p. 174
6.2 Unconstrained Optimization Problems	p. 176
6.2.1 Analytical solutions and graphical solution methods	p. 176
6.2.2 Numerical solution of unconstrained optimization using MATLAB	p. 178
6.2.3 Global minimum and local minima	p. 179
6.2.4 Solving optimization problems with gradients	p. 181
6.2.5 Optimization problems with bounded constraints	p. 182
6.3 Constrained Optimization Problems	p. 183
6.3.1 Constraints and feasibility regions	p. 184
6.3.2 Solving linear programming problems	p. 185
6.3.3 Solving quadratic programming problems	p. 187
6.3.4 Solving general nonlinear programming problems	p. 188
6.4 Mixed Integer Programming Problems	p. 191
6.4.1 Solving mixed integer programming problems	p. 191
6.4.2 Solving binary programming problems	p. 194
6.5 Linear Matrix Inequalities	p. 195
6.5.1 A general introduction to LMIs	p. 196
6.5.2 Lyapunov inequalities	p. 196
6.5.3 Classification of LMI problems	p. 198
6.5.4 LMI problem solutions with MATLAB	p. 199
6.5.5 Optimization of LMI problems by YALMIP Toolbox	p. 201
Exercises	p. 203
7 Differential Equation Problems	p. 207
7.1 Analytical Solution Methods for Special Classes of ODEs	p. 208
7.1.1 Mathematical descriptions	p. 208
7.1.2 Analytical solution methods	p. 210
7.1.3 Applications of Laplace transforms	p. 212
7.1.4 Analytical solutions to LTI state-space equations	p. 214
7.1.5 Analytical solutions to special nonlinear differential equations	p. 215
7.2 Numerical Solutions to ODEs	p. 215
7.2.1 Overview of numerical solution algorithms	p. 216
7.2.2 Fixed-step Runge-Kutta algorithm and its MATLAB implementation	p. 218
7.2.3 Numerical solution to first-order vector ODEs	p. 219
7.2.4 Transforms to standard ODEs	p. 224
7.2.5 Validation of numerical solutions to ODEs	p. 231
7.3 Numerical Solutions to Special Ordinary Differential Equations	p. 232
7.3.1 Solutions of stiff ODEs	p. 232
7.3.2 Solutions of implicit differential equations	p. 235
7.3.3 Solutions to differential algebraic equations	p. 239
7.3.4 Solutions to delay differential equations	p. 241
7.4 Solving Boundary Value Problems	p. 243
7.4.1 Solutions to two-point boundary value problems	p. 243
7.4.2 Solutions to general boundary value problems	p. 245
7.5 Introduction to Partial Differential Equations	p. 247
7.5.1 Solving a set of 1D PDEs	p. 248
7.5.2 Mathematical description to 2D PDEs	p. 249
7.5.3 The GUI for the PDE Toolbox - an introduction	p. 251
7.6 Solving ODEs with Block Diagrams in Simulink	p. 258
7.6.1 A brief introduction to Simulink	p. 258
7.6.2 Simulink - relevant blocks	p. 258
7.6.3 Using Simulink for modeling and simulation of ODEs	p. 260
Exercises	p. 263
8 Data Interpolation and Functional Approximation Problems	p. 269
8.1 Interpolation and Data Fitting	p. 270
8.1.1 One-dimensional data interpolation	p. 270
8.1.2 Definite integral evaluation from given samples	p. 273
8.1.3 Two-dimensional grid data interpolation	p. 275
8.1.4 Two-dimensional scattered data interpolation	p. 277
8.1.5 High-dimensional data interpolations	p. 280
8.2 Spline Interpolation and Numerical Calculus	p. 281
8.2.1 Spline interpolation in MATLAB	p. 281
8.2.2 Numerical differentiation and integration with splines	p. 284
8.3 Data Modeling	p. 287
8.3.1 Polynomial fitting	p. 287
8.3.2 Approximation by continued fraction expansions	p. 290
8.3.3 Pade rational approximations	p. 292
8.3.4 Curve fitting by linear combination of basis functions	p. 294
8.3.5 Least squares curve fitting	p. 296
8.4 Signal Analysis and Digital Signal Processing	p. 298
8.4.1 Correlation analysis	p. 298
8.4.2 Fast Fourier transforms	p. 300
8.4.3 Filtering techniques and filter design	p. 302
Exercises	p. 306
9 Probability and Mathematical Statistics Problems	p. 309
9.1 Distributions and Pseudo-Random Number Generators	p. 309
9.1.1 Introduction to PDFs and CDFs	p. 309
9.1.2 PDFs/CDFs of commonly used distributions	p. 310
9.1.3 Solving probability problems	p. 317
9.1.4 Random numbers and pseudo-random numbers	p. 318
9.2 Statistics	p. 319
9.2.1 Mean and variance of random variables	p. 319
9.2.2 Moments of random variables	p. 321
9.2.3 Covariance analysis of multivariate random variables	p. 322
9.2.4 Multivariate normal distributions	p. 323
9.2.5 Monte Carlo solutions to mathematical problems	p. 324
9.3 Statistical Analysis	p. 326
9.3.1 Parametric estimation and interval estimation	p. 326
9.3.2 Multivariable linear regression and interval estimation	p. 328
9.3.3 Nonlinear parametric and interval estimations	p. 330
9.4 Statistic Hypothesis Tests	p. 333
9.4.1 Basic concept and procedures for statistic hypothesis test	p. 333
9.4.2 Solving hypothesis test problems in MATLAB	p. 334
9.5 Analysis of Variance and Its Computation	p. 337
9.5.1 One-way ANOVA	p. 337
9.5.2 Two-way ANOVA	p. 339
9.5.3 n-way ANOVA	p. 341
Exercises	p. 341
10 Nontraditional Solution Methods	p. 345
10.1 Fuzzy Logic and Fuzzy Inference	p. 346
10.1.1 Classical set theory and fuzzy sets	p. 346
10.1.2 Membership function and fuzzification	p. 349
10.1.3 An interactive membership function editor	p. 351
10.1.4 Building fuzzy inference systems	p. 351
10.1.5 Fuzzy rules and fuzzy inference	p. 353
10.2 Neural Network and Its Applications in Data Fitting Problems	p. 356
10.2.1 Fundamentals of neural networks	p. 357
10.2.2 Graphical user interface for neural networks	p. 364
10.3 Evolution Algorithms and Their Applications in Optimization Problems	p. 366
10.3.1 Basic idea of genetic algorithms	p. 366
10.3.2 MATLAB solutions to optimization problems with genetic algorithms	p. 368
10.3.3 Particle swarm optimizations	p. 373
10.3.4 Solving optimization problems with GADS Toolbox	p. 374
10.3.5 Towards accurate global minimum solutions	p. 377
10.4 Wavelet Transform and Its Applications in Data Processing	p. 378
10.4.1 Wavelet transform and waveforms of wavelet bases	p. 378
10.4.2 Wavelet transform in signal processing problems	p. 383
10.4.3 Graphical user interface in wavelets	p. 386
10.5 Rough Set Theory and Its Applications	p. 388
10.5.1 Introduction to rough set theory	p. 388
10.5.2 Data processing problem solutions using rough sets	p. 391
10.6 Fractional-Order Calculus	p. 395
10.6.1 Definitions of fractional-order calculus	p. 395
10.6.2 Evaluating fractional-order differentiation	p. 400
10.6.3 Solving fractional-order differential equations	p. 405
Exercises	p. 412
References and Bibliography	p. 415
MATLAB Functions Index	p. 419
Index	p. 425

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