 اسم المؤلف
Alexander Stanoyevitch
التاريخ
3 مارس 2023
المشاهدات
249
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Introduction to Numerical Ordinary and Partial Differential Equations Using MATLAB
Alexander Stanoyevitch
Contents
Preface ix
PART I: Introduction to MATLAB and
Numerical Preliminaries
Chapter 1: MATLAB Basics 1
Section 1.1: What Is MATLAB?
Section 1.2: Starting and Ending a MATLAB Session
Section 1.3: A First MATLAB Tutorial
Section 1.4: Vectors and an Introduction to MATLAB Graphics
Section 1.5: A Tutorial Introduction to Recursion on MATLAB
Chapter 2: Basic Concepts of Numerical Analysis 23
with Taylor’s Theorem
Section 2.1: What Is Numerical Analysis?
Section 2.2: Taylor Polynomials
Section 2.3: Taylor’s Theorem
Chapter 3: Introduction to M-Files 45
Section 3.1: What Are M-files?
Section 3.2: Creating an M-file for a Mathematical Function
Chapter 4: Programming in MATLAB 57
Section 4.1: Some Basic Logic
Section 4.2: Logical Control Flow in MATLAB
Section 4.3: Writing Good Programs
Chapter 5: Floating Point Arithmetic and Error Analysis 85
Section 5.1: Floating Point Numbers
Section 5.2: Floating Point Arithmetic: The Basics
Section 5.3:1 Floating Point Arithmetic: Further Examples and Details
Chapter 6: Rootfinding 107
Section 6.1: A Brief Account of the History of Rootfinding
Section 6.2: The Bisection Method
Section 6.3: Newton’s Method
Section 6.4: The Secant Method
Section 6.5: Error Analysis and Comparison of Rootfmding Methods
Chapter 7: Matrices and Linear Systems 143
Section 7.1: Matrix Operations and Manipulations with MATLAB
Section 7.2: Introduction to Computer Graphics and Animation
Section 7.3: Notations and Concepts of Linear Systems
Section 7.4: Solving General Linear Systems with MATLAB
Section 7.5: Gaussian Elimination, Pivoting, and LU Factorization
Section 7.6: Vector and Matrix Norms, Error Analysis, and Eigendata
Section 7.7: Iterative Methods
PART II: Ordinary Differential Equations
Chapter 8: Introduction to Differential Equations 285
Section 8.1: What Are Differential Equations?
Section 8.2: Some Basic Differential Equation Models and Euler’s
Method
Section 8.3: More Accurate Methods for Initial Value Problems
Section 8.4: Theory and Error Analysis for Initial Value Problems
Section 8.5: Adaptive, Multistep, and Other Numerical Methods for
Initial Value Problems
Chapter 9: Systems of First-Order Differential Equations 355
and Higher-Order Differential Equations
Section 9.1: Notation and Relations
Section 9.2: Two-Dimensional First-Order Systems
Section 9.3: Phase-Plane Analysis for Autonomous First-Order Systems
Section 9.4: General First-Order Systems and Higher-Order Differential
Equations
Chapter 10: Boundary Value Problems for Ordinary 399
Differential Equations
Section 10.1: What Are Boundary Value Problems and How Can They
Be Numerically Solved?
Section 10.2: The Linear Shooting Method
Section 10.3: The Nonlinear Shooting Method
Section 10.4: The Finite Difference Method for Linear BVPs
Section 10.5: Rayleigh-Ritz Methods Contents vii
PART III: Partial Differential Equations
Chapter 11: Introduction to Partial Differential Equations 459
Section 11.1: Three-Dimensional Graphics with MATLAB
Section 11.2: Examples and Concepts of Partial Differential Equations
Section 11.3: Finite Difference Methods for Elliptic Equations
Section 11.4: General Boundary Conditions for Elliptic Problems and
Block Matrix Formulations
Chapter 12: Hyperbolic and Parabolic Partial Differential 523
Equations
Section 12.1: Examples and Concepts of Hyperbolic PDEs
Section 12.2: Finite Difference Methods for Hyperbolic PDEs
Section 12.3: Finite Difference Methods for Parabolic PDEs
Chapter 13: The Finite Element Method 599
Section 13.1: A Nontechnical Overview of the Finite Element Method
Section 13.2: Two-Dimensional Mesh Generation and Basis Functions
Section 13.3: The Finite Element Method for Elliptic PDEs
Appendix A: Introduction to MATLAB’s Symbolic 691
Toolbox
Appendix B: Solutions to All Exercises for the Reader 701
References 799
MATLAB Command Index 805
General Index 809
General Index
Abel, Niels Henrik, 108,109
Actual error, 24
Affine transformation, 163
Algebraic multiplicity, 243
Approximation, 24
Associated matrix norm, 226
Asymptotic error constant, 133
Augmented matrix, 195
Autonomous, 357
Auxiliary condition, 286
Back substitution, 206
Backward difference approximation, 509
Backward Euler method, 332
Base, 85, 94
Basin of attraction, 382
Basis theorem, 193
Bendixson, Ivar, 382
Big-0 notation, 320
Binary arithmetic, 85
Birthrate, 290
Birthday problem, 70
Bisection method, 110
Boundary condition, 475
-Cauchy, 527
-Dirichlet, 476
-Neumann, 476
-Robin, 637
Boundary value problem, 355, 399
Bracket, 116
Bunyakowsky, Viktor Yakovlevich, 455
Cantor, Georg F.L.P., 169
Cantor square, 184
Cardano, Girolamo, 108
Carrying capacity, 292
Cauchy, Augustin Louis, 455
Cauchy-Bunyakowski-Schwarz inequality, 455
Cauchy problem, 527
Center, 362
Central difference formula, 43,418-419, 544
Characteristic polynomial, 241, 342
Chopped arithmetic, 89
Clay Foundation, 68
Cofactor expansion, 76
Collatz, Lothar, 77
Collatz conjecture, 67,68
Column, 143
Combinatorics: alternating power sums, 202
Combinatorics: power sums, 202
Compatibility condition, 508
Component-wise operation, 9
Computed solution, 231
Computer graphics, 157
Condition number, 228-230
Conservation of energy, 537
Convergence order, 132
Convergence theorem, 262,264, 265, 266
Convex hull, 609
Contact rate, 366
Counter, 60
Courant, Richard, 597-598
Courant-Friedrichs-Levy condition, 548
Cramer, Gabriel, 203
Cramer’s rule, 203
Crank, John, 575
Crank-Nicolson method, 575-577
Cycling, 122
D’Alembert, Jean Le Rond, 525
Death rate, 290
Degree, 25
Delaunay triangulation, 608
Determinant, 75,222
Diagonal matrix, 147
Diameter, 71
Differential Equation (DE), 285
Diffusivity, 469
Dilation, 172
Dimension, 171
Direct method, 252
Dirichlet’s principle, 520
Discriminant, 379, 474
Divergence theorem, 519
Divided difference, 131
Domain of dependence, 531
Dot product, 22, 144
Double root, 130
Eigendata, 240
Eigenfunction, 441,496
Eigenspace, 243
Eigenvalue, 240,496, 497
Eigenvector, 240
Element, 598
-Standard, 633

• Standard rectangular, 629
Elementary row operation (ERO), 207
Epicycloids, 13
809 810
Epidemie, 366
Equilibrium solution, 299, 362, 373
-Isolated, 377
Equivalent linear system, 195
Error bound via residual, 233
Error function, 42
Error term, 231
Essentially disjoint, 172
Euclidean length, 224
Euler, Leonhard, 292-293
Euler’s method, 292-294
Expected value, 83
Existence theorem, 314, 376, 402,494, 496,
507,515,585
Explicit method, 332
False, 57
Fern leaf fractal, 185
Finite difference schemes
• Crank-Nicolson, 575-577
-Elliptic, Sec. 11.3, 11.4
-Explicit, 542-543
• Forward-time central-space, 573
• Backward-time central-space, 574
-Implicit, 558
-ODEs, 418-425
• Richardson, 575
Finite element interpolant, 607
Finite element method, 597
First generation, 170
Fixed point iteration, 140
Floating point number, 85
Flop, 74
Flop counts (for Gaussian elimination), 226
Flow, 323
Fontana, Niccolo 108
Forward substitution, 207
Forward difference formula, 43, 508
Fractals (fractal sets), 169
Future value annuities, 72, 73
Galerkin, Boris Grigorievich, 440
Galerkin method, 440
Galois, Evariste, 109
Gauss, Carl Friedrich, 204
Gauss-Seidel iteration, 256
Gaussian elimination, 203-213
General solution, 286
Generalized minimum residual method, 273-274
Geometric multiplicity, 243
Ghost node, 509
Global solution, 315
Global variables, 46
Gomperz law, 300
Gosper island fractal, 185,186
Green ‘s identities, 519
-First, 519
General Index
-Second, 520
Growth rate, 290
Hamming, Richard Wesley, 348
Hamming method, 348
Harmonic function, 476
Hat function, 432
Heat conductivity, 469
Heat (diffusion) equation, 469,470
• Fundamental solution, 478
• With source term, 470
Heun’s method, 303
Higher-order Taylor methods, 318
Hubert, David, 193
Homogeneous, 401,472
Homogeneous coordinates, 163,164
Hyper convergence of order or, 133
IEEE double precision standard, 86
Ill-conditioned, 102
Ill-posed, 187
Implicit method, 332
Improved Euler method, 303-304
Infectivity, 364
Initial condition (IC), 286
Initial value problem (IVP), 286
Inline function, 51
Infinite loop, 16
Infinity matrix norm, 227
Infinity (vector) norm, 225
Initial population, 290
Inner product, 427
Input-output analysis, 200,201
Internal demand matrix, 201
Internal elastic energy, 428
Inverse of a matrix, 148
Invertible (nonsingular), 148
Iterative, 109
Iterative method, 252
Iterative refinement, 249
Jacobi-Gauss convergence theorem, 262
Jacobi iteration, 253
Jacobian matrix, 378
Julia, Gastón, 169
Kinetic energy, 535
Kronecker delta, 608
Kutta, Martin W., 305
Lagrange, Joseph Louis, 471
Laplace, Pierre Simon, 471
Laplace equation, 471
Laplace operator (Laplacian), 470
• in polar coordinates, 686
Leontief, Wassily, 200
Linear convergence, 133 General Index 811
Linear operator, 473
Linear ODE, 401
Linear PDE, 472
Linear transformation, 160
Linearization, 378
Lipschitz condition, 313, 376
Local basis, 607
Local solution, 315
Local truncation error, 317
Local variables, 46
Logic, 57
Logical operators, 58
Logistical growth model, 291
Lorenz, Edward N., 387
Lorenz strange attractor, 387
Lotka, Alfred, 359
Lower triangular, 205
LU decomposition (or factorization), 213
M-file, 45
• Function M-files, 45
• Script M-files, 45
Machín, John, 43
Machine epsilon, 86
Maclaurin, Colin, 39
Maclaurin series, 38
Malthus, Thomas, 290
Malthus growth model, 290
Mandelbrot, Benoit, 170
Mantissa, 87
Matrix, 143
-Banded, 152,420
• Block, 502
• Diagonally-dominant, 221, 264, 422
• Elementary, 208
-Hubert, 192
-Identity, 148
• Nonsingular (in vertible), 148
• Positive definite, 265
-Sparse, 151,269-278,420
• Stiffness, 434,443
-Technology, 201
• Tridiagonal, 420
Matrix arithmetic, 144
Max norm, 225
Maximum principle, 477, 586, 595
Midpoint method, 343
Monte-Carlo method, 173
Mother loop, 62
Multiple root, 125
Multiplicity 1, 55
Multistep method, 337
Natural growth rate, 292
Nearly singular (poorly conditioned), 228
Necrotic, 300
Nested loop, 61
Newton’s method, 118, 119
Newton-Coates formula, 669
Nicolson, Phyllis, 575
Node, 602
Nonautonomous, 357
Nullclines, 373
Numerical differentiation, 43
Numerically stable, 334
Numerically unstable, 335
One-step method, 303
Orbit, 362
-Closed, 382
Order, 130,285,308,317
Ordinary Differential Equation (ODE), 285
Output matrix, 201
Overflow, 88
Parallel, 239
Parametric equations, 11
Partial Differential Equation (PDE), 285, 459
• Divergence form, 637
-Elliptic, 474
-Hyperbolic, 474
• Parabolic, 474
Partial pivoting, 211
Path (MATLAB’s), 45
Peano, Guiseppe, 169
Pendulum model, 389
Perfect number, 81
Phase-plane, 362
Piecewise smooth, 496
Pivot, 211
Poincaré, Henri, 378
Poincaré-Bcndixson theorem, 382
Poisson, Siméon-Denis, 649-649
Poisson’s integral formula, 649
Poisson equation, 479
Polynomial, 25
Polynomial interpolation, 189, 197-199
Poorly conditioned matrix, 150, 228
Potential energy, 536
Potential theory, 476
Preconditioning, 273
Predator-prey model, 358-360
Predictor-corrector scheme, 339
Prime number, 81
Principle of minimum potential energy, 428
Principle of virtual work, 428
Prompt, 2
Pyramid function, 603
Quartic, 108
Quintic, 108 812 General Index
Random integer matrix generator, 152
Random walk, 82
Rayleigh-Ritz method, 426-458
Recursion formulas, 15
Reduced row echelon form, 195
Reflection, 162
-Methodof, 531
Region of numerical stability, 335
Relative error bound (via residual), 233
Relaxation parameter, 258
Remainder (Taylor’s), 35
Repelling, 382
Reproduction rate, 367
Residual, 116
Residual matrix, 250
Residual vector, 232
Rhind Mathematical Papyrus, 107
Richardson’s method, 575
Ritz, Walter, 426
Root, 110
Rossler, Otto, 395
Rotation, 161
Rounded arithmetic, 89
Row, 143
Runge, Carle D. T., 205
Runge-Kutta method,
-Classical, 304-305
• Higher order, 350
Runge-Kutta-Fehlberg method (RKF45), 327
Scalar, 240
Scalar multiplication, 144
Scaling, 161
Schwarz, Hermann Amandus, 455
Secant method, 128, 129
Self-similarity property, 169
Separation of variables, 302
Shearing, 181
Shift transformation, 162
Shooting method, 399
-Linear, 403-411
-Nonlinear, 411-418
Sierpinski, Waclaw, 170
Sierpinski carpet fractal, 184, 185
Significant digits, 85
Similarity transformation, 172
Simple root, 125
Simpson’s Rule, 325
Simulation, 79
Single-step method, 337
Singularity, 527
SIR model, 363
SIRS mode, 367
Solution, 285
SOR (successive over relaxation), 258
SOR convergence theorem, 264
Special function, 693
Specific heat, 468
Spectrum, 251,497
Spline, 449
Stability, 323,381
Stability condition, 574
Stable, 299, 323, 376
-Conditionally, 586
-Neutrally, 323
-Unconditionally, 586
• Weakly, 342
Standard local basis, 608
Statement, 57
Stencil, 481,542,576
Step size, 293
Stiff, 335
Strutt, John William, 426
Submatrix, 76
Superposition principle, 473
Symbolic computation, 689
Symmetric matrix, 243
Tartaglia, 108
Taylor, Brook, 34
Taylor polynomial, 25
Taylor series, 38
Taylor’s theorem,
• One variable, 35
• Two variables, 350
Tessellation, 186
Thomas, Llewellyn H., 220
Thomas method, 220
Three-body problem, 391
Tolerance, 24
Torricelli, Evangelista, 312
Torricelli’s law, 312
Traffic logistics, 199,200
Transient part, 335
Transpose, 7
Trapezoid method, 337
Triangulation, 598
Tridiagonal matrix, 150
Triple root, 126
True, 57
Truth value, 57
Two-body problem, 391
Unconditional numerical stability, 336
Underflow, 88
Uniqueness theorem, 314, 376,402,421, 494,
496,507,515,585
Unit roundoff, 86
Unstable, 299, 323, 376, 548
Upper triangular matrix, 204
van der Pol, Balthasar, 396
van der Pol equation, 396
Vandermonde matrix, 197
Variable precision arithmetic, 689
Vector, 7 General Index
Vector norm, 225
Verhulst, Peirre Francois, 292
Volterra, Vito, 358-359
von Koch, Niels F.H., 179
von Koch snowflake, 179
Voronoi diagram, 610
Voronoi region, 609
Vortex, 362
Wave equation, 474, 523, 524
Weierstrass, Karl, 179
Weights, 662
Well-conditioned, 102
Well-posed, 187
Zero divisors, 105
Zeroth generation, 170

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