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E-raamat: Nonstandard Finite Difference Schemes: Methodology And Applications

(Clark Atlanta Univ, Usa)
  • Formaat: 332 pages
  • Ilmumisaeg: 11-Nov-2020
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789811222559
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Nonstandard Finite Difference Schemes: Methodology And ApplicationsSuurem pilt
  • Formaat: 332 pages
  • Ilmumisaeg: 11-Nov-2020
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789811222559
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This second edition of Nonstandard Finite Difference Models of Differential Equations provides an update on the progress made in both the theory and application of the NSFD methodology during the past two and a half decades. In addition to discussing details related to the determination of the denominator functions and the nonlocal discrete representations of functions of dependent variables, we include many examples illustrating just how this should be done.Of real value to the reader is the inclusion of a chapter listing many exact difference schemes, and a chapter giving NSFD schemes from the research literature. The book emphasizes the critical roles played by the 'principle of dynamic consistency' and the use of sub-equations for the construction of valid NSFD discretizations of differential equations.
Preface vii
0 A Second Edition Why?
1(8)
0.1 Purpose
1(1)
0.2 Ambiguities with the Discretization Process
2(5)
0.3 The Nonstandard Finite Difference Methodology
7(1)
References
8(1)
1 Introduction
9(14)
1.1 Numerical Integration
9(1)
1.2 Standard Finite Difference Modeling Rules
10(2)
1.3 Examples
12(7)
1.3.1 Decay Equation
13(1)
1.3.2 Logistic Equation
14(1)
1.3.3 Harmonic Oscillator
15(1)
1.3.4 Unidirectional Wave Equation
16(2)
1.3.5 Diffusion Equation
18(1)
1.3.6 Burgers' Equation
18(1)
1.4 Critique
19(1)
References
20(3)
2 Numerical Instabilities
23(36)
2.1 Introduction
23(1)
2.2 Decay Equation
24(6)
2.3 Harmonic Oscillator
30(5)
2.4 Logistic Differential Equation
35(11)
2.5 Unidirectional Wave Equation
46(4)
2.6 Burgers' Equation
50(2)
2.7 Summary
52(4)
References
56(3)
3 Nonstandard Finite Difference Schemes
59(22)
3.1 Introduction
59(1)
3.2 Exact Finite Difference Schemes
60(3)
3.3 Examples of Exact Schemes
63(7)
3.4 Nonstandard Modeling Rules
70(3)
3.5 Best Finite Difference Schemes
73(5)
References
78(3)
4 First-Order ODE's
81(22)
4.1 Introduction
81(1)
4.2 A New Finite Difference Scheme
82(3)
4.3 Examples
85(8)
4.3.1 Decay Equation
85(1)
4.3.2 Logistic Equation
86(2)
4.3.3 ODE with Three Fixed-Points
88(5)
4.4 Nonstandard Schemes
93(6)
4.4.1 Logistic Equation
94(1)
4.4.2 ODE with Three Fixed-Points
94(5)
4.5 Discussion
99(2)
References
101(2)
5 Second-Order, Nonlinear Oscillator Equations
103(22)
5.1 Introduction
103(2)
5.2 Mathematical Preliminaries
105(1)
5.3 Conservative Oscillators
106(7)
5.4 Limit-Cycle Oscillators
113(5)
5.5 General Oscillator Equations
118(1)
5.6 Response of a Linear System
119(2)
References
121(4)
6 Two First-Order, Coupled Ordinary Differential Equations
125(18)
6.1 Introduction
125(1)
6.2 Background
126(2)
6.3 Exact Scheme for Linear Ordinary Differential Equations
128(2)
6.4 Nonlinear Equations
130(1)
6.5 Examples
131(10)
6.5.1 Harmonic Oscillator
131(1)
6.5.2 Damped Harmonic Oscillator
132(2)
6.5.3 Duffing Oscillator
134(1)
6.5.4 X + x + x2 = 0
135(1)
6.5.5 Van der Pol Oscillator
136(2)
6.5.6 Lewis Oscillator
138(1)
6.5.7 General Class of Nonlinear Oscillators
139(1)
6.5.8 Batch Fermentation Processes
140(1)
6.6 Summary
141(1)
References
141(2)
7 Partial Differential Equations
143(24)
7.1 Introduction
143(1)
7.2 Wave Equations
144(6)
7.2.1 Ut + ux = 0
144(1)
7.2.2 Ut - ux - 0
145(1)
7.2.3 Utt - uxx = 0
146(1)
7.2.4 Ut + ux = u(1 - u)
147(1)
7.2.5 Uk + ux = buxx
147(3)
7.3 Diffusion Equations
150(7)
7.3.1 Ut = auxx + bu
150(1)
7.3.2 Ut = uuxx
151(3)
7.3.3 Ut = uuxx + λu(1 - u)
154(1)
7.3.4 Ut = uxx + λu(1 - u)
155(2)
7.4 Burgers' Type Equations
157(5)
7.4.1 Ut + uux = 0
158(1)
7.4.2 Ut + uux = λu(1 - u)
159(1)
7.4.3 Ut + uux --- uuxx
160(2)
7.5 Discussion
162(1)
References
163(4)
8 Schrodinger Differential Equations
167(22)
8.1 Introduction
167(1)
8.2 Schrodinger Ordinary Differential Equations
168(4)
8.2.1 Numerov Method
168(1)
8.2.2 Mickens-Ramadhani Scheme
169(2)
8.2.3 Combined Numerov-Mickens Scheme
171(1)
8.3 Schrodinger Partial Differential Equations
172(12)
8.3.1 Ut = iuxx
172(9)
8.3.2 Ut = i[ uxx + f(x)u]
181(2)
8.3.3 Nonlinear, Cubic Schrodinger Equation
183(1)
References
184(5)
9 The NSFD Methodology
189(14)
9.1 Introduction
189(1)
9.2 The Modeling Process
190(1)
9.3 Intrinsic Time and Space Scales
191(3)
9.4 Dynamical Consistency
194(1)
9.5 Numerical Instabilities
195(1)
9.6 Denominator Functions
196(1)
9.7 Nonlocal Discretization of Functions
197(1)
9.8 Method of Sub-Equations
198(1)
9.9 Constructing NSFD Schemes
199(1)
9.10 Final Comments
199(1)
References
200(3)
10 Some Exact Finite Difference Schemes
203(20)
10.1 Introduction
203(1)
10.2 General, Linear, Homogeneous, First-Order ODE
203(2)
10.3 Several Important Exact Schemes
205(5)
10.3.1 Decay Equation
205(1)
10.3.2 Harmonic Oscillator
205(1)
10.3.3 Logistic Equation
205(1)
10.3.4 Quadratic Decay Equation
206(1)
10.3.5 Nonlinear Equation
206(1)
10.3.6 Cubic Decay Equation
206(1)
10.3.7 Linear Velocity Force Equation
206(1)
10.3.8 Damped Harmonic Oscillator
206(1)
10.3.9 Unidirectional Wave Equations
207(1)
10.3.10 Full Wave Equation
207(1)
10.3.11 Nonlinear, Fisher-Type Unidirection Wave Equation
207(1)
10.3.12 Unidirectional, Spherical Wave Equation
208(1)
10.3.13 Steady-State Wave Equation with Spherical Symmetry
208(1)
10.3.14 Wave Equation having Spherical Symmetry
208(1)
10.3.15 Two-Dimensional, Linear Advection Equation
209(1)
10.3.16 Two-Dimensional, Nonlinear (Logistic) Advection Equation
209(1)
10.4 Two Coupled, Linear ODE's with Constant Coefficients
210(1)
10.5 Jacobi Cosine and Sine Functions
211(2)
10.6 Cauchy-Euler Equation
213(1)
10.7 Michaelis-Menten Equation
214(2)
10.8 Weierstrass Elliptic Function
216(2)
10.9 Modified Lotka-Volterra Equations
218(1)
10.10 Comments
219(1)
References
220(3)
11 Applications and Related Topics
223(74)
11.1 Introduction
223(1)
11.2 Stellar Structures
223(4)
References
227(1)
11.3 The x - y - z Model
228(3)
References
231(1)
11.4 Mickens' Modified Newton's Law of Cooling
231(4)
References
235(1)
11.5 NSFD Schemes for dx/dt = -λxα
236(1)
Reference
237(1)
11.6 Exact Scheme for Linear ODE's with Constant Coefficients
237(3)
References
240(1)
11.7 Discrete 1-Dim Hamiltonian Systems
240(6)
11.7.1 Discrete Hamiltonian Construction
243(1)
11.7.2 Discrete Equations of Motion for Eq. (11.7.32)
244(1)
11.7.3 Non-Polynomial Potential Energy
245(1)
11.7.4 Two Interesting Results
246(1)
References
246(1)
11.8 Cube Root Oscillators
247(4)
11.8.1 Cube Root Oscillator
247(2)
11.8.2 Inverse Cube-Root Oscillator
249(2)
References
251(1)
11.9 Alternative Methodologies for Constructing Discrete-Time Population Models
251(6)
11.9.1 Comments
251(1)
11.9.2 Modified Anderson-May Models
251(3)
11.9.3 Discrete Exponentialization
254(3)
References
257(1)
11.10 Interacting Populations with Conservation Laws
257(14)
11.10.0 Comments
257(1)
11.10.1 Conservation Laws
258(2)
11.10.2 Chemostate Model
260(1)
11.10.3 SIR Model
261(1)
11.10.4 SEIR Model with Net Birthrate
262(1)
11.10.5 Criss-Cross Model
263(2)
11.10.6 Brauer-van den Driessche SIR Model
265(1)
11.10.7 Spatial Spread of Rabies
266(2)
11.10.8 Fisher Equation
268(3)
References
271(1)
11.11 Black-Scholes Equations
271(2)
References
273(1)
11.12 Time-Independent Schrodinger Equations
274(3)
References
277(1)
11.13 Linear, Damped Wave Equation
277(2)
References
279(1)
11.14 NSFD Constructions for Burgers and Burgers-Fisher PDE's
280(4)
11.14.1 Burgers Equations: ut + uux = uxx
280(2)
11.14.2 Burgers-Fisher Equations: ut + uux = uxx + u(1 - u)
282(2)
References
284(1)
11.15 Cross-Diffusion
284(4)
References
288(1)
11.16 Delay Differential Equations
289(2)
References
291(1)
11.17 Fractional Differential Equations
292(2)
References
294(1)
11.18 Summary
295(2)
Appendix A Difference Equations
297(4)
A.1 Linear Equations
297(1)
A.2 Riccati Equations
298(1)
A.3 Separation-of-Variables
299(1)
Reference
300(1)
Appendix B Linear Stability Analysis
301(4)
B.1 Ordinary Differential Equations
301(1)
B.2 Ordinary Difference Equations
302(1)
References
303(2)
Appendix C Discrete WKB Method
305(2)
References
306(1)
Bibliography 307(4)
Index 311
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