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E-raamat: Advanced Fractional Differential and Integral Equations

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  • Formaat: 388 pages
  • Ilmumisaeg: 01-Feb-2018
  • Kirjastus: Nova Science Publishers Inc
  • ISBN-13: 9781634631327
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Advanced Fractional Differential and Integral EquationsSuurem pilt
  • Formaat: 388 pages
  • Ilmumisaeg: 01-Feb-2018
  • Kirjastus: Nova Science Publishers Inc
  • ISBN-13: 9781634631327
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Fractional calculus deals with extensions of derivatives and integrals to non-integer orders. It represents a powerful tool in applied mathematics to study a myriad of problems from different fields of science and engineering, with many break-through results found in mathematical physics, finance, hydrology, biophysics, thermodynamics, control theory, statistical mechanics, astrophysics, cosmology and bioengineering. This book is devoted to the existence and uniqueness of solutions and some Ulam's type stability concepts for various classes of functional differential and integral equations of fractional order. Some equations present delay which may be finite, infinite or state-dependent. Others are subject to multiple time delay effect. The tools used include classical fixed point theorems. Other tools are based on the measure of non-compactness together with appropriates fixed point theorems. Each chapter concludes with a section devoted to notes and bibliographical remarks and all the presented resultsare illustrated by examples. The content of the book is new and complements the existing literature in Fractional Calculus. It is useful for researchers and graduate students for research, seminars and advanced graduate courses, in pure and applied mathematics, engineering, biology and other applied sciences.
Preface xv
Introduction xvii
1 Preliminary Background
1(20)
1.1 Notations and Definitions
1(1)
1.2 Preliminaries from General Topology
2(3)
1.2.1 The Weak Topology σ(X,X*)
3(1)
1.2.2 The Space C([ 0,T],Ew)
4(1)
1.3 Frechet Spaces
5(1)
1.4 Properties of Fractional Calculus
5(10)
1.4.1 Preliminary Lemmas
7(1)
1.4.2 Riemann-Liouville and Caputo Partial Fractional Derivatives
8(1)
1.4.3 Fractional Pettis Integral
9(4)
1.4.4 The Weak Derivative of Fractional-Order
13(1)
1.4.5 Riemann-Liouville Stieltjes Fractional Integral
14(1)
1.5 Multi-valued Analysis
15(2)
1.6 Concept of the Stability
17(1)
1.7 Fixed Point Theorems
18(2)
1.8 Auxiliary Lemmas
20(1)
2 Nonlinear Differential Equations of Fractional Order
21(28)
2.1 Introduction
21(1)
2.2 Local Attractivity for Delay Partial Differential Equations
21(6)
2.2.1 Introduction
21(1)
2.2.2 Main Results
21(5)
2.2.3 An Example
26(1)
2.3 Existence and Attractivity for the Darboux Problem of Neutral Differential Equations
27(6)
2.3.1 Introduction
27(1)
2.3.2 Main Results
27(5)
2.3.3 An Example
32(1)
2.4 Asymptotic Stability in Nonlinear Delay Differential Equations
33(8)
2.4.1 Introduction
33(1)
2.4.2 Main Results
34(5)
2.4.3 An Example
39(2)
2.5 Global Asymptotic Stability for Nonlinear Multi-delay Differential Equations
41(6)
2.5.1 Introduction
41(1)
2.5.2 Main Results
42(4)
2.5.3 An Example
46(1)
2.6 Notes and Remarks
47(2)
3 Fractional Order Riemann-Liouville Integral Equations
49(22)
3.1 Introduction
49(1)
3.2 Partial Integral Equations
49(6)
3.2.1 Introduction
49(1)
3.2.2 Main Results
50(4)
3.2.3 Examples
54(1)
3.3 Qualitative Theory for Riemann-Liouville Integral Equations in Two Independent Variables
55(9)
3.3.1 Introduction
55(1)
3.3.2 Main Results
56(5)
3.3.3 An Example
61(3)
3.4 Attractivity Results for Functional Partial Integral Equations
64(6)
3.4.1 Introduction
64(1)
3.4.2 Main Results
64(5)
3.4.3 An Example
69(1)
3.5 Notes and Remarks
70(1)
4 Fractional Order Riemann-Liouville Volterra-Stieltjes Quadratic Integral Equations
71(44)
4.1 Introduction
71(1)
4.2 Existence and Stability Results for Nonlinear Riemann-Liouville Volterra-Stieltjes Quadratic Integral Equations
71(13)
4.2.1 Introduction
71(1)
4.2.2 Main Results
72(4)
4.2.3 Global Existence and Stability of Solutions
76(6)
4.2.4 Examples
82(2)
4.3 Asymptotic Behavior of Solutions of Nonlinear Riemann-Liouville Volterra-Stieltjes Quadratic Integral Equations
84(8)
4.3.1 Introduction
84(1)
4.3.2 Main Results
85(6)
4.3.3 An Example
91(1)
4.4 Global Stability Results for Nonlinear Partial Riemann-Liouville Volterra-Stieltjes Functional Integral Equations
92(6)
4.4.1 Introduction
92(1)
4.4.2 Main Results
93(4)
4.4.3 An Example
97(1)
4.5 On the Existence and Global Asymptotic Stability of Solutions of Riemann-Liouville Volterra-Stieltjes Integral Equations
98(7)
4.5.1 Introduction
98(1)
4.5.2 Main Results
99(5)
4.5.3 An Example
104(1)
4.6 Global Attractivity of Solutions for Nonlinear Riemann-Liouville Volterra-Stieltjes Partial Integral Equations
105(9)
4.6.1 Introduction
105(1)
4.6.2 Main Results
106(6)
4.6.3 An Example
112(2)
4.7 Notes and Remarks
114(1)
5 Fractional Order Riemann-Liouville Volterra-Stieltjes Quadratic Delay Integral Equations
115(18)
5.1 Introduction
115(1)
5.2 Asymptotic Behavior of Solutions of Nonlinear Riemann-Liouville Volterra-Stieltjes Multi-delay Integral Equations
115(9)
5.2.1 Introduction
115(1)
5.2.2 Main Results
116(6)
5.2.3 An Example
122(2)
5.3 Existence and Attractivity of Solutions for Nonlinear Riemann-Liouville Volterra-Stieltjes Multi-delay Integral Equations
124(7)
5.3.1 Introduction
124(1)
5.3.2 Main Results
124(6)
5.3.3 An Example
130(1)
5.4 Notes and Remarks
131(2)
6 Fractional Order Riemann-Liouville Integro-Differential Equations
133(20)
6.1 Introduction
133(1)
6.2 Integro-Differential Equations
133(6)
6.2.1 Introduction
133(1)
6.2.2 Main Results
133(4)
6.2.3 Examples
137(2)
6.3 Perturbed Integro-Differential Equations
139(5)
6.3.1 Introduction
139(1)
6.3.2 Main Results
139(4)
6.3.3 An Example
143(1)
6.4 Impulsive Partial Integro-Differential Equations
144(7)
6.4.1 Introduction
144(1)
6.4.2 Main Results
144(6)
6.4.3 An Example
150(1)
6.5 Notes and Remarks
151(2)
7 Fractional Order Riemann-Liouville Delay Integro-Differential Equations
153(30)
7.1 Introduction
153(1)
7.2 Partial Neutral Functional Integro-Differential Equations with Delay
153(9)
7.2.1 Introduction
153(1)
7.2.2 Existence Results with Finite Delay
154(4)
7.2.3 The Phase Space B
158(1)
7.2.4 Existence Results with Infinite Delay
159(2)
7.2.5 An Example
161(1)
7.3 Integro-Differential Equations with Multiple Time Delay
162(5)
7.3.1 Introduction
162(1)
7.3.2 Main Results
162(4)
7.3.3 Examples
166(1)
7.4 Local Attractivity for Delay Partial Integro-Differential Equations
167(6)
7.4.1 Introduction
167(1)
7.4.2 Main Results
167(4)
7.4.3 An Example
171(2)
7.5 Existence and Attractivity Results for Partial Integro-Differential Equations with Delay
173(9)
7.5.1 Introduction
173(1)
7.5.2 Existence and Uniqueness
173(3)
7.5.3 Estimates on the Solutions
176(2)
7.5.4 Global Asymptotic Stability of Solutions
178(2)
7.5.5 An Example
180(2)
7.6 Notes and Remarks
182(1)
8 Abstract Integral Equations
183(8)
8.1 Introduction
183(1)
8.2 A Local Uniqueness Result for Abstract Integral Equations of Volterra Type in Banach Spaces
184(3)
8.2.1 Introduction
184(1)
8.2.2 Existence of Solutions
185(2)
8.3 A Global Uniqueness Result for Abstract Integral Equations of Volterra Type in Banach Spaces
187(2)
8.3.1 Introduction
187(1)
8.3.2 Existence of Solutions
187(2)
8.4 Notes and Remarks
189(2)
9 Abstract Integro-Differential Equations
191(24)
9.1 Introduction
191(2)
9.2 A local Uniqueness Result for Functional Integro-Differential Equations in Banach Spaces
193(5)
9.2.1 Introduction
193(1)
9.2.2 Existence of Solutions
193(5)
9.3 A Global Existence and Uniqueness Result for Functional Integro-Differential Equations in Frechet Spaces
198(4)
9.3.1 Introduction
198(1)
9.3.2 Existence of Solutions
198(4)
9.4 Abstract Fractional Integro-Differential Equations with State-Dependent Delay
202(6)
9.4.1 Introduction
202(1)
9.4.2 Existence of Solutions
203(4)
9.4.3 An Example
207(1)
9.5 Integro-Differential Equations with State-Dependent Delay on an Un-bounded Domain
208(6)
9.5.1 Introduction
208(1)
9.5.2 Existence of Solutions
208(5)
9.5.3 An Example
213(1)
9.6 Notes and Remarks
214(1)
10 Abstract Integro-Differential Inclusions
215(20)
10.1 Introduction
215(1)
10.2 Abstract Integro-Differential Inclusions with State-Dependent Delay
215(10)
10.2.1 Introduction
215(2)
10.2.2 Main Results
217(7)
10.2.3 An Example
224(1)
10.3 Abstract Integro-Differential Inclusions with State-Dependent Delay in Frechet Spaces
225(8)
10.3.1 Introduction
225(1)
10.3.2 Main Results
225(5)
10.3.3 Application to Control Theory
230(3)
10.4 Notes and Remarks
233(2)
11 Weak Solutions for Nonlinear Fractional Differential Equations
235(46)
11.1 Introduction
235(1)
11.2 Fractional Calculus in Banach Spaces with Weak Topologies
235(10)
11.2.1 Pettis Fractional Integrals
235(5)
11.2.2 Fractional Weak Derivative
240(3)
11.2.3 Caputo Fractional Weak Derivatives
243(2)
11.3 Weak Solutions For Nonlinear Differential Equations on Reflexive Banach Spaces
245(7)
11.3.1 Introduction
245(2)
11.3.2 Existence of Solutions
247(3)
11.3.3 An Example
250(1)
11.3.4 Concluding Remarks
251(1)
11.4 Weak Solutions of Boundary Value Problem on Unbounded Domains in Banach Spaces
252(12)
11.4.1 Introduction
252(1)
11.4.2 Existence of Solutions
252(12)
11.5 Weak Solutions for Nonlinear Differential Equations with Integral Boundary Conditions in Banach Spaces
264(5)
11.5.1 Introduction
264(1)
11.5.2 Existence of Solutions
264(4)
11.5.3 An Example
268(1)
11.6 Weak Solutions for Hyperbolic Partial Differential Equations in Banach Spaces
269(5)
11.6.1 Introduction
269(1)
11.6.2 Existence of Solutions
269(4)
11.6.3 An Example
273(1)
11.7 Pettis Integral Equations with Multiple Time Delay in Banach Spaces
274(5)
11.7.1 Introduction
274(1)
11.7.2 Existence of Solutions
274(5)
11.8 Notes and Remarks
279(2)
12 Weak Solutions for Nonlinear Fractional Differential Inclusions
281(12)
12.1 Introduction
281(1)
12.2 Weak Solutions for Boundary-Value Problems with Nonlinear Differential Inclusions
281(5)
12.2.1 Introduction
281(1)
12.2.2 Existence of Solutions
282(4)
12.3 Weak Solutions for Hyperbolic Partial Differential Inclusions in Banach Spaces
286(5)
12.3.1 Introduction
286(1)
12.3.2 Existence of Solutions
287(4)
12.4 Notes and Remarks
291(2)
13 Ulam Stabilities for the Darboux Problem for Partial Fractional Differential Equations and Inclusions
293(30)
13.1 Introduction
293(1)
13.2 Ulam Stabilities for the Darboux Problem for Partial Fractional Differential and Integro-Differential Equations
293(5)
13.2.1 Introduction
293(2)
13.2.2 Main Results
295(2)
13.2.3 An Example
297(1)
13.3 Ulam-Hyers-Rassias Stabilities for the Darboux Problem for Partial Fractional Implicit Differential Equations
298(4)
13.3.1 Introduction
298(1)
13.3.2 Main Results
298(2)
13.3.3 An Example
300(2)
13.4 Some Stability Concepts for Darboux Problem for Partial Fractional Differential Equations on Unbounded Domain
302(5)
13.4.1 Introduction
302(1)
13.4.2 Main Results
302(4)
13.4.3 An Example
306(1)
13.5 Ulam Stabilities for the Darboux Problem for Partial Fractional Differential Inclusions
307(6)
13.5.1 Introduction
307(1)
13.5.2 Main Results
308(4)
13.5.3 An Example
312(1)
13.6 Ulam Stabilities for the Darboux Problem for Impulsive Partial Fractional Differential Equations
313(8)
13.6.1 Introduction
313(3)
13.6.2 Main Results
316(4)
13.6.3 An Example
320(1)
13.7 Notes and Remarks
321(2)
A Interpretations and Motivations for Fractional Derivative
323(12)
A.1 Some Physical Interpretations
323(1)
A. 1.1 Physical Interpretation of the Stieltjes Integral
323(1)
A.1.2 Physical Interpretation of Fractional Integration: Shadows of the Past
323(1)
A.2 Physical Interpretation of the Riemann-Liouville FD
324(1)
A.2.1 Physical Interpretation of the Caputo Fractional Derivative
325(1)
A.3 Some Motivations
325(3)
A.3.1 Spring-Pot Model
326(1)
A.3.2 Impulse Response
327(1)
A.4 Some General Interpretation of Riemann-Liouville Derivative
328(5)
A.4.1 The Fractional Order Voigt Model
328(2)
A.4.2 The Fractional Order Maxwell Model
330(1)
A.4.3 The Fractional Order Zener Model
331(2)
A.5 Diffusion Phenomena
333(1)
A.6 Notes and Remarks
333(2)
References 335
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