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Topics in Fractional Differential Equations 2012 ed. [Kõva köide]

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  • Formaat: Hardback, 398 pages, kõrgus x laius: 235x155 mm, kaal: 781 g, XIV, 398 p., 1 Hardback
  • Sari: Developments in Mathematics 27
  • Ilmumisaeg: 17-Aug-2012
  • Kirjastus: Springer-Verlag New York Inc.
  • ISBN-10: 1461440351
  • ISBN-13: 9781461440352
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Topics in Fractional Differential Equations 2012 ed.Suurem pilt
  • Formaat: Hardback, 398 pages, kõrgus x laius: 235x155 mm, kaal: 781 g, XIV, 398 p., 1 Hardback
  • Sari: Developments in Mathematics 27
  • Ilmumisaeg: 17-Aug-2012
  • Kirjastus: Springer-Verlag New York Inc.
  • ISBN-10: 1461440351
  • ISBN-13: 9781461440352
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781461440352.html
Märksõnad:
??? Topics in Fractional Differential Equations is devoted to the existence and uniqueness of solutions for various classes of Darboux problems for hyperbolic differential equations or inclusions involving the Caputo fractional derivative. ??Fractional calculus generalizes the integrals and derivatives to non-integer orders. During the last decade, fractional calculus was found to play a fundamental role in the modeling of a considerable number of phenomena; in particular the modeling of memory-dependent and complex media such as porous media. It has emerged as an important tool for the study of dynamical systems where classical methods reveal strong limitations. Some equations present delays which may be finite, infinite, or state-dependent. Others are subject to an impulsive effect. The above problems are studied using the fixed point approach, the method of upper and lower solution, and the Kuratowski measure of noncompactness. This book is addressed to a wide audience of specialists such as mathematicians, engineers, biologists, and physicists. ?

This is devoted to exploration of the existence and uniqueness of solutions for various classes of Darboux problems for hyperbolic differential equations or inclusions involving the Caputo fractional derivative.
1 Introduction
1(10)
2 Preliminary Background
11(14)
2.1 Notations and Definitions
11(1)
2.2 Properties of Partial Fractional Calculus
12(6)
2.3 Properties of Set-Valued Maps
18(3)
2.4 Fixed Point Theorems
21(2)
2.5 Gronwall Lemmas
23(2)
3 Partial Hyperbolic Functional Differential Equations
25(90)
3.1 Introduction
25(1)
3.2 Partial Hyperbolic Differential Equations
25(8)
3.2.1 Introduction
25(1)
3.2.2 Existence of Solutions
26(6)
3.2.3 An Example
32(1)
3.3 Perturbed Partial Differential Equations
33(7)
3.3.1 Introduction
33(1)
3.3.2 Existence of Solutions
33(6)
3.3.3 An Example
39(1)
3.4 Neutral Partial Differential Equations
40(7)
3.4.1 Introduction
40(1)
3.4.2 Existence of Solutions
40(6)
3.4.3 An Example
46(1)
3.5 Discontinuous Partial Differential Equations in Banach Algebras
47(9)
3.5.1 Introduction
47(1)
3.5.2 Existence of Solutions
47(4)
3.5.3 Existence of Extremal Solutions
51(4)
3.5.4 An Example
55(1)
3.6 Upper and Lower Solutions Method for Partial Hyperbolic Differential Equations
56(4)
3.6.1 Introduction
56(1)
3.6.2 Main Result
56(4)
3.7 Partial Functional Differential Equations with Infinite Delay
60(13)
3.7.1 Introduction
60(1)
3.7.2 The Phase Space B
60(2)
3.7.3 Main Results
62(9)
3.7.4 An Example
71(2)
3.8 Partial Hyperbolic Differential Equations with State-Dependent Delay
73(25)
3.8.1 Introduction
73(1)
3.8.2 Existence of Solutions for Finite Delay
74(9)
3.8.3 Existence of Solutions for Infinite Delay
83(11)
3.8.4 Examples
94(4)
3.9 Global Uniqueness Results for Partial Hyperbolic Differential Equations
98(16)
3.9.1 Introduction
98(2)
3.9.2 Global Result for Finite Delay
100(5)
3.9.3 Global Result for Infinite Delay
105(7)
3.9.4 Examples
112(2)
3.10 Notes and Remarks
114(1)
4 Partial Hyperbolic Functional Differential Inclusions
115(56)
4.1 Introduction
115(1)
4.2 Partial Hyperbolic Differential Inclusions
115(10)
4.2.1 Introduction
115(1)
4.2.2 The Convex Case
116(5)
4.2.3 The Nonconvex Case
121(3)
4.2.4 An Example
124(1)
4.3 Existence Results for Partial Hyperbolic Differential Inclusions
125(18)
4.3.1 Introduction
125(1)
4.3.2 Existence of Solutions
125(5)
4.3.3 Qualitative Properties and Topological Structure of the Solution Set
130(13)
4.4 Upper and Lower Solutions Method for Partial Differential Inclusions
143(7)
4.4.1 Introduction
143(1)
4.4.2 Main Result
144(6)
4.5 Partial Functional Differential Inclusions with Infinite Delay
150(9)
4.5.1 Introduction
150(1)
4.5.2 Main Results
150(9)
4.5.3 An Example
159(1)
4.6 Fractional Order Riemann-Liouville Integral Inclusions with two Independent Variables and Multiple Time Delay
159(10)
4.6.1 Introduction
159(1)
4.6.2 Existence of Solutions
160(8)
4.6.3 An Example
168(1)
4.7 Notes and Remarks
169(2)
5 Impulsive Partial Hyperbolic Functional Differential Equations
171(80)
5.1 Introduction
171(1)
5.2 Impulsive Partial Hyperbolic Functional Differential Equations
171(8)
5.2.1 Introduction
171(1)
5.2.2 Existence of Solutions
172(6)
5.2.3 An Example
178(1)
5.3 Impulsive Partial Hyperbolic Differential Equations at Variable Times
179(9)
5.3.1 Introduction
179(1)
5.3.2 Existence of Solutions
179(7)
5.3.3 Nonlocal Impulsive Partial Differential Equations
186(1)
5.3.4 An Example
187(1)
5.4 Impulsive Discontinuous Partial Hyperbolic Differential Equations on Banach Algebras
188(11)
5.4.1 Introduction
188(1)
5.4.2 Existence of Solutions
189(4)
5.4.3 Existence of Extremal Solutions
193(5)
5.4.4 An Example
198(1)
5.5 Impulsive Partial Hyperbolic Differential Equations with Variable Times and Infinite Delay
199(12)
5.5.1 Introduction
199(1)
5.5.2 Main Result
200(9)
5.5.3 An Example
209(2)
5.6 Impulsive Partial Hyperbolic Functional Differential Equations of Fractional Order with State-Dependent Delay
211(11)
5.6.1 Introduction
211(1)
5.6.2 Impulsive Partial Differential Equations with Finite Delay
212(3)
5.6.3 Impulsive Partial Differential Equations with Infinite Delay
215(5)
5.6.4 Examples
220(2)
5.7 Impulsive Partial Hyperbolic Functional Differential Equations with Variable Times and State-Dependent Delay
222(21)
5.7.1 Introduction
222(1)
5.7.2 Impulsive Partial Differential Equations with Finite Delay
223(8)
5.7.3 Impulsive Partial Differential Equations with Infinite Delay
231(10)
5.7.4 Examples
241(2)
5.8 Upper and Lower Solutions Method for Impulsive Partial Hyperbolic Differential Equations
243(6)
5.8.1 Introduction
243(1)
5.8.2 Main Result
244(5)
5.9 Notes and Remarks
249(2)
6 Impulsive Partial Hyperbolic Functional Differential Inclusions
251(36)
6.1 Introduction
251(1)
6.2 Impulsive Partial Hyperbolic Differential Inclusions
251(14)
6.2.1 Introduction
251(1)
6.2.2 The Convex Case
252(7)
6.2.3 The Nonconvex Case
259(4)
6.2.4 An Example
263(2)
6.3 Impulsive Partial Hyperbolic Differential Inclusions with Variable Times
265(11)
6.3.1 Introduction
265(1)
6.3.2 Existence of Solutions
265(10)
6.3.3 An Example
275(1)
6.4 The Method of Upper and Lower Solutions for Partial Hyperbolic Fractional Order Differential Inclusions with Impulses
276(9)
6.4.1 Introduction
276(1)
6.4.2 Main Result
277(8)
6.5 Notes and Remarks
285(2)
7 Implicit Partial Hyperbolic Functional Differential Equations
287(54)
7.1 Introduction
287(1)
7.2 Darboux Problem for Implicit Differential Equations
287(8)
7.2.1 Introduction
287(1)
7.2.2 Riemann-Liouville and Caputo Partial Fractional Derivatives
288(1)
7.2.3 Existence of Solutions
289(5)
7.2.4 An Example
294(1)
7.3 A Global Uniqueness Result for Implicit Differential Equations
295(4)
7.3.1 Introduction
295(1)
7.3.2 Existence of Solutions
295(3)
7.3.3 An Example
298(1)
7.4 Functional Implicit Hyperbolic Differential Equations with Delay
299(17)
7.4.1 Introduction
299(1)
7.4.2 Existence Results with Finite Delay
300(5)
7.4.3 Existence Results for Infinite Delay
305(3)
7.4.4 Existence Results with State-Dependent Delay
308(4)
7.4.5 Examples
312(4)
7.5 Darboux Problem for Implicit Impulsive Partial Hyperbolic Differential Equations
316(10)
7.5.1 Introduction
316(1)
7.5.2 Existence of Solutions
316(9)
7.5.3 An Example
325(1)
7.6 Implicit Impulsive Partial Hyperbolic Differential Equations with State-Dependent Delay
326(13)
7.6.1 Introduction
326(1)
7.6.2 Existence Results with Finite Delay
327(5)
7.6.3 Existence Results with Infinite Delay
332(5)
7.6.4 Examples
337(2)
7.7 Notes and Remarks
339(2)
8 Fractional Order Riemann-Liouville Integral Equations
341(42)
8.1 Introduction
341(1)
8.2 Uniqueness Results for Fredholm-Type Fractional Order Riemann-Liouville Integral Equations
341(12)
8.2.1 Introduction
341(1)
8.2.2 Main Results
342(11)
8.3 Fractional Order Riemann-Liouville Integral Equations with Multiple Time Delay
353(6)
8.3.1 Introduction
353(1)
8.3.2 Existence of Solutions
353(5)
8.3.3 Examples
358(1)
8.4 Nonlinear Quadratic Volterra Riemann-Liouville Integral Equations of Fractional Order
359(6)
8.4.1 Introduction
359(1)
8.4.2 Existence of Solutions
359(5)
8.4.3 An Example
364(1)
8.5 Asymptotic Stability of Solutions of Nonlinear Quadratic Volterra Integral Equations of Fractional Order
365(9)
8.5.1 Introduction
365(1)
8.5.2 Main Results
366(7)
8.5.3 An Example
373(1)
8.6 Attractivity Results for Nonlinear Fractional Order Riemann-Liouville Integral Equations in Banach Algebras
374(8)
8.6.1 Introduction
374(1)
8.6.2 Main Results
375(6)
8.6.3 An Example
381(1)
8.7 Notes and Remarks
382(1)
References 383(12)
Index 395