Muutke küpsiste eelistusi

Volterra Integral Equations: An Introduction to Theory and Applications [Kõva köide]

4.00/5 (2 hinnangut Goodreads-ist)
(Hong Kong Baptist University)
Teised raamatud teemal:
Volterra Integral Equations: An Introduction to Theory and ApplicationsSuurem pilt
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781107098725.html
Märksõnad:
This book offers a comprehensive introduction to the theory of linear and nonlinear Volterra integral equations (VIEs), ranging from Volterra's fundamental contributions and the resulting classical theory to more recent developments that include Volterra functional integral equations with various kinds of delays, VIEs with highly oscillatory kernels, and VIEs with non-compact operators. It will act as a 'stepping stone' to the literature on the advanced theory of VIEs, bringing the reader to the current state of the art in the theory. Each chapter contains a large number of exercises, extending from routine problems illustrating or complementing the theory to challenging open research problems. The increasingly important role of VIEs in the mathematical modelling of phenomena where memory effects play a key role is illustrated with some 30 concrete examples, and the notes at the end of each chapter feature complementary references as a guide to further reading.

This book offers a comprehensive introduction to the theory of linear and nonlinear Volterra integral equations (VIEs), from Volterra's fundamental contributions and resulting classical theory to more recent developments. It includes Volterra functional integral equations with various kinds of delays, VIEs with highly oscillatory kernels, and VIEs with non-compact operators.

Arvustused

'One of the strengths of the book is the attention given to the history of the subject and the large number of references to older literature. At the same time the author succeeds in giving an introduction to the current state of the art in the theory of Volterra integral equations and the notes at the end of each chapter are very helpful in this respect as they point the reader to the relevant papers.' Gustaf Gripenberg, Zentralblatt MATH 'In summary, the book is a very clear and thorough presentation of the theory. It is an excellent compendium and text with a simple exposition and few lengthy proofs. It is very thoroughly referenced (with 39 pages of references), with most of the references explained or commented on in the text. The audience will include students for a specialized course and researchers. I highly recommend the book.' John A. DeSanto, Mathematical Reviews

Muu info

This book offers a comprehensive introduction to the theory of linear and nonlinear Volterra integral equations. It includes applications and an extensive bibliography.
Preface xi
1 Linear Volterra Integral Equations
1(56)
1.1 Introduction
1(1)
1.2 Second-Kind VIEs with Smooth Kernels
2(16)
1.2.1 Existence and Uniqueness of Solutions
2(9)
1.2.2 Linear VIEs with Convolution Kernels
11(2)
1.2.3 Adjoint VIEs
13(1)
1.2.4 Systems of Linear VIEs
14(3)
1.2.5 Comparison Theorems
17(1)
1.3 Second-Kind VIEs with Weakly Singular Kernels
18(11)
1.3.1 The Mittag-Leffler Function and Weakly Singular VIEs
19(2)
1.3.2 Existence and Uniqueness of Solutions
21(5)
1.3.3 Other Types of Singular Kernels
26(1)
1.3.4 Comparison Theorems
27(2)
1.4 VIEs of the First Kind: Smooth Kernels
29(7)
1.4.1 Existence and Uniqueness of Solutions
29(4)
1.4.2 First-Kind VIEs are Ill-Posed Problems
33(3)
1.5 VIEs of the First Kind with Weakly Singular Kernels
36(9)
1.5.1 Abel's Integral Equation
36(3)
1.5.2 General First-Kind VIEs: Volterra's Nota II of 1896
39(3)
1.5.3 Other Types of Kernel Singularities
42(3)
1.6 VIEs of the Third Kind (I)
45(3)
1.7 Exercises and Research Problems
48(3)
1.8 Notes
51(6)
2 Regularity of Solutions
57(46)
2.1 VIEs of the Second Kind
57(10)
2.1.1 VIEs with Smooth Kernels
57(1)
2.1.2 VIEs with Weakly Singular Kernels
58(5)
2.1.3 Bounded but Non-Smooth Kernels
63(1)
2.1.4 Kernels with Boundary Singularities
64(1)
2.1.5 Kernel Singularities of the Form (t2 - s2)-1/2
65(2)
2.2 VIEs of the First Kind
67(6)
2.2.1 VIEs with Smooth Kernels
67(1)
2.2.2 VIEs with Weakly Singular Kernels
68(2)
2.2.3 Other Types of Kernel Singularities
70(1)
2.2.4 The Generalised Abel Integral Equation
71(2)
2.3 Linear Volterra Functional Integral Equations
73(22)
2.3.1 Introduction
73(3)
2.3.2 Second-Kind VFIEs with Vanishing Delays
76(5)
2.3.3 First-Kind VFIEs with Vanishing Delays
81(3)
2.3.4 Second-Kind VFIEs with Non-Vanishing Delays
84(6)
2.3.5 First-Kind VFIEs with Non-Vanishing Delays
90(5)
2.4 Exercises and Research Problems
95(5)
2.5 Notes
100(3)
3 Non-Linear Volterra Integral Equations
103(72)
3.1 Non-Linear Second-Kind VIEs
103(18)
3.1.1 General Existence Theorems
103(8)
3.1.2 VIEs of Hammerstein Type
111(1)
3.1.3 Maximal Solutions and a Comparison Theorem
112(4)
3.1.4 VIEs with Multiple Solutions
116(3)
3.1.5 Regularity Results
119(2)
3.2 Solutions with Finite-Time Blow-Up
121(13)
3.2.1 Introduction
121(6)
3.2.2 Blow-Up Theory for General Hammerstein VIEs
127(5)
3.2.3 Multiple Solutions -- Revisited
132(2)
3.3 Quenching of Solutions
134(10)
3.3.1 Quenching in Differential Equations
134(4)
3.3.2 Quenching in VIEs of Hammerstein Type
138(6)
3.4 Other Types of Non-Linear VIEs
144(13)
3.4.1 Non-Standard Second-Kind VIEs
145(1)
3.4.2 VIEs of Auto-Convolution Type
146(7)
3.4.3 Implicit VIEs
153(4)
3.5 Non-Linear First-Kind VIEs
157(2)
3.6 Non-Linear Volterra Functional Integral Equations
159(5)
3.6.1 State-Independent Delays
159(2)
3.6.2 Blow-Up Theory for Non-Linear VFIEs
161(2)
3.6.3 State-Dependent Delays
163(1)
3.7 Exercises and Research Problems
164(5)
3.8 Notes
169(6)
4 Volterra Integral Equations with Highly Oscillatory Kernels
175(23)
4.1 Introduction
175(1)
4.2 VIEs of the Second Kind
176(8)
4.2.1 Smooth Kernels
176(3)
4.2.2 VIEs with Weakly Singular Kernels
179(4)
4.2.3 Comparison with Highly Oscillatory Fredholm Integral Equations
183(1)
4.3 VIEs of the First Kind
184(9)
4.3.1 VIEs with Smooth Kernels
184(5)
4.3.2 VIEs with Weakly Singular Kernels
189(3)
4.3.3 Other Types of Oscillators
192(1)
4.4 General Oscillators eiωg(t,s)
193(1)
4.5 Exercises and Research Problems
193(3)
4.6 Notes
196(2)
5 Singularly Perturbed and Integral-Algebraic Volterra Equations
198(25)
5.1 Singularly Perturbed VIEs
198(9)
5.1.1 Examples
199(3)
5.1.2 VIEs with Smooth Kernels
202(3)
5.1.3 VIEs with Weakly Singular Kernels
205(2)
5.1.4 Non-Linear VIEs
207(1)
5.2 Integral-Algebraic Equations with Smooth Kernels
207(10)
5.2.1 Introduction
207(3)
5.2.2 ν-smoothing Volterra Integral Operators
210(1)
5.2.3 The Tractability Index of a System of Linear IAEs
211(4)
5.2.4 The Decoupling of Index-1 IAEs
215(2)
5.3 Open Problems
217(1)
5.4 Exercises and Research Problems
218(2)
5.5 Notes
220(3)
6 Qualitative Theory of Volterra Integral Equations
223(18)
6.1 Introduction
223(1)
6.2 Asymptotic Properties of Resolvent Kernels
224(5)
6.2.1 VIEs with Convolution Kernels
224(5)
6.2.2 VIEs with General Kernels
229(1)
6.3 Asymptotic Behaviour of Solutions
229(2)
6.3.1 VIEs with Convolution Kernels
229(2)
6.4 VIEs of Hammerstein Form
231(5)
6.4.1 Non-Linear Perturbations of Linear VIEs
231(2)
6.4.2 Hammerstein VIEs with Convolution Kernels
233(3)
6.5 Exercises and Research Problems
236(2)
6.6 Notes
238(3)
7 Cordial Volterra Integral Equations
241(38)
7.1 Cordial Volterra Integral Operators
241(11)
7.1.1 Basic Properties of Cordial Volterra Integral Operators
242(7)
7.1.2 The Spectrum of a Cordial Volterra Integral Operator
249(3)
7.2 Linear Cordial Volterra Integral Equations
252(14)
7.2.1 Cordial VIEs of the Second Kind
252(3)
7.2.2 Cordial VIEs of the First Kind
255(6)
7.2.3 VIEs of the Third Kind (II)
261(5)
7.3 Non-Linear Cordial VIEs
266(4)
7.4 Cordial VIEs with Highly Oscillatory Kernels
270(4)
7.4.1 The Spectra of Highly Oscillatory Cordial Volterra Operators
270(2)
7.4.2 Second-Kind Cordial VIEs with Highly Oscillatory Kernels
272(2)
7.4.3 Cordial First-Kind VIEs with Highly Oscillatory Kernels
274(1)
7.5 Exercises and Research Problems
274(3)
7.6 Notes
277(2)
8 Volterra Integral Operators on Banach Spaces
279(25)
8.1 Mapping Properties
279(5)
8.1.1 Volterra Integral Operators on C (I)
279(1)
8.1.2 Volterra Integral Operators on Holder and Lp-spaces
280(4)
8.2 Quasi-Nilpotency
284(1)
8.3 Resolvent Kernels and Resolvent Operators
285(3)
8.3.1 Resolvent Kernels for Second-Kind VIEs
285(1)
8.3.2 Resolvent Kernels for First-Kind VIEs
286(2)
8.4 Singular Values of Volterra Integral Operators
288(4)
8.5 Norms of Powers of ν
292(5)
8.5.1 The Basic Volterra Integral Operator ν
292(5)
8.5.2 Volterra Integral Operators with Convolution Kernels
297(1)
8.6 Exercises and Research Problems
297(3)
8.7 Notes
300(4)
9 Applications of Volterra Integral Equations
304(21)
9.1 VIEs of the First Kind
304(3)
9.1.1 Integral Equations of Abel Type
304(1)
9.1.2 General First-Kind VIEs
305(2)
9.2 VIEs of the Second Kind
307(13)
9.2.1 The Renewal Equation
307(1)
9.2.2 Population Growth Models
308(1)
9.2.3 Heat Transfer, Diffusion Models and Shock Wave Problems
309(3)
9.2.4 Blow-Up and Quenching Phenomena
312(3)
9.2.5 American Option Pricing
315(1)
9.2.6 Optimal Control Problems
316(1)
9.2.7 A Brief Review of Further Applications
317(3)
9.3 VIEs of the Third Kind
320(1)
9.4 Systems of Integral-Algebraic VIEs
320(2)
9.5 Notes
322(3)
Appendix A Review of Banach Space Tools
325(19)
A.1 Banach Spaces in the Theory of VIEs
325(5)
A.1.1 The Spaces Cd(I)
325(1)
A.1.2 The Holder Spaces Cd,β(I)
326(1)
A.1.3 The Lebesgue Spaces Lp(O, T)
327(2)
A.1.4 The Sobolev Spaces Wd'P(Ω)
329(1)
A.2 Linear Operators on Banach Spaces
330(9)
A.2.1 Bounded Operators
330(2)
A.2.2 Compact Operators
332(3)
A.2.3 The Spectrum of Bounded Linear Operators
335(3)
A.2.4 Quasi-Nilpotent Operators
338(1)
A.3 Non-Linear Operators on Banach Spaces
339(3)
A.3.1 The Frechet Derivative
339(1)
A.3.2 The Implicit Function Theorem
340(1)
A.3.3 The Fixed-Point Theorems of Banach and Schauder
341(1)
A.4 Notes
342(2)
References 344(39)
Index 383
Hermann Brunner is a Research Professor of Mathematics at the Hong Kong Baptist University, and in 2006 he won the David Borwein Distinguished Career Award of the Canadian Mathematical Society. His previous books include Collocation Methods for Volterra Integral and Related Functional Differential Equations (Cambridge, 2004) and The Numerical Solution of Volterra Equations (1986).