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Modelling with Ordinary Differential Equations: A Comprehensive Approach [Kõva köide]

(University of Wurzburg, Germany)
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Modelling with Ordinary Differential Equations: A Comprehensive ApproachSuurem pilt
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Püsilink: https://www.kriso.ee/db/9780815392613.html
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Modelling with Ordinary Differential Equations: A Comprehensive Approach aims to provide a broad and self-contained introduction to the mathematical tools necessary to investigate and apply ODE models. The book starts by establishing the existence of solutions in various settings and analysing their stability properties. The next step is to illustrate modelling issues arising in the calculus of variation and optimal control theory that are of interest in many applications. This discussion is continued with an introduction to inverse problems governed by ODE models and to differential games.

The book is completed with an illustration of stochastic differential equations and the development of neural networks to solve ODE systems. Many numerical methods are presented to solve the classes of problems discussed in this book.

Features:











Provides insight into rigorous mathematical issues concerning various topics, while discussing many different models of interest in different disciplines (biology, chemistry, economics, medicine, physics, social sciences, etc.)





Suitable for undergraduate and graduate students and as an introduction for researchers in engineering and the sciences





Accompanied by codes which allow the reader to apply the numerical methods discussed in this book in those cases where analytical solutions are not available

Arvustused

"Expertly written, organized and presented, Modelling with Ordinary Differential Equations: A Comprehensive Approach is an ideal textbook for college and university Numerical Analysis & Scientific Computing curriculums. [ . . . ] unreservedly recommended as a critically important addition to academic library collections"

Midwest Book Review

Preface xi
Author xv
1 Introduction
1(12)
1.1 Ordinary differential equations
1(3)
1.2 The modelling process
4(9)
2 Elementary solution methods for simple ODEs
13(12)
2.1 Simple ODEs
13(3)
2.1.1 Simple ODE of
1. type, y' = f(x)
13(1)
2.1.2 Simple ODE of
2. type, y' = f(y)
14(1)
2.1.3 Simple ODE of
3. type, y' = f{x) g(y)
14(1)
2.1.4 Simple ODE of
4. type, y' = f(ax + by + d)
15(1)
2.1.5 Simple ODE of
5. type, y' = f(y/x)
15(1)
2.2 Linear ODEs
16(1)
2.3 Method of variation of the constants
16(2)
2.4 Bernoulli's differential equation
18(1)
2.5 Riccati's differential equation
19(1)
2.6 Exact differential equations
20(5)
3 Theory of ordinary differential equations
25(14)
3.1 The Cauehy problem and existence of solutions
25(1)
3.2 Euler's method
26(4)
3.3 Uniqueness of solutions
30(4)
3.4 The Caratheodory theorem
34(5)
4 Systems of ordinary differential equations
39(38)
4.1 Systems of first-order ODEs
39(2)
4.2 Dependence of solutions on the initial conditions
41(7)
4.3 Systems of linear ODEs
48(1)
4.4 Systems of linear homogeneous ODEs
49(3)
4.5 The d'Alembert reduction method
52(3)
4.6 Nonhomogeneous linear systems
55(2)
4.7 Linear systems with constant coefficients
57(8)
4.8 The exponential matrix
65(6)
4.9 Linear systems with periodic coefficients
71(6)
5 Ordinary differential equations of order n
77(12)
5.1 Ordinary differential equations of order n in normal form
77(1)
5.2 Linear differential equations of order n
78(2)
5.3 The reduction method of d'Alembert
80(1)
5.4 Linear ODEs of order n with constant coefficients
81(2)
5.5 Nonhomogeneous ODEs of order n
83(2)
5.6 Oscillatory solutions
85(4)
6 Stability of ODE systems
89(50)
6.1 Local stability of ODE systems
89(5)
6.2 Stability of linear ODE systems
94(2)
6.3 Stability of nonlinear ODE systems
96(1)
6.4 Remarks on the stability of periodic ODE problems
97(2)
6.5 Autonomous systems in the plane
99(7)
6.6 The Lyapunov method
106(3)
6.7 Limit points and limit cycles
109(5)
6.8 Population dynamics
114(11)
6.9 The Lorenz model
125(5)
6.10 Synchronisation
130(9)
7 Boundary and eigenvalue problems
139(10)
7.1 Linear boundary-value problems
139(3)
7.2 Sturm-Liouville eigenvalue problems
142(7)
8 Numerical solution of ODE problems
149(40)
8.1 One-step methods
149(17)
8.2 Motion in special relativity
166(8)
8.3 The Kepler problem
174(5)
8.4 Approximation of Sturm-Liouville problems
179(6)
8.5 The shape of a drop on a flat surface
185(4)
9 ODEs and the calculus of variations
189(36)
9.1 Existence of a minimum
189(5)
9.2 Optimality conditions
194(7)
9.2.1 First-order optimality conditions
197(1)
9.2.2 Second-order optimality conditions
198(3)
9.3 The Euler-Lagrange equations
201(11)
9.3.1 Direct and indirect numerical methods
206(1)
9.3.2 Unilateral constraints
207(1)
9.3.3 Free boundaries
208(1)
9.3.4 Equality constraints
209(3)
9.4 The Legendre condition
212(4)
9.5 The Weierstrass-Erdmann conditions
216(4)
9.6 Optimality conditions in Hamiltonian form
220(5)
10 Optimal control of ODE models
225(50)
10.1 Formulation of ODE optimal control problems
225(2)
10.2 Existence of optimal controls
227(5)
10.3 Optimality conditions
232(7)
10.4 Optimality conditions in Hamiltonian form
239(4)
10.5 The Pontryagin's maximum principle
243(15)
10.6 Numerical solution of ODE optimal control problems
258(5)
10.7 A class of bilinear optimal control problems
263(7)
10.8 Linear-quadratic feedback-control problems
270(5)
11 Inverse problems with ODE models
275(14)
11.1 Inverse problems with linear models
275(4)
11.2 Tikhonov regularisation
279(4)
11.3 Inverse problems with nonlinear models
283(1)
11.4 Parameter identification with a tumor growth model
284(5)
12 Differential games
289(20)
12.1 Finite-dimensional game problems
289(6)
12.2 Infinite-dimensional differential games
295(3)
12.3 Linear-quadratic differential Nash games
298(6)
12.4 Pursuit-evasion games
304(5)
13 Stochastic differential equations
309(22)
13.1 Random variables and stochastic: processes
309(5)
13.2 Stochastic differential equations
314(7)
13.3 The Euler-Maruyama method
321(3)
13.4 Stability
324(3)
13.5 Piecewise deterministic processes
327(4)
14 Neural networks and ODE problems
331(26)
14.1 The perceptron and a learning scheme
331(8)
14.2 Approximation properties of neural networks
339(6)
14.3 The neural network solution of ODE problems
345(5)
14.4 Parameter identification with neural networks
350(4)
14.5 Deep neural networks
354(3)
Appendix Results of analysis
357(12)
A.1 Some function spaces
357(5)
A.1.1 Spaces of continuous functions
357(2)
A.1.2 Spaces of integrable functions
359(1)
A.1.3 Sobolev spaces
359(3)
A.2 The Arzela-Ascoli theorem
362(2)
A.3 The Gronwall inequality
364(1)
A.4 The implicit function theorem
364(2)
A.5 The Lebesgue dominated convergence theorem
366(3)
Bibliography 369(12)
Index 381
Alfio Borzì, born 1965 in Catania (Italy), is the professor and chair of Scientific Computing at the Institute for Mathematics of the University of Würzburg, Germany. He studied Mathematics and Physics in Catania and Trieste where he received his PhD in Mathematics from Scuola Internazionale Superiore di Studi Avanzati (SISSA).

He served as Research Officer at the University of Oxford (UK) and as assistant professor at the University of Graz (Austria) where he completed his Habilitation and was appointed as Associate Professor. Since 2011 he has been Professor of Scientific Computing at University of Würzburg.

Alfio Borzi is the author of 3 mathematics books and numerous articles in journals. The main topics of his research and teaching activities are modelling and numerical analysis, optimal control theory and scientific computing. He is member of the editorial board for the SIAM Journal on Scientific Computing and for SIAM Review.