**Integral Equations on Time Scales** This book offers the reader an overview of recent developments of integral equations on time scales It also contains elegant analytical and numerical methods This book is primarily intended for senior undergraduate students and beginning graduate students of engineering and science courses The students in mathematical and physical sciences will find many sections of direct relevance The book contains nine chapters and each chapter is pedagogically organized This book is specially designed f

Publisher : Atlantis Press

Author : Svetlin G. Georgiev

ISBN : 9789462392274

Year : 2016

Language: en

File Size : 2.52 MB

Category : Science Math

Atlantis Studies in Dynamical Systems

Series Editors: Henk Broer · Boris Hasselblatt

Svetlin G.Georgiev

Integral Equations

on Time Scales

Atlantis Studies in Dynamical Systems

Volume 5

Series editors

Henk Broer, Groningen, The Netherlands

Boris Hasselblatt, Medford, USA

The “Atlantis Studies in Dynamical Systems” publishes monographs in the area of

dynamical systems, written by leading experts in the ﬁeld and useful for both

students and researchers. Books with a theoretical nature will be published

alongside books emphasizing applications.

More information about this series at http://www.atlantis-press.com

Svetlin G. Georgiev

Integral Equations on Time

Scales

Svetlin G. Georgiev

Department of Differential Equations

Soﬁa University

Soﬁa

Bulgaria

Atlantis Studies in Dynamical Systems

ISBN 978-94-6239-227-4

ISBN 978-94-6239-228-1

DOI 10.2991/978-94-6239-228-1

(eBook)

Library of Congress Control Number: 2016951703

© Atlantis Press and the author(s) 2016

This book, or any parts thereof, may not be reproduced for commercial purposes in any form or by any

means, electronic or mechanical, including photocopying, recording or any information storage and

retrieval system known or to be invented, without prior permission from the Publisher.

Printed on acid-free paper

Preface

Many problems arising in applied mathematics or mathematical physics, can be

formulated in two ways namely as differential equations and as integral equations.

In the differential equation approach, the boundary conditions have to be imposed

externally, whereas in the case of integral equations, the boundary conditions are

incorporated within the formulation, and this confers a valuable advantage to the

latter method. Moreover, the integral equation approach leads quite naturally to

the solution of the problem as an inﬁnite series, known as the Neumann expansion,

the Adomian decomposition method, and the series solution method in which the

successive terms arise from the application of an iterative procedure. The proof

of the convergence of this series under appropriate conditions presents an interesting exercise in an elementary analysis.

This book encompasses recent developments of integral equations on time

scales. For many population models biological reasons suggest using their difference analogues. For instance, North American big game populations have discrete

birth pulses, not continuous births as is assumed by differential equations.

Mathematical reasons also suggest using difference equations—they are easier to

construct and solve in a computer spreadsheet. North American large mammal

populations do not have continuous population growth, but rather discrete birth

pulses, so the differential equation form of the logistic equation will not be convenient. Age-structured models add complexity to a population model, but make

the model more realistic, in that essential features of the population growth process

are captured by the model. They are used difference equations to deﬁne the population model because discrete age classes require difference equations for simple

solutions. The discrete models can be investigated using integral equations in the

case when the time scale is the set of the natural numbers. A powerful method

introduced by Poincaré for examining the motion of dynamical systems is that of a

Poincaré section. This method can be investigated using integral equations on the

set of the natural numbers. The total charge on the capacitor can be investigated

with an integral equation on the set of the harmonic numbers.

This book contains elegant analytical and numerical methods. This book is

intended for the use in the ﬁeld of integral equations and dynamic calculus on time

v

vi

Preface

scales. It is also suitable for graduate courses in the above ﬁelds. This book contains

nine chapters. The chapters in this book are pedagogically organized. This book is

specially designed for those who wish to understand integral equations on time

scales without having extensive mathematical background.

The basic deﬁnitions of forward and backward jump operators are due to Hilger.

In Chap. 1 are given examples of jump operators on some time scales. The

graininess function, which is the distance from a point to the closed point on the

right, is introduced in this chapter. In this chapter, the deﬁnitions for delta derivative

and delta integral are given and some of their properties are deducted. The basic

results in this chapter can be found in [2]. Chapter 2 introduces the classiﬁcation of

integral equations on time scales and necessary techniques to convert dynamic

equations to integral equations on time scales. Chapter 3 deals with the generalized

Volterra integral equations and the relevant solution techniques. Chapter 4 is

concerned with the generalized Volterra integro-differential equations and also

solution techniques. Generalized Fredholm integral equations are investigated in

Chap. 5. Chapter 6 is devoted on Hilbert–Schmidt theory of generalized integral

equations with symmetric kernels. The Laplace transform method is introduced in

Chap. 7. Chapter 8 deals with the series solution method. Nonlinear integral

equations on time scales are introduced in Chap. 9.

The aim of this book was to present a clear and well-organized treatment of the

concept behind the development of mathematics and solution techniques. The text

material of this book is presented in highly readable, mathematically solid format.

Many practical problems are illustrated displaying a wide variety of solution

techniques. Nonlinear integral equations on time scales and some of their applications in the theory of population models, biology, chemistry, and electrical

engineering will be discussed in a forthcoming book “Nonlinear Integral Equations

on Time Scales and Applications.”

The author welcomes any suggestions for the improvement of the text.

Paris, France

June 2016

Svetlin G. Georgiev

Contents

1 Elements of the Time Scale Calculus . . . . . . . . . . . . . . . . . . . . . .

1.1 Forward and Backward Jump Operators, Graininess Function

1.2 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3 Mean Value Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.4 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.5 The Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.5.1 Hilger’s Complex Plane . . . . . . . . . . . . . . . . . . . . . . .

1.5.2 Deﬁnition and Properties of the Exponential Function

1.5.3 Examples for Exponential Functions . . . . . . . . . . . . . .

1.6 Hyperbolic and Trigonometric Functions . . . . . . . . . . . . . . . .

1.7 Dynamic Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.8 Advanced Practical Exercises . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

1

1

8

19

22

39

39

49

63

65

67

74

2 Introductory Concepts of Integral Equations on Time Scales . .

2.1 Reducing Double Integrals to Single Integrals . . . . . . . . . . . .

2.2 Converting IVP to Generalized Volterra Integral Equations . .

2.3 Converting Generalized Volterra Integral Equations to IVP . .

2.4 Converting BVP to Generalized Fredholm Integral Equation .

2.5 Converting Generalized Fredholm Integral Equation to BVP .

2.6 Solutions of Generalized Integral Equations and Generalized

Integro-Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . .

2.7 Advanced Practical Exercises . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

77

80

82

88

95

105

....

....

116

121

3 Generalized Volterra Integral Equations . . . . . . . . . . . . . . . . . . .

3.1 Generalized Volterra Integral Equations of the Second Kind .

3.1.1 The Adomian Decomposition Method . . . . . . . . . . . . .

3.1.2 The Modiﬁed Decomposition Method . . . . . . . . . . . . .

3.1.3 The Noise Terms Phenomenon . . . . . . . . . . . . . . . . . .

3.1.4 Differential Equations Method . . . . . . . . . . . . . . . . . . .

3.1.5 The Successive Approximations Method . . . . . . . . . . .

.

.

.

.

.

.

.

131

131

131

138

144

146

158

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

vii

viii

Contents

3.2 Conversion of a Generalized Volterra Integral Equation

of the First Kind to a Generalized Volterra Integral

Equation of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3 Existence and Uniqueness of Solutions . . . . . . . . . . . . . . . . . . . . .

3.3.1 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3.2 Existence of Solutions of Generalized Volterra Integral

Equations of the Second Kind . . . . . . . . . . . . . . . . . . . . . . .

3.3.3 Uniqueness of Solutions of Generalized Volterra Integral

Equations of Second Kind. . . . . . . . . . . . . . . . . . . . . . . . . .

3.3.4 Existence and Uniqueness of Solutions of Generalized

Volterra Integral Equations of the First Kind . . . . . . . . . . .

3.4 Resolvent Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5 Application to Linear Dynamic Equations . . . . . . . . . . . . . . . . . . .

3.6 Advanced Practical Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Generalized Volterra Integro-Differential Equations . . . . . . . . .

4.1 Generalized Volterra Integro-Differential Equations

of the Second Kind. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1.1 The Adomian Decomposition Method . . . . . . . . . . . . .

4.1.2 Converting Generalized Volterra Integro-Differential

Equations of the Second Kind to Initial

Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1.3 Converting Generalized Volterra Integro-Differential

Equations of the Second Kind to Generalized

Volterra Integral Equations . . . . . . . . . . . . . . . . . . . . .

4.2 Generalized Volterra Integro-Differential Equations

of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3 Advanced Practical Exercises . . . . . . . . . . . . . . . . . . . . . . . . .

5 Generalized Fredholm Integral Equations . . . . . . . . . . . . . . . . . .

5.1 Generalized Fredholm Integral Equations of the Second Kind

5.1.1 The Adomian Decomposition Method . . . . . . . . . . . . .

5.1.2 The Modiﬁed Decomposition Method . . . . . . . . . . . . .

5.1.3 The Noise Terms Phenomenon . . . . . . . . . . . . . . . . . .

5.1.4 The Direct Computation Method . . . . . . . . . . . . . . . . .

5.1.5 The Successive Approximations Method . . . . . . . . . . .

5.2 Homogeneous Generalized Fredholm Integral Equations

of the Second Kind. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3 Fredholm Alternative Theorem . . . . . . . . . . . . . . . . . . . . . . . .

Z bZ b

jKðx; YÞj2 DXDY\1 . . . . . . .

5.3.1 The Case When

163

166

166

172

175

177

178

184

192

....

197

....

....

197

197

....

206

....

212

....

....

219

223

.

.

.

.

.

.

.

.

.

.

.

.

.

.

227

227

227

232

237

241

246

....

....

251

258

.

.

.

.

.

.

.

.

.

.

.

.

.

.

....

258

5.3.2 The General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

5.3.3 Fredholm’s Alternative Theorem . . . . . . . . . . . . . . . . . . . . . 274

5.4 The Schmidth Expansion Theorem and the Mercer

Expansion Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

a

a

Contents

ix

5.4.1 Operator-Theoretical Notations . . . . . . . . . . . . . . . . . . . . . .

5.4.2 The Schmidt Expansion Theorem . . . . . . . . . . . . . . . . . . . .

5.4.3 Application to Generalized Fredholm Integral Equation

of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.4.4 Positive Deﬁnite Kernels. Mercer’s Expansion

Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.5 Advanced Practical Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 Hilbert-Schmidt Theory of Generalized Integral Equations

with Symmetric Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.1 Schmidt’s Orthogonalization Process . . . . . . . . . . . . . . . . . . .

6.2 Approximations of Eigenvalues . . . . . . . . . . . . . . . . . . . . . . .

6.3 Inhomogeneous Generalized Integral Equations . . . . . . . . . . .

7 The Laplace Transform Method . . . . . . . . . . . . . . . . . . . . . . . . .

7.1 The Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.1.1 Deﬁnition and Examples . . . . . . . . . . . . . . . . . . . . . . .

7.1.2 Properties of the Laplace Transform . . . . . . . . . . . . . .

7.1.3 Convolution and Shifting Properties

of Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . .

7.2 Applications to Dynamic Equations . . . . . . . . . . . . . . . . . . . .

7.3 Generalized Volterra Integral Equations of the Second Kind .

7.4 Generalized Volterra Integral Equations of the First Kind . . .

7.5 Generalized Volterra Integro-Differential Equations of the

Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.6 Generalized Volterra Integro-Differential Equations

of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.7 Advanced Practical Exercises . . . . . . . . . . . . . . . . . . . . . . . . .

8 The

8.1

8.2

8.3

Series Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Generalized Volterra Integral Equations of the Second Kind .

Generalized Volterra Integral Equations of the First Kind . . .

Generalized Volterra Integro-Differential Equations of the

Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

275

282

288

289

298

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

301

301

307

314

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

321

321

321

325

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

333

347

352

357

....

360

....

....

366

370

....

....

....

375

375

384

....

387

9 Non-linear Generalized Integral Equations . . . . . . . . . . . . . . . . . . . . . 395

9.1 Non-linear Generalized Volterra Integral Equations . . . . . . . . . . . . 395

9.2 Non-linear Generalized Fredholm Integral Equations . . . . . . . . . . . 396

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

Chapter 1

Elements of the Time Scale Calculus

This chapter is devoted to a brief exposition of the time scale calculus that provide

the framework for the study of integral equations on time scales. Time scale calculus

is very interesting in itself, this challenging subject has been developing very rapidly

in the last decades. A detailed discussion of the time scale calculus is beyond the

scope of this book, for this reason the author confine to outlining a minimal set of

properties needed in the further proceeding. The presentation in this chapter follows

the book [2]. A deep and thorough insight into the time scale calculus, as well as the

discussion of the available bibliography on this issue, can be found in the book [2].

1.1 Forward and Backward Jump Operators, Graininess

Function

Definition 1 A time scale is an arbitrary nonempty closed subset of the real numbers.

We will denote a time scale by the symbol T .

We suppose that a time scale T has the topology that inherits from the real

numbers with the standard topology.

Example 1 [1, 2], R, N are time scales.

Example 2 [a, b), (a, b], (a, b) are not time scales.

Definition 2 For t ∈ T we define the forward jump operator σ : T −→ T as

follows

σ (t) = inf{s ∈ T : s > t}.

We note that σ (t) ≥ t for any t ∈ T .

© Atlantis Press and the author(s) 2016

S.G. Georgiev, Integral Equations on Time Scales, Atlantis Studies

in Dynamical Systems 5, DOI 10.2991/978-94-6239-228-1_1

1

2

1 Elements of the Time Scale Calculus

Definition 3 For t ∈ T we define the backward jump operator ρ : T −→ T by

ρ(t) = sup{s ∈ T : s < t}.

We note that ρ(t) ≤ t for any t ∈ T .

Definition 4 We set

inf Ø = sup T , sup Ø = inf T .

Definition 5 For t ∈ T we have the following cases.

If σ (t) > t, then we say that t is right-scattered.

If t < sup T and σ (t) = t, then we say that t is right-dense.

If ρ(t) < t, then we say that t is left-scattered.

If t > inf T and ρ(t) = t, then we say that t is left-dense.

If t is left-scattered and right-scattered at the same time, then we say that t is

isolated.

6. If t is left-dense and right-dense at the same time, then we say that t is dense.

√

√

Example 3 Let T = { 2n + 1 : n ∈ N }. If t = 2n + 1 for some n ∈ N , then

t2 − 1

n=

and

2

1.

2.

3.

4.

5.

√

√

√

σ (t) = inf{l ∈ N : 2l + 1 > 2n + 1} = 2n + 3 = t 2 + 2 f or n ∈ N ,

√

√

√

ρ(t) = sup{l ∈ N : 2l + 1 < 2n + 1} = 2n − 1 = t 2 − 2 f or n ∈ N , n ≥ 2.

For n = 1 we have

√

√

ρ( 3) = sup Ø = inf T = 3.

Since

t2 − 2 < t <

t2 + 2

for

n ≥ 2,

√

2n + 1, n ∈ N , n ≥ 2, is right-scattered and leftwe conclude that every point

√

scattered, i.e., every point 2n + 1, n ∈ N , n ≥ 2, is isolated.

Because

√

√

√

√

3 = ρ( 3) < σ ( 3) = 5,

we have that the point

√

Example 4 Let T =

1. t =

1

. Then

2

3 is right-scattered.

1

: n ∈ N ∪ {0} and t ∈ T be arbitrarily chosen.

2n

1.1 Forward and Backward Jump Operators, Graininess Function

3

1

1

1

1

1

, 0 : , 0 > , l ∈ N = inf Ø = sup T = ,

= inf

2

2l

2l

2

2

1

1

1

1

1

,0 : ,0 < ,l ∈ N = 0 < ,

= sup

ρ

2

2l

2l

2

2

σ

1

is left-scattered.

2

1

, n ∈ N , n ≥ 2. Then

2. t =

2n

1 1

1

1

1

1

:

>

,l ∈ N =

>

,

= inf

σ

2n

2l 2l

2n

2(n − 1)

2n

1

1

1

1

1

1

,0 : ,0 <

,l ∈ N =

<

.

= sup

ρ

2n

2l

2l

2n

2(n + 1)

2n

i.e.,

1

Therefore all points

, n ∈ N , n ≥ 2, are right-scattered and left-scattered,

2n

1

, n ∈ N , n ≥ 2, are isolated.

i.e., all points

2n

3. t = 0. Then

σ (0) = inf{s ∈ T : s > 0} = 0,

ρ(0) = sup{s ∈ T : s < 0} = sup Ø = inf T = 0.

Example 5 Let T =

n

: n ∈ N0 and t = , n ∈ N0 , be arbitrarily chosen.

3

3

n

1. n ∈ N . Then

σ

ρ

n

3

n

3

l

l

n

n

n+1

, 0 : , 0 > , l ∈ N0 =

> ,

3

3

3

3

3

n−1

l

l

n

n

, 0 : , 0 < , l ∈ N0 =

< .

3

3

3

3

3

= inf

= sup

n

Therefore all points t = , n ∈ N , are right-scattered and left-scattered, i.e., all

3

n

points t = , n ∈ N , are isolated.

3

2. n = 0. Then

l

1

l

σ (0) = inf

, 0 : , 0 > 0, l ∈ N0 = > 0,

3

3

3

l l

: , 0 < 0, l ∈ N0 = sup Ø = inf T = 0,

ρ(0) = sup

3 3

4

1 Elements of the Time Scale Calculus

i.e., t = 0 is right-scattered.

√

3

Exercise 1 Classify each point t ∈ T = { 2n − 1 : n ∈ N0 } as left-dense, leftscattered, right-dense, or right-scattered.

√

3

Answer. The points 2n − 1, n ∈ N , are isolated, the point −1 is right-scattered.

Definition 6 The numbers

H0 = 0,

Hn =

n

1

k=1

k

, n∈N,

will be called harmonic numbers.

Exercise 2 Let

H = {Hn : n ∈ N0 }.

Prove that H is a time scale. Find σ (t) and ρ(t).

Answer. σ (Hn ) = Hn+1 , n ∈ N0 , ρ(Hn ) = Hn−1 , n ∈ N , ρ(H0 ) = H0 .

Definition 7 The graininess function μ : T −→ [0, ∞) is defined by

μ(t) = σ (t) − t.

: n ∈ N . Let also, t = 2n+1 ∈ T for some n ∈ N .

Example 6 Let T = 2n+1

Then

σ (t) = inf 2l+1 : 2l+1 > 2n+1 , l ∈ N = 2n+2 = 2t.

Hence,

μ(t) = σ (t) − t = 2t − t = t or μ 2n+1 = 2n+1 , n ∈ N .

Example 7 Let T =

√

√

n + 1 : n ∈ N . Let also, t = n + 1 for some n ∈ N .

Then n = t 2 − 1 and

√

√

√

√

l + 1 : l + 1 > n + 1, l ∈ N = n + 2 = t 2 + 1.

σ (t) =

Hence,

μ(t) = σ (t) − t =

√

√

√

t 2 + 1 − t or μ( n + 1) = n + 2 − n + 1, n ∈ N .

n

n

: n ∈ N0 . Let also, t = for some n ∈ N0 . Then n = 2t

Example 8 Let T =

2

2

and

n+1

l l

n

1

σ (t) = inf

: > , l ∈ N0 =

=t+ .

2 2

2

2

2

1.1 Forward and Backward Jump Operators, Graininess Function

Hence,

μ(t) = σ (t) − t = t +

5

n

1

1

1

= .

−t =

or μ

2

2

2

2

Example 9 Suppose that T consists of finitely many different points: t1 , t2 , . . . , tk .

Without loss of generality we can assume that

t1 < t 2 < . . . < t k .

For i = 1, 2, . . . , k − 1 we have

σ (ti ) = inf{tl ∈ T : tl > ti , l = 1, 2, . . . , k} = ti+1 .

Hence,

μ(ti ) = ti+1 − ti , i = 1, 2, . . . , k − 1.

Also,

σ (tk ) = inf{tl ∈ T : tl > tk , l = 1, 2, . . . , k} = inf Ø = sup T = tk .

Therefore

μ(tk ) = σ (tk ) − tk = tk − tk = 0.

From here,

k

μ(ti ) =

i=1

Exercise 3 Let T =

Answer. μ

√

3

k−1

μ(ti ) + μ(tk ) =

i=1

k−1

(ti+1 − ti ) = tk − t1 .

i=1

√

3

n + 2 : n ∈ N0 . Find μ(t), t ∈ T .

√

√

n + 2 = 3 n + 3 − 3 n + 2.

Definition 8 If f : T −→ R is a function, then we define the function

f σ : T −→ R by

f σ (t) = f (σ (t)) for any t ∈ T . i.e.,

f σ = f ◦ σ.

Below, for convenience, we will use the following notation σ k (t) = (σ (t))k ,

f (t) = ( f (t))k , k ∈ R.

Example 10 Let T = t = 2n+2 : n ∈ N , f (t) = t 2 + t − 1. Then

k

σ (t) = inf 2l+2 : 2l+2 > 2n+2 , l ∈ N = 2n+3 = 2t.

© 2018-2019 uberlabel.com. All rights reserved