Open access peer-reviewed chapter

The Fourier Transform Method for Second-Order Integro-Dynamic Equations on Time Scales

By Svetlin G. Georgiev

Submitted: September 14th 2020Reviewed: December 22nd 2020Published: September 8th 2021

DOI: 10.5772/intechopen.95622

Downloaded: 17

Abstract

In this chapter we introduce the Fourier transform on arbitrary time scales and deduct some of its properties. In the chapter are given some applications for second-order integro-dynamic equations on time scales.

Keywords

  • time scale
  • Fourier transform
  • generalized shift problem
  • integro-dynamic equation

1. Introduction

Starting with the pioneering work of Hilger [1], the measure chains and in particular, the time scales have gained a great attention in the last decades. Especially, theoretical studies on dynamic equations on general time scales, which can be regarded as generalization of the differential equations, achieved big progress [2, 3].

The main aim of this chapter is to introduce the Fourier transform on arbitrary time scales and to deduct some of its properties. We give applications for solving of second-order integro-dynamic equations on time scales.

The chapter is organized as follows. In the next section we give some basic definitions and facts from time scale calculus, Laplace, bilateral Laplace transform. In Section 3 we define the Fourier transform and deduct some of its properties. In Section 4 we give applications for second-order integro-dynamic equations on time scales.

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2. Preliminaries and auxiliary results

2.1 Time scales

Throughout this paper, we will assume that the reader is familiar with the basics of the time scale calculus. A detailed introduction to the time scale calculus is given in [2, 3]. Here, we collect the definitions and theorems that will be most useful in this paper.

Definition 2.1.A time scale, denoted byT, is a nonempty, closed subset ofR. Fora,bT, we letabdenote the setabT.

Definition 2.2.LetTbe a time scale. FortT, we define the forward jump operatorσ:TRbyσt=infsT:s>t, and the backward jump operatorρ:TTis given byρt=supxT:s<t.

By convention, we take inf=supT, sup=infT. For a function f:TR, we will use the notation fσtfor the composition fσt.

Definition 2.3.The graininess functionμ:T0is defined byμt=σtt,tT.

Definition 2.4.LettT. Ifσt=tandt<supT, thentis right-dense. Ifσt>t, thentis right-scattered. Similarly, ifρt=tandt>infT, thentis left-dense. Ifρt<t, thentis left-scattered.

Definition 2.5.IfsupT=msuch thatmis left-scattered, then defineTκ=T\m, otherwise, defineTκ=T.

Definition 2.6.A functionf:TRis rd-continuous provided it is continuous at right-dense points inTand its left-sided limits exist and are finite at all left-dense points inT. A functionp:TRis regressive provided1+μtpt0,tTκ. The set of all regressive and rd-continuous functions on a time scaleTis denoted byR=RT. We use the notationR+to denote the subgroup of thosepRfor which1+μtpt>0for alltTκ.

Definition 2.7.The delta derivative off:TRattTκ, is defined to be

fΔt=limstfσtfsσtsE1

provided this limit exists.

Definition 2.8.ForpR, the generalized exponential functionep:T×TRis defined by

epts=expstξμτpτΔτ,E2

fors,tT, where the cylinder transformation, ξhz, is defined by

ξhz=1hLog1+zh,h>0,z,h=0.E3

Definition 2.9.Forp,qR, we define the operationandas follows

pqt=pt+qt+μtptqt,pt=pt1+μtpt.E4

The proof of the next theorem is given in [2, 3].

Theorem 2.1.Ifp,qRandt,s,rT, then

  1. e0ts=1, eptt=1.

  2. epσts=1+μtptepts.

  3. epst=1epts=epts.

  4. eptsepsr=eptr.

  5. eptseqts=epqts.

  6. eptt0>0for anyt0,tTifpRand1+μtpt>0for anytTκ.

Definition 2.10.Forh>0, the Hilger complex plane is defined byCh=C\1hand we takeC0=CandC=C\0.

Definition 2.11.For givenh0, the Hilger real part of a numberzCis given by the formula

Rehz=Rez,h=0,1+hz1h,0<h<,z,h=.E5

It is known, see [4], that for a fixed zand 0<h<, Rehzis a nondecreasing function of h. This relationship extends to h=because for any 0<h<,

Rehz=1+hz1h1+hz1h=z=Rez.E6

2.2 The Laplace transform

Here we suppose that supT=and sT.

Definition 2.12.For0handλR, we define

Chλ=zCh:Rehz>λE7

and

C¯hλ=zCh:0<Rehz<λ.E8

Definition 2.13.Define minimal graininess as followsμs=inftsμt.

If λis positively regressive, then for any zCμsλ, it is known (see [4]) that

eλztseλReμszts,t[s,),limtReμszts=0andlimteλzts=0.E9

Definition 2.14.IfXTandαR+is a constant, then we say thatfCrdTis of exponential orderαonXif there exists a constantKsuch that for alltX, the boundftKeαtsholds.

If fCrd([s,))is of exponential order α, then for any zCμsα(see [4]) limtftezts=0.

Definition 2.15.Iff:TCandzCis a complex number such that for alltswe have1+μtz0, then the Laplace transform is defined by the improper integral

Lfzs=sfτezστsΔτ,E10

whenever the integral exists.

Significant work has been conducted in [4, 5] and references therein to understand the analytical properties of the Laplace transform.

2.3 The bilateral Laplace transform

Here we suppose that supT=, infT=and sT. Denote μs=suptsμt, μ¯s=inftsμt. For λR, define

Mλts=ts11+λμτΔτ.E11

For λR+(s], λR, it is known (see [6])

  1. MλΔts<0for all ts, where the differentiation is with respect to t.

  2. limtMλts=.

  3. eλztseλReμszts.

  4. limteλReμszts=0.

  5. limteλzts=0.

Definition 2.16.Suppose thatf:TRis regulated. Then the bilateral Laplace transform offis defined by

Lbfzs=ftezσtsΔt,E12

for regressivezCwhere the improper integral exists.

Definition 2.17.Letα,γR. We say that a functionfCrdThas double exponential orderαγonTif the restrictionsfsandfsare of exponential orderαandγ, respectively.

If fCrdTis of double exponential order αγ, in [6], they are proved the following properties

  1. for any zCμsγ, limtftezts=0.

  2. for any zC¯μsα, limtftezts=0.

For zC, we define

μ¯¯sz=μs,Reμ¯sz0,μ¯s,Reμ¯sz>0.E13

Definition 2.18.LetαR+(s]andγR+([s,)),α,γR. We say thatsαγis an admissible triple if

Cs,α,γ=zC:Reμsz<αReμsz>γ1+μ¯¯(sz)Reμ¯sz0.E14

If sαγis an admissible triple and if fCrdTis of double exponential order αγ, then in [6] it is proved that Lbsexists on Cs,α,γ, converges absolutely and uniformly, and

limzLbfzs=0.E15

3. The Fourier transform

Suppose that Tis a time scale so that infT=, supT=and sT.

Definition 3.1.Suppose thatf:TRis regulated. Then the Fourier transform of the functionfis defined by

Ffxs=fteixσtsΔtE16

forxRfor which1+ixμt0for anytTκand the improper integral exists.

Definition 3.2.LetαR+([s,)),γR+(s]. We say thatsγαis a real admissible triple if

Rs,γ,α=xR:Reμsix<γReμsix>α1+μ¯¯sReμ¯six0.E17

If fCrdT, then the triple sγαis a real admissible triple and fis of double exponential order αγ, then Ffsexists on Rs,γ,αand converges absolutely and uniformly on Rs,γ,α. Below we will list some of the properties of the Fourier transform.

Theorem 3.1.Letf,g:TR,α,βC. Then

Fαf+βgxs=αFfxs+βFgxsE18

for thosexRfor which1+t0, tTκ, and the respective integrals exist.

Proof.We have

Fαf+βgxs=αf+βgteixσtsΔt=αfteixσtsΔt+βgteixσtsΔt=αFfxs+βFgxs.E19

This completes the proof.□

Theorem 3.2.Letf:TRbe enough timesΔ-differentiable. For anykN, we have

FfΔkxs=ixkFfxsE20

for thosexRfor which1+t0, tTκ, and the respective integrals exist and

limt±fΔlteixts=0,l0k1.E21

Proof.We will use the principle of mathematical induction.

  1. For k=1, we have

FfΔxs=fΔteixσtsΔt=limtfteixtslimtfteixtsixtfteixtsΔt=ixfteixσtsΔt=ixFfxs.E22

  1. Assume that

FfΔkxs=ixkFfxsE23

for some kN.

  1. We will prove that

FfΔk+1xs=ixk+1Ffxs.E24

Really, we have

FfΔk+1xs=ixFfΔkxs=ixk+1Ffxs.E25

This completes the proof.□

Theorem 3.3.Letf:TR. Then

Ffxs¯=FfxsE26

for thosexRfor which1±t0, tTκ, and the respective integrals exist.

Proof.From the definition of the Fourier transform, we have

Ffxs¯=esσt1μτLog1+μτixτΔτftΔt¯=esσt1μτLog1+μτixτΔτft¯Δt=esσt1μτLog1+μτixτΔτftΔt=Ffxs.E27

This completes the proof.□

Theorem 3.4.Letf:TRbe regulated and

Ft=atfτΔτ,tT,E28

for some fixed aT. Then

FFxs=ixFfxsE29

for those xR, x0, for which

limt±Fteixts=0.E30

Proof.We have

FFxs=FteixσtsΔt=Ft1+μtixteixtsΔt=Ft11+txeixtsΔt=1ixFtix1+txeixtsΔt=ixFtixteixtsΔt=ixFteixΔtsΔt=ixlimtFteix(ts)limtFteix(ts)ixfteixσtsΔt=ixFfxsE31

for those xR, x0, for which

limt±Fteixts=0.E32

This completes the proof.□

4. Applications to second-order integro-dynamic equations

Consider the equation

yΔ2+a1yΔ+a2y=atfsΔs,E33

where a1,a2R, fCrdT, f:TR. Let sTbe fixed. Let also, xRbe such that

x2ia1xa20E34

and

limt±yΔlteixts=0,l=0,1,E35

and

limt±Fteixts=0,E36

where

Ft=atfsΔs,tT.E37

Here aTis a fixed constant. Set

Yx=Fyxs.E38

Then

FyΔxs=ixFyxs=ixYx,FyΔ2xs=ix2Fyxs=x2YxE39

and

Ffxs=ixFxs.E40

Then the Eq. (33) takes the form

x2Yx+ia1xYx+a2Yx=ixFxs,E41

or

x2ia1xa2Yx=ixFfxs,E42

or

Yx=ixx2ia1xa2Ffxs.E43

Consequently

yt=F1ixx2ia1xa2Ff(s)t,tT,E44

provided that F1exists.

Additional classifications

AMS Subject Classification:39A10, 39A11, 39A12

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Svetlin G. Georgiev (September 8th 2021). The Fourier Transform Method for Second-Order Integro-Dynamic Equations on Time Scales, Recent Developments in the Solution of Nonlinear Differential Equations, Bruno Carpentieri, IntechOpen, DOI: 10.5772/intechopen.95622. Available from:

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