Introduction to Numerical Ordinary and Partial Differential Equations Using MATLAB

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
Adams family methods, 338
Adams-Bashforth method, 338
Adams-Moulton method, 339
Abel, Niels Henrik, 108,109
Actual error, 24
Adaptive method, 327
Admissible function, 427
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
  Exact answer, 24,231
  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 quadrature, 662
  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
  Leading one, 195
  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
  Load potential, 428
  Load vector, 434,443
  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
  Preconditioned conjugate gradient method, 273
  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
  Quadratic convergence, 139
  Quadrature, 51
  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
  Sierpinski gasket fractal, 170
  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
  Steady-state solution, 336
  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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