Advances in the Applications of Nonstandard Finite Difference Schemes

CONTENTS......Page 10
Preface......Page 6
1 General Comments......Page 16
2 NSFD Methods: Basic Principles......Page 17
3 Dynamic Consistency......Page 19
4 Some Outstanding Problems......Page 21
References......Page 22
2 Application of Nonstandard Finite Difference Schemes to the Simulation Studies of Robotic Systems R. F. Abo-Shanab. N . Sepehri and C. Q. Wu......Page 26
1 Introduction......Page 27
2 Dynamic Model of the Robotic Systems......Page 28
3 Constructing the Discrete Derivatives......Page 30
4 Simulation Studies......Page 33
4.1.1. Description of the Case......Page 34
4.1.2. Numerical Results......Page 35
4.2.1. Description of the Case......Page 45
4.2.2. Numerical Results......Page 47
5 Conclusions......Page 59
References......Page 60
1 Introduction......Page 62
2 The Buckmire MFD Scheme......Page 63
2.1.1. The cylindrical case......Page 65
2.1.2. The spherical case......Page 66
2.2. Informal Derivation of the Nonstandard Schemes......Page 67
3.1. The First Model Problem......Page 68
3.1.1. The m # 0 cylindrical case......Page 69
3.2.1. The n # 0 spherical case......Page 70
3.3. Numerical Results......Page 71
Conclusions......Page 73
Introduction......Page 75
4.1. Numerical Solutions of the Bratu- Gelfand Problem......Page 78
Introduction......Page 81
5.1. The 1-dimensional Planar Bratu Problem......Page 82
5.2.1. Finite difference methods......Page 85
5.2.2. Boyd collocation......Page 86
5.2.3. Adomian polynomial decomposition......Page 89
5.2.4. Shooting method......Page 91
5.3. Numerical Results......Page 92
6 Conclusion......Page 99
References......Page 100
1 Introduction......Page 104
2.1. Finite-Diflerence Models and the FDTD Algorithm......Page 106
2.2. A Nonstandard Finite Difference Model......Page 108
2.3. Extension to Two and Three Dimensions......Page 111
3.1. Exact Algorithm in One Dimension......Page 115
3.2. Extension to Two and Three Dimensions......Page 117
4.1. The Yee Algorithm......Page 118
4.2. Maxwell’s Equations and the Wave Equation......Page 119
4.3. NSFD Version of the Yee Algorithm......Page 123
5.1. Introduction......Page 127
5.2. Absorbing Boundary Conditions in One Dimension......Page 129
5.3. Nonstandard Finite Difference Model of Engquist-Majda One- Way Wave Equations......Page 133
5.4. Comparisons and Practical Tests......Page 140
5.5. Summary of the NS-ABC Method......Page 141
Sampling Density......Page 142
Numerical Stability......Page 144
6.2. Stability of the Wave Equation FDTD Algorithm......Page 147
NSFD Stability......Page 148
6.3. Stability of the Absorbing Wave Equation FDTD Algorithm......Page 149
One Dimension......Page 151
Two Dimensions......Page 152
Three Dimensions......Page 154
NSFD Algorithms......Page 155
Shape Representation......Page 158
Magnetic Permeability and Electric Permittivity......Page 160
Sources in the Wave Equation and Maxwell’s Equations......Page 161
Computing Scattered Fields......Page 162
Time Dependence of Sources......Page 164
8.1. Introduction......Page 165
8.2. Structural Color of Butterfly Wings......Page 168
8.3. Improving Light Transmission Across Interfaces with Subwavelength Structures......Page 172
8.4. Classical Diffraction by Conducting Grating......Page 174
9.1. Introduction......Page 178
General Development......Page 186
Extension to the Conducting Maxwell Equations......Page 188
9.3. Stability Analysis......Page 189
Stability of the Recursive Convolution Algorithm......Page 191
9.4. Effective Current Algorithm......Page 192
Stability of the Effective Current Algorithm......Page 194
Stability of the Hybrid Frequency-Domain Time-Domain Algorithm......Page 195
10 Summary......Page 196
Acknowledgments......Page 200
References......Page 202
1 Introduction......Page 206
2 Governing Equations and an Energy Estimate......Page 210
3 Nonstandard Finite Difference Scheme......Page 213
4 Nonlinear Interfacial Problem......Page 231
5 Conclusion......Page 240
Acknowledgments......Page 242
References......Page 261
6 Reliable Finite Difference Schemes with Applications in Mathematical Ecology D. T. Dimitrow. H . V . Kojouharov and B. M. Chen-Charpentier......Page 264
1 Introduction......Page 265
2 Definitions and Preliminaries......Page 266
3 Numerical Methods for Single-Species Population Models......Page 268
4 Numerical Methods for Multi-Species Population Models......Page 276
4.1. Elementary Stable Nonstandard (ESN) Methods......Page 277
4.2. Positive and Elementary Stable Nonstandard (PESN) Methods......Page 279
Rosenzweig-MacArthur predator-prey models......Page 280
Phytoplankton-Nutrient Systems with Nutrient Loss......Page 283
General Multi-Dimensional Systems......Page 286
5.1. Single-Species Population Simulations......Page 287
5.2. Multi-Species Population Simulations......Page 290
6 Conclusions......Page 297
References......Page 299
1 Introduction......Page 302
2 A Short Introduction to the Theory of Differential Inclusion......Page 304
2.1. A general existence result for differential inclusions......Page 306
2.1.1. Differential equations with discontinuous right-hand side......Page 307
2.2. Differential inclusions for friction problems......Page 308
2.3. Differential inclusion in vibro-impacts......Page 309
3 Numerical Methods for Differential Inclusions......Page 312
3.1. Numerical methods for vibro-impact problems......Page 314
3.1.1. The PS-scheme......Page 316
3.1.2. The Moreau-Monteiro Marques-Mabrouk scheme......Page 317
4 Frictional Oscillators......Page 318
4.1. The harmonic linear oscillator......Page 319
4.2. Bictional oscillators with velocity-dependent frictions......Page 320
5 Vibro-Impact Oscillators......Page 326
5.1.1. A non-standard PS-scheme for conservative oscillators......Page 331
5.1.2. A non-standard MMM-scheme......Page 333
5.2. I. The NSMMM-exact-scheme......Page 338
5.2.2. The NSPS-exact-scheme......Page 339
5.3. The duffing oscillator with impacts......Page 345
6 Conclusions......Page 349
References......Page 352
1 Introduction......Page 358
2.1. The Black-Scholes Equation for American Options......Page 360
2.2. The Schrodinger-Poisson System......Page 363
2.3. The Standard "Parabolic Equation"......Page 364
3 The Finite Difference Equations......Page 366
3.2. Schrodinger-Poisson System......Page 367
3.3. The Standard “Parabolic Equation”......Page 369
4.1. The Black-Scholes Equation for American Options......Page 373
4.2. The Schrodinger-Poisson System......Page 374
4.3. The Standard "Parabolic Equation"......Page 375
5 Discrete ABCs through Asymptotic Solutions......Page 377
5.1. The Schrodinger-Poisson System......Page 379
5.2. The Standard “Parabolic Equation”......Page 381
6 The Continued Fraction Approach......Page 383
6.1. The Schrodinger-Poisson System......Page 384
7 The Approximation by the Sum of Exponentials......Page 385
8 Numerical Example......Page 388
Conclusion......Page 392
Acknowledgments......Page 393
Appendix A . The Z-Transformation......Page 394
References......Page 397
9 Asymptotically Consistent Non-Standard Finite-Difference Methods for Solving Mathematical Models Arising in Population Biology A . B . Gumel, K . C . Patidar and R . J . Spiteri......Page 400
1 Introduction......Page 401
2 SIS Model......Page 402
2.1.1. Disease-free equilibrium......Page 403
2.1.2. Endemic equilibrium......Page 404
2.2.2. Analysis of fixed points......Page 405
2.2.3. Second-order implicitly-derived explicit method......Page 407
2.3. Asymptotically consistent finite-difference method......Page 409
3 Transmission Dynamics of Two HIV Subtypes......Page 411
3.1. Existence and stability of equilibria......Page 412
3.2.1. Method 1......Page 413
3.2.2. Method 2: Non-standard positivity-preserving......Page 415
4 A Predator-Prey Model......Page 417
4.2. Positivity-preserving finite-difference method......Page 418
4.3. Numerical simulations......Page 419
4.4. A modified predator-prey model......Page 421
5 Concluding Remarks and Challenges......Page 422
References......Page 435
1 Introduction......Page 438
2.1. SI and SIS models......Page 441
2.2. A simple SIR model......Page 446
2.3. A crisscross disease model......Page 450
2.4. An SIS model with immigration of infectives......Page 453
3 A Multiple Populations Competition Model......Page 463
4 Discussion......Page 469
References......Page 471
1 Introduction......Page 474
2.1. Poincare section and bifurcation diagram......Page 478
2.2. Template and linking numbers......Page 481
3 Motivations for Nonstandard Schemes......Page 484
4 The Nonstandard Mickens Scheme......Page 489
4.1. A finite difference model for the Rossler system......Page 490
4.2. The Genesio and Tesi system......Page 494
4.3. Discretization of the Lorenz system......Page 499
5.1. The scheme......Page 506
5.2. The Genesio and Tesi system......Page 507
5.3. The Lorenx system......Page 516
6 Conclusion......Page 522
References......Page 524
12 Contributions to the Theory of Non-Standard Finite Difference Methods and Applications to Singular Perturbation Problems J. M.-S. Luburna and K. C. Patidar......Page 528
1 Introduction......Page 529
2 Towards a Definition of the Nonstandard Finite Difference Method......Page 531
3.1. Generalities......Page 544
3.2. Case of hyperbolic fixed points......Page 545
3.3. The case of non-hyperbolic fixed-points......Page 550
3.4. The case of linear non-autonomous differential equations......Page 553
4.1. A class of nonlinear problems......Page 556
4.2. Linear problems......Page 559
5 Singularly Perturbed Advection-Reaction Problems......Page 563
6 Singularly Perturbed Reaction-Diffusion Problems......Page 568
7 Conclusions......Page 571
References......Page 573
13 Frequency Accurate Finite Difference Methods A. L. Perkins P. A. Orlin and F. Zand......Page 576
2 Average Value Spatial Difference......Page 577
2.1. Frequency Domain Formulation......Page 579
2.2. Frequency Domain Solutions......Page 580
3 Temporal Derivative......Page 587
3.1. Average Value Approximation......Page 589
3.2. Frequency Domain Formulation......Page 592
3.3. Temporal Frequency Specification......Page 593
3.4. Frequency Domain Solutions......Page 594
3.5. Fitting Example......Page 596
4 Higher Order Spatial Derivatives......Page 599
4.1. Frequency Domain Formulation......Page 602
4.2. Heuristic Formulation......Page 603
4.3. Approximation Fitting......Page 606
5.1. Dispersion Relations......Page 612
5.2. Frequency Domain Fitting......Page 614
5.3. Nonlinear Stabilization......Page 618
5.4. Numerical Example......Page 619
5.5. Conclusion......Page 626
6 Historical Notes......Page 627
References......Page 629
1 Introduction......Page 630
2.1. Predator-prey System......Page 631
2.2. Competitive System......Page 637
3 The May-Leonard Competitive System......Page 641
3.1.1. Symmetric May-Leonard competition model......Page 642
3.1.2. Asymmetric May-Leonard competition model......Page 645
3.2.1. Symmetric May-Leonard competition model......Page 647
3.2.2. Asymmetric May-Leonard Competition Model......Page 649
3.3. Numerical Examples......Page 651
4 Three-Dimensional Lotka-Volterra Competitive System......Page 655
4.1. Kahan Competitive Model......Page 656
4.2. The Discrete Model of the Exponential Type......Page 660
4.3. The Discrete Model of the Fractional Type......Page 661
5 Conclusions and Open Problems......Page 663
References......Page 664