Properties of the Generalized Laplace Transform and Transport Partial Dynamic Equation on Time Scales

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1 University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Dissertations, Theses, and Student Research Papers in Mathematics Mathematics, Department of 200 Properties of the Generalized Laplace Transform and Transport Partial Dynamic Equation on Time Scales Chris R. Ahrendt University of Nebraska at Lincoln, ahrendcr@uwec.edu Follow this and additional works at: Part of the Analysis Commons, Ordinary Differential Equations and Applied Dynamics Commons, and the Science and Mathematics Education Commons Ahrendt, Chris R., "Properties of the Generalized Laplace Transform and Transport Partial Dynamic Equation on Time Scales" 200. Dissertations, Theses, and Student Research Papers in Mathematics This Article is brought to you for free and open access by the Mathematics, Department of at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Dissertations, Theses, and Student Research Papers in Mathematics by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln.

2 PROPERTIES OF THE GENERALIZED LAPLACE TRANSFORM AND TRANSPORT PARTIAL DYNAMIC EQUATION ON TIME SCALES by Chris Ahrendt A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfilment of Requirements For the Degree of Doctor of Philosophy Major: Mathematics Under the Supervision of Professors Lynn Erbe and Allan Peterson Lincoln, Nebraska August, 200

3 PROPERTIES OF THE GENERALIZED LAPLACE TRANSFORM AND TRANSPORT PARTIAL DYNAMIC EQUATION ON TIME SCALES Chris Ahrendt, Ph.D. University of Nebraska, 200 Advisors: Lynn Erbe and Allan Peterson In this dissertation, we first focus on the generalized Laplace transform on time scales. We prove several properties of the generalized exponential function which will allow us to explore some of the fundamental properties of the Laplace transform. We then give a description of the region in the complex plane for which the improper integral in the definition of the Laplace transform converges, and how this region is affected by the time scale in question. Conditions under which the Laplace transform of a power series can be computed term-by-term are given. We develop a formula for the Laplace transform for periodic functions on a periodic time scale. Regressivity and its relationship to the Laplace transform is examined, and the Laplace transform for several functions is explicitly computed. Finally, we explore some inversion formulas for the Laplace transform via contour integration. In Chapter 4, we develop two recursive representations for the unique solution of the transport partial dynamic equation on an isolated time scale. We then use these representations to explicitly find the solution of the transport equation in several specific cases. Finally, we compare and contrast the behavior with that of the wellknown behavior of the solution to the transport partial difference equation in the case where T Z.

4 iii DEDICATION This dissertation is dedicated to my wife Carol and my daughter Michelle.

5 iv ACKNOWLEDGMENTS I would like to thank my advisors, Dr. Lynn Erbe and Dr. Allan Peterson. Your innumerable suggestions, insights and corrections have been extremely helpful in not only preparing this dissertation, but in helping me to mature in my mathematical ability. Further, your ability to motivate and encourage me throughout this entire process has made this work possible. Dr. Peterson has given me the opportunity to work with his REU students as a graduate mentor for the last three summers, and I would like to thank you especially for that fantastic opportunity to learn what undergraduate research is all about. To the other members of my committee, Dr. Mikil Foss, Dr. John Orr, and Dr. Ted Pardy, thank you. To all the professors who I have had the opportunity to work with here at UNL, I truly appreciate the support you have given me and for never once making me feel like I was just a lowly grad student. And finally, to my friends and my entire family, especially my wife, I thank you for your constant support and prayers. Not only has my wife Carol put up with being married to me for the last four years, she has put up with being married to me while I have attended graduate school, and for that I am extremely grateful.

6 v Contents Contents v List of Figures vii List of Tables ix Introduction 2 Time Scale Preliminaries 5 2. The Time Scale Calculus Properties of the Generalized Exponential Function The Generalized Laplace Transform Introduction Region of Convergence of the Laplace Transform Power Series on Time Scales Basic Properties of the Laplace Transform Periodic Time Scales Regressivity Inversion Integrals Some Calculations

7 vi 4 The Transport Equation Introduction The Transport Equation on an Isolated Time Scale Examples Bibliography 0

8 vii List of Figures 3. The region of convergence guaranteed by Theorem for an arbitrary time scale T The region of convergence guaranteed by Theorem for a time scale such tha < µ min µt for all t T The region of convergence guaranteed by Example with γ and 2 δ The region of convergence guaranteed by Example with γ and 2 δ The region of convergence guaranteed by Example with γ and 2 δ The region of convergence guaranteed by Example with γ and 2 δ The domain of the solution of found in Example The solution of found in Example on T {0,, 3, 4, 6, } with the initial function given by ft t The domain of the solution of found in Example The solution of found in Example on T {0,, 3, 6, 7, 9, 2, } with the initial function given by ft t

9 viii 4.5 The domain of the solution of found in Example The solution of found in Example on the so-called harmonic time scale with the initial function given by ft t

10 ix List of Tables 3. Laplace Transform formulas for α β Laplace Transform formulas for α

11 Chapter Introduction In this dissertation we will use the tools of the time scales calculus to explore several properties of the generalized exponential function, the generalized Laplace transform on time scales, and the transport partial dynamic equation on time scales. In 988 Stefan Hilger introduced the concept of the time scale calculus in his Ph.D. dissertation [6] as a means to unify continuous and discrete analysis. Many results one encounters in the study of both differential and difference equations have analogs in the time scale case. However, the time scale result encompasses both the discrete and continuous results as special cases. In addition, the time scale calculus is a rich source of interesting problems that do not have any natural equivalent in either the continuous or discrete case. While unification is certainly a goal in studying time scales, another focus is to extend and explain the results of differential and difference equations. For example, if we consider the second-order self-adjoint differential equation px t + qtx 0, it is well known that nonzero solutions to this equation correspond, via the so-called

12 2 Riccati substitution, to solutions of the differential Riccati equation, z + qt + z2 pt 0.. In the difference equation case, the well-known second-order self-adjoint equation p xt + qtxt + 0 has nonzero solutions that correspond to the solutions of the discrete Riccati equation, zt + qt + z 2 t zt + pt 0..2 It is through analysis using the techniques of time scales that we can not only unify these two concepts, but also explain why the discrete Riccati equation has a different form than the continuous Riccati equation. It turns out that nonzero solutions for the dynamic equation px t + qtx σ t 0 on a time scale T correspond to the solutions of the time scale Riccati equation, z t + qt + z 2 t µtzt + pt 0..3 Since z t z t if T R, and z t zt if T Z, we find that the µt term explains the differences between the continuous and discrete cases. If T R, then µt 0 and.3 reduces to the well-known continuous case.; and if T Z, then µt and.3 reduces to the well-known discrete case.2. Note that in Section 2. the notation used here will be introduced and explained. After a quick introduction to the time scale calculus, we will look at four new results that deal with the generalized exponential function on time scales see Section 2.2. These results will be fundamental in many of the proofs we give in Chapter 3. The results pertaining to the generalized exponential function range from asymptotic

13 3 properties to determining the region of analyticity. In Chapter 3, we will turn our attention to the generalized Laplace transform on time scales. The definition of the Laplace transform that we will be concerned with here was first given by Bohner and Peterson in the paper Laplace transform and Z- transform: Unification and extension, see [5]. Note that this definition differs from that given by Stefan Hilger in [7]. A special case of this transform in the discrete setting is given in Donahue s honors thesis [] written under the supervision of Paul Eloe. A thorough introduction to the Laplace transform we will define and examine here is also given in [4], and some further properties that we will make use of are given in [3]. A generalization of the Laplace transform to so-called α-derivatives on generalized time scales can be found in [2]. We will define the generalized Laplace transform in Section 3. and proceed to examine its properties in the sections that follow. Specifically, we will examine the region in the complex plane for which the Laplace transform converges with various assumptions on the time scale in question. In this section, the conclusions of Theorems and are similar to those obtained in [0] by Davis, et al.; however, we have extended and clarified these results by relaxing the assumptions necessary on the time scale. Further, the proofs given here are substantially different than those in [0]. We then work through several results of the Laplace transform when applied to power series and with respect to periodic time scales. We show that in situations involving initial value problems in which the dynamic equation is not regressive on certain time scales, we can still apply the Laplace transform in order to solve these problems. We conclude our examination of some of the properties of the transform by directly calculating several Laplace transforms. Finally, we develop two inversion formulas which are markedly different from the one given by Davis, et al., in [0] as we will use the techniques of contour integration to achieve the results in this

14 4 dissertation. In Chapter 4, we change gears and focus on the transport partial dynamic equation. We will look at the specific case when the time scale is isolated. In this case, we show that the solution to the transport partial dynamic equation with a given initial condition is unique. Further, we develop two different recursive representations of this solution. Then, using these representations, we find explicitly the unique solution for several specific time scales. These particular examples prove to be quite interesting since they depart significantly from the behavior observed in the continuous and discrete cases.

15 5 Chapter 2 Time Scale Preliminaries 2. The Time Scale Calculus A detailed introduction to the time scale calculus is given in [4] and [5]. In this section we collect the definitions and theorems that will be most useful to us. Definition 2... A time scale, denoted T, is a nonempty, closed subset of R. For a, b T such that a < b, we let [a, b] T denote the set [a, b] T. Definition Let T be a time scale. For t T, we define the forward jump operator σ : T T by σt : inf{s T : s > t}, and the backward jump operator ρ : T T is defined by ρt : sup{s T : s < t}. In these definition we use the convention that inf supt and sup inft. For f : T R, we frequently use the notation f σ t for the composition fσt.

16 6 Definition The graininess function µ : T [0, is defined by µt : σt t. The above definitions for the forward and backward jump operators lend to a natural classification of the points in a time scale: Definition Let t T. If σt t and t supt, then t is right-dense. If σt > t, then t is right-scattered. Similarly, if ρt t and t inft, then t is leftdense, and if ρt < t, then t is left-scattered. If a point t T is both right-scattered and left-scattered, we say that t is an isolated point. Definition If a time scale T is composed completely of isolated points, we say that T is an isolated time scale. Definition If supt m such that m is left-scattered, then define T κ : T\{m}; otherwise, define T κ : T. Definition A function f : T R is rd-continuous provided it is continuous at right-dense points in T and its left-sided limits exist and are finite at all left-dense points in T. A function p : T R is regressive provided + µtpt 0 for all t T κ. The set of all regressive and rd-continuous functions on a time scale T is denoted by R RT. A function p : T R is positively regressive provided +µtpt > 0 for all t T κ. The set of all positively regressive and rd-continuous functions is denoted by R + R + T. Throughout, we will use the following abuse of notation: we will write z C R to mean z C and + zµt 0 for all t T. In other words, when z is viewed as a

17 7 constant function from T to C, it is indeed an element of R. However, we will always take C R C. We are now in a position to define the generalization of the classical derivative, the so-called -derivative, as well as the generalization of the classical integral, the -integral, for an arbitrary time scale. Definition Let f : T R and t T κ. If t is a right-scattered point, the -derivative of f is defined to be f t : fσt ft σt t fσ t ft. µt Otherwise, we define it to be f t : lim s t ft fs t s provided this limit exists. Definition The Cauchy -integral of a function f : T R is defined as b a ft t : Fb Fa, where F : T R is an antiderivative of f. Of course, defining the -integral to be antidifferentiation as is done in Definition 2..9 is extremely restrictive. It turns out that the -integral can be developed as a Riemann integral see, for example, [4]. In fact, this development is almost identical to what one would encounter in an introductory analysis course for the classical Riemann integral. Further, the -integral can be developed as a Lebesgue integral. Since this will prove useful to us in this dissertation, some of this development is given

18 8 below starting with Theorem A thorough treatment of building the -integral as a Lebesgue integral can be found in [8] and [3]. Definition The generalized Taylor monomials, h k : T T R for k N 0, are defined recursively as follows: h 0 t, s, and h k+ t, s t s h k τ, s τ, for all s, t T, and k N 0. In the next definition, we define an important function on a time scale: the generalized exponential function, e p t,. Properties of this function will play a pivotal role in obtaining many of the results found in Chapter 3. Definition 2... For p R, the generalized exponential function e p : T T R is defined by t e p t, s exp ξ µτ pτ τ, s for s, t T, where the cylinder transformation, ξ h : C\ { h} C for h > 0, is given by ξ h z Log + zh, h and ξ 0 z z. The proofs of the following theorems can be found in [4]. Theorem Let T be a time scale and T. If p R +, then + t pτ τ e p t, for all t [, T.

19 9 Definition For p, q R, we define circle plus addition, denoted, and circle minus, denoted, as follows: p qt : pt + qt + µtptqt and pt : pt + ptµt, for all t T κ. It should be noted that R, is an abelian group. With and in hand, we are able to obtain several basic properties of the generalized exponential function. The proofs of the next two theorems are given in [4], Theorem 2.36 and Theorem 2.44, respectively. Theorem If p, q R and t, s, r T, then a e 0 t, s and e p t, t, b e p σt, s + µtpte p t, s, c e p s, t e pt,s e pt, s, d e p t, se p s, r e p t, r, and e e p t, se q t, s e p q t, s. Note that d is often referred to as the semigroup property. Theorem Assume p R + and T. Then e p t, > 0 for all t T. Definition For p : T R such that µp 2 R, we define the trigonometric functions by cos p t, e ipt, + e ip t, 2 and sin p t, e ipt, e ip t,. 2i

20 0 Definition For each t [, T, we define T rd t : {s [, t T : σs s}, and T rs t : {s [, t T : σs > s}. Note that if is right-scattered, we take T rs { }, and similarly, if is rightdense, we take T rd { }. So, T rd t is the set of all right-dense points in [, T strictly less than t. For consistency in notation, we will denote the set of all right-dense points in [, T by T rd, and similarly, the set of all right-scattered points in [, T will be denoted by T rs. As noted above, -integration can be realized via the Lebesgue -measure, µ, on T. This measure is briefly introduced in [6, Section 5.7] and fleshed out in detail by Cabada and Vivero in [8]. Hence, we can apply general measure theory results such as the Dominated Convergence Theorem to the -integral. Throughout this work, we would like to evaluate the -integral over [, t T by integrating over the sets T rd t and T rs t, and then adding. This is only legal if T rd t and T rs t are -measurable. Theorem The set of all right-scattered points, T rs, is countable. Proof. For each k Z and n N, define A n,k : { t [, T [k, k + : µt }. n Note that A n,k [k, k + is bounded, and for any distinct s, t A n,k, we have t s n. Therefore, A n,k has only finitely many elements. Hence, for any fixed k Z, A k : n A n,k is countable. Further, by construction, A k is precisely the set

21 of all right-scattered points in [, T [k, k +. Therefore, T rs k A k, which is the countable union of countable sets, and hence is countable. Note that in the proceeding theorem, we proved the set of all right-scattered points in the unbounded set [, T is countable. Cabada and Vivero prove an identical result for a bounded time scale via a different method. They define a monotone increasing function on a real interval that has jump discontinuities precisely at the right-scattered points of T, and hence the set of right-scattered points is countable see [8, Lemma 3.]. Corollary For any t [, T, the sets T rs t and T rd t are -measurable. Further, the sets T rs and T rd are -measurable. Proof. Since for any t [, T, the singleton set {t} is -measurable see [6, Section 5.7], it follows that T rs, a countable union of singletons, is -measurable. Since T rd [, T [T rs ] c, we have that T rd is -measurable. Finally, fix t [, T. Since all intervals of the form [a, b T for a, b T are -measurable, T rd t T rd [, t T and T rs t T rs [, t T are -measurable. Theorem Let E [, T be nonempty such that E is -measurable and contains no right-scattered points. Let f : E R be a -measurable function. Then, ftdµ t ftdmt, E E where m is the usual Lebesgue measure on R. Proof. Let A E be an arbitrary, -measurable set. Since A is -measurable, for each k N 0, we have that each A k : A [ + k, + k + is also -measurable. Since each A k is bounded, +k + A k, and A k contains no right-scattered points,

22 2 by Proposition 3. in [8], it follows that µ A k ma k for all k N 0. Noting that A i A j for i j, we have µ A µ A k k0 µ A k k0 ma k m A k ma. k0 k0 Therefore, since µ A ma for all -measurable A E, it follows from standard measure theory arguments that ftdµ t ftdmt. E E We conclude this section by stating a very useful result from Cabada and Vivero see [8, Theorem 5.2]. We have rephrased the statement slightly to match our notation. Theorem Let E [, T be any bounded, -measurable set, and f : T R be -measurable. If supe E and is a right-scattered point, then take Ẽ E\{supE}. Otherwise, take Ẽ E. Then, ftdµ t ftdmt + E E t Ẽ Trs ftµt. From this point forward, when writing integrals, we will stop explicitly mentioning the measures in question. i.e., ft t denotes integration with respect to the µ E measure, and ftdt denotes integration with respect to the Lebesgue measure. E

23 3 2.2 Properties of the Generalized Exponential Function In addition to the basic properties of the generalized exponential function outlined in the previous section and in Chapter 2 of [4], we will utilize several other properties throughout. The proofs of these four results first appeared in my paper []. In the first lemma, we compute the limit as t goes to infinity of e α x t, where α, x R such that x > α and + αµt > 0 for all t T. Of course, this result is obvious when the time scale in question is T R since, in this case, e α x t, e α xt t0. However, since the constant α x generalizes to a function dependent on t, namely α xt, some care is required in obtaining the desired result on an arbitrary time scale. This lemma and Lemma which relates a generalized exponential involving a complex number z to a generalized exponential involving Rez, the real part of z will be used in tandem to generalize several well-known results about the classical Laplace transform on R. In Lemma 2.2.3, we will give a detailed proof that the generalized exponential function is analytic on the domain Ω : C\{x R : x 0}. Finally, we show that complex conjugation behaves as expected with the generalized exponential function; if p : T R, then e ip t, e ip t,. Lemma Let T be unbounded above. Fix T and let x > α such that α R + be given. Then, lim e x αt, +, and lim e α x t, 0. t t Proof. We first consider the limit of e x α t, as t. Since T is unbounded above, at least one of the following must hold:

24 4 a mt rd +, or b t T rs µt +. Here m is the classical Lebesgue measure on R. Note that x αt x α +αµt Thus, x α R +. By Theorem 2..2, we have > 0, and hence + x αtµt > 0 for all t T. t e x α t, + x ατ τ t 0 x α + T rd t + αµτ τ + + x α dτ + x α T rd t T rst + x αmt rd t + x α x α + αµτ τ µτ + αµτ τ T rst τ T rst by Theorem 2..2 µτ + αµτ. 2. In the above calculation, when applying Theorem 2..2 to the integral over T rs t, we note that sup{t rs t} t and t T rs t by the definition of the set T rs t, and hence the integral reduces to a sum over the entire set T rs t. Case : Assume mt rd + holds. From 2., since every term is nonnegative, we have e x α t, x αmt rd t. Since mt rd, it follows that T rd t as t. Therefore, e x α t, as t. Case 2: Assume t T rs µt + holds. Again from 2., since every term is nonnegative, we have that e x α t, x α τ T rst µτ + αµτ. 2.2 Because t T rs µt, we can choose {t, t 2, } T rs such that

25 5 a t < t 2 <, b lim n t n, and c µt n. n Note that for each t T, T rs t {t, t 2, } is a finite set as sup{t rs t} t and lim n t n. Hence, for each t T where t t, there exists n 0 n 0 t N such that T rs t {t, t 2, } {t, t 2,, t n0 }. Further, by the definition of T rs t, the construction of {t n } n, and our choice of n 0, we have that as t, n 0. Therefore, from 2.2, for t t, e x α t, x α τ T rst k µτ + αµτ n 0 µt k x α + αµt k. 2.3 We aim to show that the series µt k k +αµt k diverges. We will consider two subcases. Subcase : Assume lim n µt n 0. Note that in this case lim n µt n lim + αµt n µt n n + αµt n > 0, so by the Limit Comparison Test, since n µt n diverges, n also diverges. Subcase 2: Assume lim n µt n 0. µt n +αµt n In this case, there exists a subsequence {t nk } k such that µt n k ǫ 0 > 0, for some ǫ 0. Note that if we define fx : x +αx, then f x +αx 2. So, in particular,

26 6 if α 0, then f x > 0 for all x 0,, and if α < 0, then f x > 0 for all x 0, α. If α 0, then + αx 0 for all x [0, and f is continuous and increasing on the entire interval 0,. Therefore, since µt nk ǫ 0, it follows that fµt nk fǫ 0. If α < 0, then f is continuous on 0, α and is increasing on this interval. Now, α R + implies + αµt > 0 which implies µt < for all t T since α < 0. α Therefore, µt [ 0, α for all t T. So again, we have that µtnk ǫ 0 for all k N which implies fµt nk fǫ 0. So, regardless of the value of α, we have that fµt nk fǫ 0. Thus, fµt nk µt nk + αµt nk ǫ 0 + αǫ 0 fǫ 0 > 0, for all k N. Therefore, lim n µt n +αµt n 0 and so n µt n +αµt n diverges. As t, we noted above that n 0 n 0 t. Therefore, n 0 µt k k +αµt k as t. It follows from 2.3 that e x α t, as t. As for the lim t e α x t,, note that using Theorem 2..4c, lim e α xt, lim e x α t, lim t t t e x α t, 0. When working with the standard exponential function defined on the complex numbers, a commonly used identity is e z e x where z x + iy. Unfortunately, this does not necessarily hold for the generalized exponential function on an arbitrary time scale. However, we are guaranteed the following inequality in the case where Rez R + R.

27 7 Lemma Let p : T C such that pt ut + ivt where u, v : T R and u R +. Then, a e p t, e u t,, and b e p t, e u t,, for all t [, T. Proof. First note that p R. To see this, consider for any t T, + ptµt + [ut + ivt]µt [ + utµt] + ivtµt 0, since u R +. Now for any t T, + ptµt , + utµt + vtµt + utµt and hence + ptµt + utµt. Therefore, using the cylinder transform given in Definition 2.., [ ] Re µt Reξ µt pt Log + ptµt, µt > 0, Rept, µt 0 ln + ptµt, µt > 0, µt ut, µt 0 ln + utµt, µt > 0, µt ut, µt 0

28 8 Log + utµt, µt > 0, µt ut, µt 0 ξ µt ut, where the second to last equality follows since u R + ; i.e., + utµt > 0 for all t T. Finally, note that by the definition of the generalized exponential function Definition 2.., we have t e p t, exp ξ µτ pτ τ t 0 t t exp Reξ µτ pτ τ + i Imξ µτ pτ τ t 0 t t exp Reξ µτ pτ τ exp ξ µτ uτ τ e u t,.. e ut, The second statement of the lemma follows immediately as u R + and e u t, In practice, at least in this dissertation, we will apply Lemma by taking z C R with z x + iy and concluding that e z t, e x t,. Lemma Let T be a time scale. Define Ω : C\{x R : x 0} and fix any s, t T. Then, e z t, s is analytic on the domain Ω. Further, d t dz [e τ zt, s] e z t, s s + zµτ, for all z Ω. Proof. Let s, t T be arbitrary. First consider fz : t s ξ µτz τ. We claim that f is analytic on Ω.

29 9 Fix z 0 Ω. Let z 0 x 0 + iy 0. If x 0 > 0, then take R : 2 z 0. If x 0 0, then take R : 2 y 0. Note that since z 0 Ω, if x 0 0, then y 0 0. So, in either case, R > 0. Since Ω is open, there exists 0 < r < R such that B r z 0 Ω. Note that fz fz 0 lim z z 0 z z 0 lim z z0 z z 0 t lim z z0 s [ t ξ µτ z τ s ξ µτ z ξ µτ z 0 τ. z z 0 t s ] ξ µτ z 0 τ We will apply the Dominated Convergence Theorem in order to interchange this integral and limit. Fix τ T. Since the principle logarithm is analytic on Ω and z Ω implies + zµτ Ω, by direct calculation, d dz ξ µτz d dz [ ] Log + zµτ µτ, µτ > 0, d [z], µτ 0 dz, µτ > 0, +zµτ, µτ 0 + zµτ, for any z Ω. Hence, for any fixed τ T, ξ µτ z ξ µτ z 0 lim z z 0 z z 0 d dz ξ µτz zz0 + z 0 µτ. We will now show that ξ µτz ξ µτ z 0 z z 0 is dominated by a -integrable function on

30 20 [s, t] T. If µτ 0, we have that ξ µτ z ξ µτ z 0 z z 0 z z 0 z z Assume µτ > 0. In this case, note that ξ µτ z ξ µτ z 0 z z 0 Log + zµτ Log + z 0 µτ µτz z 0 + zµτ µτz z 0 Log + z 0 µτ µτz z 0 Log + z z 0µτ z 0 µτ We claim that z z 0µτ +z 0 µτ define g : [0, R by < for z B r z 0 and any τ T. To show this, we gx : x + z 0 x x + xx0 2 + xy 0 2, where z 0 x 0 +iy 0. Note that g is continuous on [0, because z 0 Ω which implies +z 0 x 0 for all x [0,. Also, g x +xx 0 +xx 0 2 +xy x x 0 is a possible critical point. We will now consider two cases. Case: Assume x 0 0., and so, provided x 0 0, It follows that g has no critical points in 0,. Further, x 0 0 implies g x > 0 for x [0, ; hence, g is increasing on [0,. Also, note that lim x gx z 0 and g is continuous on [0,. Therefore, in this case, gx z 0 for all x [0,. Thus, for z B r z 0, z z 0 µτ + z 0 µτ z z 0 gµτ z z 0 z 0 < r z 0 < 2,

31 2 by the choice of r. Case: Assume x 0 < 0. Since z 0 Ω, y 0 0. Then x 0 is a critical point in 0, which corresponds to a local maximum. Note that g x 0 x 0 x 0 y 0 y 0. Therefore, in this case, gx y 0 for all x [0,. Thus, for z B r z 0, z z 0 µτ + z 0 µτ z z 0 gµτ z z 0 y 0 < r y 0 < 2, again by the choice of r. Thus, we have shown for any τ T and z B r z 0, z z 0 µτ + z 0 µτ < 2. Thus, for z B r z 0, z z 0µτ +z 0 is in the radius of convergence for the Taylor series µτ expansion of Log + z z 0µτ +z 0. µτ So, from 2.5, we have for z B r z 0, ξ µτ z ξ µτ z 0 z z 0 µτz z 0 Log + z z 0µτ + z 0 µτ k k+ z z0 µτ z z 0 µτ k + + z 0 µτ k0 k+ z z0 µτ z z 0 µτ k + + z 0 µτ k0

32 + z 0 µτ + z 0 µτ + z 0 µτ z z0 µτ k + + z 0 µτ k0 k k + 2 k0 k 2 k0 k z 0 µτ. 2.6 From 2.4 and 2.6, we have shown that for z B r z 0 and any τ T, ξ µτ z ξ µτ z 0 z z z 0 µτ. We will now show that 2 +z 0 µτ is -integrable on [s, t] T. Consider h : [0, R given by hx 2 +z 0 x. Since z 0 Ω, + z 0 x Ω for all x [0,, which implies + z 0 x 0 for all x [0,. Therefore, h is continuous on [0,. So, h satisfies the following properties: a h is continuous on [0,, b h0 2, and c lim x hx 0. Therefore, it must be the case that hx is bounded on [0,. Thus, hµτ 2 +z 0 µτ is bounded for τ T. In particular, 2 +z 0 µτ is bounded on the interval [s, t] T. Since bounded functions on a finite interval are -integrable see [6, Theorem 5.20], 2 +z 0 µτ is -integrable on [s, t] T. Applying the Dominated Convergence Theorem, we have fz fz 0 lim z z 0 z z 0 lim t z z0 s t s ξ µτ z ξ µτ z 0 τ z z 0 ξ µτ z ξ µτ z 0 lim τ z z 0 z z 0 t s τ + z 0 µτ.

33 23 Since z 0 Ω was arbitrary, we have that for any z Ω, the derivative exists and [ d t ] t τ ξ µτ z τ dz s s + zµτ. Finally, since e z is entire, by the chain rule, d dz [e zt, s] d [ exp dz t exp for all z Ω. s t ] ξ µτ z τ s [ d t ξ µτ z τ dz s ] t ξ µτ z τ e z t, s s τ + zµτ, Lemma Let T be any time scale. If p : T R, then e ip t, e ip t,. Proof. Since pt is real-valued, ±iµtpt 0 for t T, and hence ±ip R. Note that for any t T, Log iptµt ln iptµt + i Arg iptµt ln + iptµt + i arctan ptµt ln + iptµt i arctanptµt ln + iptµt i Arg + iptµt Log + iptµt.

34 24 We have used here the fact that arctan is an odd function. So, Log iptµt, µt > 0, µt ξ µt ipt ipt, µt 0 Log + iptµt, µt > 0, µt ipt, µt 0. ξ µt ipt, which implies t t ξ µτ ipτ τ ξ µτ ipτ τ. have Since e z e z, by using the definition of the generalized exponential function, we t t e ip t, exp ξ µτ ipτ τ exp ξ µτ ipτ τ t exp ξ µτ ipτ τ e ip t,.

35 25 Chapter 3 The Generalized Laplace Transform 3. Introduction Throughout, we let T be a time scale that is unbounded above with T fixed. Definition 3... Assume that f : T R is a locally -integrable function. Then the generalized Laplace transform of f is defined by L{f}z : fte σ zt, t for z D{f}, where D{f} C consists of all z C R for which the improper integral exists. Recall that we are using the notation z R C to mean z C and +zµt 0 for all t T. Thus, R C C see the explanation after Definition In Section 3.2 we will discuss the set D{f} C. We will show that for reasonable assumptions on the function f, D{f} is nonempty regardless of the time scale in

36 26 question. Further, if we put some restrictions on the time scale, we can say even more about D{f}. Definition A function f : T R is said to be of exponential order α provided there exist constants M > 0, α R + R such that ft Me α t, for all t [, T. We will always assume that f : T R is a locally -integrable function, sometimes without explicit mention. By making this assumption, we are assured that the only problem we must worry about in the definition of L{f}z is whether or not the improper integral converges for a particular z C. 3.2 Region of Convergence of the Laplace Transform The following theorem appears in [4, Example 3.9]. Here, Bohner and Peterson via direct computation obtain the following result: Theorem Let T be any time scale that is unbounded above. Fix T. Then for z C R and α R, L{e α, }z z α provided lim t e α z t, 0. This limit defines precisely the set of z C R for which the improper integral in the definition of the Laplace transform converges to. In other words, given any z α time scale T that is unbounded above and a fixed T, we have that D{e α, } { } z C R : lim e α z t, 0. t

37 27 Depending on the complexity of the time scale in question, computing the set in the complex plane that this limit describes can be a difficult task. In fact, for a given f : T R and choice of α, it may not be immediately clear that D{f} is nonempty. In this section, we will show that with the standard assumption that f : T R is of exponential order α, the set D{f} is nonempty for any time scale; and further, D{f} contains the right-half plane z C such that Rez > α. If we restrict the time scale in question such that the graininess is bounded below away from 0, we can say even more about D{f}. Finally, if we have a specific time scale in hand, we will show via an example, that we can determine D{f} even more explicitly. Theorem Let T be any time scale that is unbounded above. Fix T. If f : [, T R is of exponential order α where α R + R, then {z C : Rez > α} D{f}. Furthermore, the improper integral in the definition of the Laplace transform converges absolutely for z in this region. Proof. Fix z C such that z x + iy and x > α. Here x, y R. Since α R +, + xµt > + αµt > 0 for any t T. Thus, + zµt + xµt + iyµt 0 and so z R. We aim to show that fte σ z t, t converges. Since x > α and α R +, by Lemma 2.2., lim t e α x t, 0, which implies, by Theorem 3.2., that L{e α, }x exists and equals x α.

38 28 For any L [, T, we have L fte σ z t, t M M M L e α t, e σ z t, t L e α t, e σ x t, t by Lemma e α t, e σ x t, t ML{e α, }x M x α. As L, the real-valued function of L, L fte σ z t, t, is monotone increasing and bounded above, and so it follows that lim L L fte σ z t, t converges. i.e., the desired improper integral converges, and we have that L{f}z converges absolutely. Hence, z D{f}. Example Consider the time scale T R and ft e α t, 0 e αt for some α R. In this case, the Laplace transform is simply the classical Laplace transform. It is well-known that in the classical case, L{e αt }z exists if and only if Rez > α. i.e., we have found a function of exponential order α and a time scale such that {z C : Rez > α} D{f}. Remark In light of Example and Theorem 3.2.2, for an arbitrary time scale and an arbitrary function of exponential order, Theorem is the best we can do. The set {z C : Rez > α} is the largest set not depending on the time scale T for which we can say {z C : Rez > α} D{f}. So, for the generalized Laplace transform, if we restrict our attention to functions

39 29 Figure 3.: The region of convergence guaranteed by Theorem for an arbitrary time scale T. of exponential order, the region of convergence is extremely similar to the classical Laplace transform on R. However, we have shown that we have a half-plane of convergence for any time scale T. This half-plane is shown in Figure 3.. We will now show that if we put some restrictions on the time scale, we can obtain an even larger region in the complex plane independent of T for which the Laplace transform of a function of exponential order converges. Theorem Let T be unbounded above such tha < µ min µt for all t T. If f : [, T R is of exponential order α > 0, then {z C : + zµ min > + αµ min } D{f}.

40 30 Furthermore, the improper integral in the definition of the Laplace transform converges absolutely for z in this region. To prove this theorem, we use the following lemma. Lemma Let T be unbounded above such tha < µ min µt for all t T. Let α > 0 be given. Then, for z C such that + zµ min > + αµ min, there exists a constant δ δα, z, µ min > 0 such that + αµt δ < for all t T. + zµt Proof. Fix z C such that + zµ min > + αµ min. 3. This implies +αµ min +zµ min <. Let z u + iv and define f : [0, R by fx + αx + zx + αx + ux2 + vx 2. Thus, f x α u + xαu u2 + v 2 [ + ux 2 + vx 2 ] 3 2 α Rez + xα Rez z 2 + zx 3. Note that it follows from assumption 3. that + αµ min < + zµ min + z µ min α < z. 3.2

41 3 If α Rez z 2 0, then using 3.2 we have 0 α Rez z 2 < α Rez α 2 αrez α Rez α > 0, where the final implication is obtained by dividing by α > 0. Thus, in this case, f x α Rez + zx 3 < 0, and hence f is strictly decreasing on 0, and f0. We now assume α Rez z 2 0. Thus, the only possible critical point, x 0, of f is given by x 0 Rez α α Rez z We will now consider two cases. Case: Assume Rez α. Then α Rez [Rez] 2 [Rez] 2 + [Imz] 2 z 2 and so α Rez z 2 < 0. We have strict inequality here since z α which follows from assumption 3.. Therefore, x 0 0 and so f does not have a critical point on the interval 0,. Further, it is straightforward to see that f is continuous on 0,, f is strictly decreasing on 0,, and f0. Thus, 0 < µ min implies fµ min <. Further, µ min µt for all t T implies fµ min fµt for all t T. Define δ : fµ min, and so, in this case we have fµt + αµt + zµt + αµ min + zµ min fµ min δ < for all t T. Case: Assume Rez < α.

42 32 If Rez 0, then Imz 0 by assumption 3.. Thus, α Rez z 2 [Imz] 2 < 0. If Rez < 0, then α Rez < 0 α Rez z 2 < 0. If Rez > 0, then from 3.2 we have α Rez < Rez z z 2 α Rez z 2 < 0. So, for Rez < α, α Rez z 2 < 0, and hence x 0 > 0 given by 3.3 is a critical point for f in the interval 0,. Again, it is straightforward to see that f is continuous on 0,, f is strictly increasing on 0, x 0, and f is strictly decreasing on x 0,. Since f0, it follows that for x [0, x 0, fx. Further, lim x fx α z < by 3.2. By assumption 3., fµ min <, and so µ min x 0,. So, in particular, f is decreasing on the interval µ min,. Thus, µ min µt for all t T implies fµ min fµt for all t T. Define δ : fµ min, and so again we have fµt + αµt + zµt + αµ min + zµ min fµ min δ < for all t T. Proof of Theorem Since 0 < µ min µt for all t T, the time scale T is an isolated time scale. Let T {, t, } where < t <. Fix z C such that + zµ min > + αµ min. Note that by the definition of the generalized exponential function, for any n N 0, tn e α z t n, exp exp exp t 0 tn t 0 tn ξ µτ α zτ τ Re[ξ µτ α zτ] τ + αµτ τ µτ ln + zµτ

43 33 n exp ln + αµt i + zµt i0 i n + αµt i + zµt i i0 n δ by Lemma i0 by Theorem 2..2 δ n, where δ <. In an argument that is very similar to that given in Lemma and also given in the proof of Lemma 2.2.3, it follows that there exists a constant C Cz > 0 µt such that C for all t [, T. Thus, for any k N 0, we have +zµt tk tk fte σ z t, t M e α t, e σ z t, t t 0 tk M + zµt e α zt, t k M µt i + zµt i e α zt i, by Theorem 2..2 i0 k MC δ i MC i0 i0 δ i MC δ. As k, the real-valued sequence t k fte σ z t, t is monotone increasing and bounded above, thus lim k tk fte σ z t, t converges. i.e., the desired improper integral converges, and we have that L{f}z converges absolutely. Hence, z D{f}. Example Let T hz, h > 0, and ft e α t, 0 for some α > 0. On this time scale, it is straightforward to see that e α z t, 0 t +αh h. Using Theorem +zh

44 , we have L{e α, 0}z 0 t0 e α t, 0e σ z t, 0 t h + zh e α zt, 0 h + zh t0 t + αh h. + zh Thus, the Laplace transform converges if and only if +αh < if and only if + αh < + zh. Therefore, in conjunction with Theorem 3.2.5, we can conclude {z C : + zh > + αh} D{f}. +zh Remark Similar to what we saw before, because of Example and Theorem 3.2.5, for a time scale such tha < µ min µt for all t T and an arbitrary function of exponential order α, Theorem is the best we can do. The set {z C : + zµ min > + αµ min } is the largest set not depending on the time scale T for which we can say {z C : + zµ min > + αµ min } D{f}. So, for the generalized Laplace transform, if we restrict the graininess of our time scale as indicated and only focus on functions of exponential order α, the region of convergence is extremely similar to the classical Z-transform from difference equations see, for example, [2, Section 6.]. This region of convergence for the generalized Laplace transform is shown in Figure 3.2. Note that this region is slightly different from what we expect for the Z-transform from difference equations. For the Z-transform, the transform converges for all points outside a ball centered at the origin. However here, we have convergence for all points outside a ball centered at µ min. The reason for this is that the generalized Laplace

45 35 Figure 3.2: The region of convergence guaranteed by Theorem for a time scale such tha < µ min µt for all t T. transform on Z is a shift of the classical Z-transform. In other words, z + L{f}z Z{f}z +. So far we have shown that for any time scale, the region of convergence of the generalized Laplace transform behaves very similar to that of the classical Laplace transform on R. We have also shown that by bounding the graininess below away from zero, the region of convergence of the generalized Laplace transform behaves very similar to that of the classical Z-transform on Z. We conclude this section with an interesting example that shows that when we take a time scale that is not either of the classical cases R or Z, then we find that the region of convergence of the

46 36 generalized Laplace transform behaves in a substantially different way than the two classical cases. Example In this example we will consider the function ft e α t,, for some α > 0, on the time scale defined by T : {0, γ, γ +δ, 2γ +δ, 2γ +2δ, }, where γ > 0 and δ > 0. We will enumerate T {, t, t 2, }. 0 γ γ + δ 2γ + δ 2γ + 2δ 3γ + 2δ It is straightforward to see that on this time scale µt n γ if n is even, µt n δ if n is odd, and e α z t n, +αδ +zδ +αδ +zδ n 2 n+ 2 +αγ +zγ n 2 +αγ +zγ, n even, n 2, n odd. Thus, L{e α, }z n0 n even e α t, e σ z t, t µt n + zµt n0 n e α zt n, by Theorem 2..2 n γ n + αδ αγ + zγ + zδ + zγ γ + zγ + n0 n odd δ + zδ n+ + αδ 2 + zδ k + αδ + αγ + zδ + zγ k0 δ + αδ + + zδ + zδ k0 n + αγ 2 + zγ k k+ + αγ + zγ k

47 37 [ γ + zγ + δ + αδ + zδ 2 ] k0 [ + αδ + zδ ] k + αγ. + zγ Noting that we have ended up with a geometric series, it follows that the Laplace transform of e α t, converges on this time scale if and only if +αδ +αγ +zδ +zγ <. This inequality defines the set of z C for which the generalized Laplace transform converges, D{e α t, }, on this particular time scale. If we fix α 4 and γ 2, Figures 3.3, 3.4, 3.5, and 3.6 show how the region of convergence varies as we modify the time scale by adjusting the value of δ. 3.3 Power Series on Time Scales From the study of the classical Laplace transform on R, it is known that we are not always able to expand a function via its power series representation, take the Laplace transform term-by-term, and obtain a representation of the Laplace transform of the original function see, for example, [9, Example.7]. The main objective of this section is to give conditions on when we can take the generalized Laplace transform of a power series representation of a function f : T R term-by-term. As in Bohner and Guseinov s paper [3], we will define F to be the set of all functions f : [, T C of the form ft c n h n t,, t [, T, 3.4 n0 where c n Kα n for all n N 0. Here K > 0, α > 0 are constants independent of n. Also note that h n t, is the generalized monomial found in Definition Bohner and Guseinov prove in [3], that for f F, the series in 3.4 converges uniformly on any compact interval [a, b] T where b > a. They also show for z C R,

48 38 Figure 3.3: The region of convergence guaranteed by Example with γ 2 and δ 3 5. Figure 3.5: The region of convergence guaranteed by Example with γ 2 and δ 2. Figure 3.4: The region of convergence guaranteed by Example with γ 2 and δ 6. Figure 3.6: The region of convergence guaranteed by Example with γ 2 and δ 8. Note that Theorem guarantees the half-plane of convergence to the right of the vertical line in these figures. Similarly, since in all four cases, the time scale satisfies 0 < 2 µt, Theorem guarantees convergence outside of a ball of radius 9 4 centered at 2 indicated by the circle in these figures.

49 39 the generalized exponential function is given by the expansion e z t, Further, Bohner and Guseinov define D n : z n h n t,. n0 { } z C R : lim h n t, e z t, 0, t and D : n0 D n. This definition relies on the fact that, via direct calculation, L{h k t, }z z k+ provided lim t h k t, e z t, 0 see [4, Theorem 3.90]. Hence, D describes the set in the complex plane for which the Laplace transform converges when applied to each term of the expansion of the function f. In [3], it is assumed that D is nonempty. However, after the work done in Section 3.2, we merely need to show that for each n N 0, h n t, is of exponential order α for some α > 0 in order to find some sets that are contained in D regardless of the time scale. Theorem Let T be unbounded above, and fix T. For any n N 0 and any α > 0, the generalized monomial h n t, is of exponential order α. Proof. Fix n N 0 and α > 0. We first claim lim t h n t, e α t, 0. To see this, we will apply a version of L Hôpital s Rule for time scales see [4, Theorem.20]: Assume f and g are -differentiable on T with lim t gt. Suppose gt > 0, g t > 0 for all t [, T. Then lim t f t g t r R implies lim t ft gt r. Using the definition of h n t,, note that h n t, 0 for all t [, T. Also, since α > 0, it follows that + αµt > 0 for all t [, T, and so e α t, > 0 for

50 40 all t [, T by Theorem Thus, for t [, T, 0 h n t, e α t, h nt, e α t,. 3.5 Since α > 0, lim t e α t, by Lemma Further, since e α t, > 0, we have e α t, αe α t, > 0. Continuing to take -derivatives shows that for every 0 k n, e k α t, αe k t, α k e α t, > 0. Therefore, α h n e n n t, h n α t, n t, αe n α t, α n e α t,. Hence, by n applications of L Hôpital s Rule, we have h n t, lim t e α t, lim t So, by 3.5, it follows that h n e n n t, α t, α lim n t e α t, 0. lim h nt, e α t, 0. t In other words, there exists a T [, T such that for all t [T, T, we have h n t, e α t, <. Define M : max { sup t [,T] T h n t, e α t,, }. Then, h n t, e α t, M for all t [, T. Therefore, h n t, Me α t, for all t [, T. Using this theorem, we can now show that for any time scale, {z C : Rez > 0} D.

51 4 Corollary For any time scale T, if z C such that Rez > 0, then z D. Proof. Let z C be such that Rez > 0. Since Rez > 0, there exists a constant α such tha < α < Rez. So for any n N 0, h n t, is of exponential order α, and by Theorem 3.2.2, z D n. This holds for any n N 0 ; therefore, z n0 D n D. We are now set to prove the main result of this section: Theorem Assume f : T R is such that there exist constants K > 0, α > 0 with ft n0 c nh n t, for all t [, T where c n Kα n for all n N 0. i.e., f F. Then, L{f}z c n L{h n t, }z n0 for all z C such that Rez > α. n0 c n z n+, Proof. By definition of the generalized monomial, for each n N 0, h n t, 0 for all t [, T. Let z C such that z x + iy and x > α. Let ǫ > 0 be given. Since x > α, we have that the geometric series α n x n0 x. Hence, x x α there exists M N such that for all N M, Now, for N M, we have x α x N α n ǫ < x K. n0 α x N { } L{f}z c n L{h n t, }z N L ft c n h n t, z n0 n0 { } L c n h n t, z nn+

52 42 K K K K nn+ nn+ e σ c n h n t, zt, t α n h n t, α n h n t, K n0 e α t, e σ xt, t N α n h n t, e σ xt, t n0 N α n h n t, e σ x t, t n0 e α t, e σ x t, t N α n h n t, e σ xt, t. n0 We have used Lemma to obtain the second inequality above. Since x > α > 0, and e α t, is of exponential order α, it follows that x D{e α t, } by Theorem Thus, L{e α, }x exists and equals x α. Similarly, since x > 0, and h n t, is of exponential order α for any n N 0, it follows that x D by Corollary Thus, L{h n, }x exists and equals for each n N 0 by [4, Theorem 3.90]. So, continuing our calculation, we have x n+ KL{e α, }x K N α n L{h n, }x n0 [ N ] K x α α n x n+ n0 [ ] K x α N α n x x < ǫ. n0 Therefore, since ǫ > 0 was arbitrary, L{f}z n0 c nl{h n t, }z. Finally,

53 43 by [4, Theorem 3.90], we have L{f}z c n L{h n t, }z n0 n0 c n z n Basic Properties of the Laplace Transform In this section, we collect some basic properties of the generalized Laplace transform on time scales. From Theorem 3.2.2, if f : T R is of exponential order α, the Laplace transform L{f}z converges absolutely for z C such that Rez > α where α R + R. In fact, the proof gives the bound L{f}z Thus, two immediate corollaries are as follows: fte σ z t, t M Rez α. Corollary Let f : T R be of exponential order α, and take x 0 > α where α R + R. Then, the Laplace transform L{f}z is uniformly bounded on {z C : Rez x 0 }. Proof. For any z x + iy such that x x 0 > α, we have that L{f}z M M x 0. Here, the bound α x 0 α is independent of z. M x α Corollary Let f : T R be of exponential order α where α R + R. Then, lim L{f}z 0. Rez

54 44 It turns out that the Laplace transform satisfies an even stronger form of convergence than absolute convergence. In the next theorem, we show that the Laplace transform converges uniformly on a closed half-plane. Theorem Let f : T R be of exponential order α, and take x 0 > α > 0. Then, the Laplace transform L{f}z converges uniformly on {z C : Rez x 0 }. Proof. First note that for x 0,, e α x t, is decreasing with respect to x. To see this, let x > 0 be given. Then x Ω as defined by Lemma Thus, using this lemma, we have d dx [e α xt, ] d dx [ ] eα t, e x t, t τ e α t, e x t, + xµτ t τ e α x t, + xµτ. Since + α xtµt + α x + αµt µt + xµt + xµt > 0 on [, T, by Theorem 2..5, e α x t, > 0 for all t [, T. Further, since x > 0, t τ > 0 for all t t d +xµτ 0, T. Therefore, [e dx α xt, ] < 0 for all x > 0. It follows that for any x x 0, e α x t, e α x0 t,. Let ǫ > 0 be given. Let z x + iy such that x x 0 be given. By Lemma 2.2., lim t e α x0 t, 0. Hence, there exists T T, sufficiently large, such that for all t [T, T, e α x0 t, < x 0 αǫ M.