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Fractional Cauchy Problem with Caputo Nabla Derivative on Time Scales
The definition of Caputo fractional derivative is given and some of its properties are discussed in detail. After then, the existence of the solution and the dependency of the solution upon the initial value for Cauchy type problem with fractional Caputo nabla derivative are studied. Also the explicit solutions to homogeneous equations and nonhomogeneous equations are derived by using Laplace transform method.
Fractional differential equation theory has gained considerable popularity and importance due to their numerous applications in many fields of science and engineering including physics, population dynamics, chemical technology, biotechnology, aerodynamics, electrodynamics of complex medium, polymer rheology, control of dynamical systems, and so on (see, e.g., [1–4], and the references therein). On the other hand, in real applications, it is not always continuous case, but also discrete case. For example, in recent papers [5–8], in order to deeply understand the background of the discrete dynamics behaviors, some interesting results are obtained by applying the discrete fractional calculus to discrete chaos behaviors. In [9–12], the delta type discrete fractional calculus is studied. In [13, 14], the nabla type discrete fractional calculus is studied. In , the theory of fractional backward difference equations (i.e., the nabla type fractional difference equations) has been studied in detail. So how to unify continuous fractional calculus and discrete fractional calculus is a natural problem. In order to unify differential equations and difference equations, Hilger  proposed firstly the time scale and then some relevant basic theories are studied by some authors (see [17–22]). Recently, some authors studied fractional calculus on time scales (see [23–25]), where Williams  gives a definition of fractional integral and derivative on time scales to unify three cases of specific time scales, which improved the results in . Bastos gives definition of fractional -integral and -derivative on time scales in . The delta fractional calculus and Laplace transform on some specific discrete time scales are also discussed in [26–28]. In the light of the above work, we further studied the theory of fractional integral and derivative on general time scales in , where -Laplace transform, fractional -power function, -Mittag-Leffler function, fractional -integrals, and fractional -differential on time scales are defined. Some of their properties are discussed in detail. After then, by using Laplace transform method, the existence of the solution and the dependency of the solution upon the initial value for Cauchy type problem with Riemann-Liouville fractional -derivative are studied. Also the explicit solutions to homogeneous equations and nonhomogeneous equations are derived by using Laplace transform method. But there is a shortcoming for Riemann-Liouville fractional -derivative. That is, Cauchy type problem with Riemann-Liouville fractional order derivative and the Laplace transform of Riemann-Liouville fractional order derivative require the initial conditions in terms of non-integer derivatives, which are very difficult to be interpreted from the physical point of view. Thus this paper’s focus on defining nabla type Caputo fractional derivative on time scales proves some useful property about Caputo fractional derivative and then studies some Caputo fractional differential equations on time scales.
The structure of this paper is as follows. In Section 2, we give some preliminaries about time scales, generalized -power function, and Riemann-Liouville -integral and -derivative. In Section 3, we present the definitions and the properties of the Caputo nabla derivative on time scales in detail. Then in Section 4, Cauchy type problem with Caputo fractional derivative is discussed. For the Caputo fractional differential initial value problem, we discuss the dependency of the solution upon the initial value. In Section 5, by applying the Laplace transform method, we study the fractional order linear differential equations with Caputo fractional derivative. We derive explicit solutions and fundamental system of solutions to homogeneous equations with constant coefficients and find particular solution and general solutions of the corresponding nonhomogeneous equations.
First, we present some preliminaries about time scales in .
Definition 1 (see ). A time scale is a nonempty closed subset of the real numbers.
Definition 2 (see ). For one defines the forward jump operator by while the backward jump operator is defined by If , we say that is right-scattered, while if , we say that is left-scattered. Points that are right-scattered and left-scattered at the same time are called isolated. Also, if and , then is called right-dense, and if and , then is called left-dense. Finally, the graininess function is defined by
Definition 3 (see ). If has a right-scattered minimum , then one defines ; otherwise . Assume is a function and let . Then one defines to be the number (provided it exists) with the property that given any , there is a neighborhood of (i.e., for some ) such that We call the nabla derivative of at .
Definition 4 (see ). A function is called regulated provided its right-sided limits exist (finite) at all right-dense points in and its left-sided limits exist (finite) at all left-dense points in . A function is called ld-continuous provided it is continuous at left-dense points in and its right-sided limits exist (finite) at right-dense points in .
Definition 5 (see [17, page 100]). The generalized nabla type polynomials are the functions , , defined recursively as follows. The function is and given for , the function is
Definition 6 (see [18, page 38]). The generalized delta type polynomials are the functions , , defined recursively as follows. The function is and given for , the function is
It is similar to the discussion in the reference [17, (page 103)] for and ld-continuous functions , , we consider the th order linear dynamic equation
Remark 8 (see ). Note that is the Cauchy function for .
Theorem 10 (see  (Taylor’s Formula)). Let . Suppose the function is such that is ld-continuous on . Let , . Then one has
Definition 11 (see ). A subset is called a time scale interval, if it is of the form for some real interval . For a time scale interval , a function is said to be left-dense absolutely continuous if for all there exist such that whenever a disjoint finite collection of subtime scale intervals for satisfies . One denotes . If , then one denotes .
Theorem 12 (see ). Let be a normed linear space, a convex set, and open in with . Let be a continuous and compact mapping. Then either(i)the mapping has a fixed point in , or(ii)there exists and with .
The following results can be found in our recent paper .
Lemma 13 (see ). Let be a measurable set. If is integrable on , then
From now on, let be a time scale such that and fix .
Definition 14 (see ). Assume that is regulated and . Then the Laplace transform of is defined by for , where consists of all complex numbers for which the improper integral exists.
Theorem 15 (see ). Assume that is such that is regulated. Thenfor those regressive satisfying , .
Definition 16 (see ). One defines fractional generalized -power function on time scales to those regressive , ; and for , .
Here we introduce generalized -derivative on time scales: Since is integral, we can consider it as a generalized function, and thus we can define for , where here means a generalized derivative. In the same way, we can define for .
For , we have
Definition 17 (see ). For a given , the solution of the shifting problem is denoted by and is called the shift of .
Definition 19 (see ). Fractional generalized -power function on time scales is defined as the shift of ; that is,
In this paper, we always denote a finite interval on a time scale .
Definition 20 (see ). Let . The Riemann-Liouville fractional -integral of order is defined by
Definition 21 (see ). Let . The Riemann-Liouville fractional -derivative of order is defined by
Throughout this paper, we denote , .
Property 1 (see ). Let . Then
Property 2 (see ). If and , then the equationis satisfied at almost every point for .
Property 3 (see ). If and , then the following equalityholds almost everywhere on .
Property 4 (see ). If , then, for , the relationholds almost everywhere on . In particular, when and , then
Property 5 (see ). Let , and let . If and , then If and , then the equality holds almost everywhere on , where .
Lemma 22 (see ). Let , and . For with . Then one has the following.
If , then If , thenfor those regressive satisfying , .
Definition 23 (see ). -Mittag-Leffler function is defined asprovided the right-hand series is convergent, where , .
Theorem 24 (see ). The Laplace transform of -Mittag-Leffler function is
By differentiating times with respect to on both sides of the formula in the theorem above, we get the following result:
3. Definition and Properties of Caputo Fractional Derivative on Time Scales
Definition 25. Let . The Caputo fractional derivative of order is defined via Riemann-Liouville fractional derivative bywhere
In particular, when , the relation (38) takes the following forms:
If , then the Caputo fractional derivative coincides with the Riemann-Liouville fractional derivative in the following case:if .
In particular, when , we have
If and the usual nabla derivative of order exists, then coincides with :
The Caputo fractional derivative is defined for functions for which the Riemann-Liouville fractional derivative of the right-hand sides of (38) exists. In particular, they are defined for belonging to the space of absolutely continuous functions defined in Definition 11. Thus the following statement holds.
Property 6. Let and let be given by (39). If , then the Caputo fractional derivative exists almost everywhere on .(a)If , is represented bywhere . Thus when , , where the notation denote the limit of as .
In particular, when and ,(b)If , then is represented by (43). In particular,
Property 7. Let and let be given by (39), , . ThenIn particular,
Property 8. Let and let or . Then
Property 9. Let and let be given by (39). If , thenIn particular, if and , then
Property 10. Assume that and . Then, for all ,for all .
Proof. For each , by Property 6, Noting that and according to Property 6, we have Thus (63) holds.
Now, for all with , we haveIn fact, from Property 6, we can get . Since , then by (41), (29), and Property 6Similarly, we have . Thus (67) holds. Then, by using (63) and (67), we have that That is, (64) holds. The results follow.
The next assertion yields the Laplace transform of the Caputo fractional nabla derivative.
Property 11. Let , be such that . Thenfor those regressive satisfying .
In particular, if , thenfor those regressive satisfying .
Remark 26. (1) For Riemann-Liouville fractional derivative, while for the Caputo fractional derivative, which shows that the Caputo fractional derivative is more near to the usual sense derivative than Riemann-Liouville fractional derivative.
(2) Comparing (34) and (70), we know that the Laplace transform of the Caputo fractional derivative involves only initial value with integer order derivative, such as , while the Laplace transform of the Riemann-Liouville fractional derivative is related to initial value with fractional order derivative which is difficult to understand the physics background, such as . Thus, the Caputo fractional derivative is used more widely in realistic applications.
4. The Cauchy Problem with Caputo Fractional Derivative
4.1. Existence and Uniqueness of the Solution to the Cauchy Type Problem
In this section we consider the nonlinear differential equation of order :involving the Caputo fractional derivative , defined in (38), with the initial conditionsWe give the conditions for a unique solution to this problem in the space . Our investigations are based on reducing the problem (76)-(77) to the integral equationFirst we establish an equivalence between the problem (76)-(77) and the integral equation (78).
Theorem 27. Let and let be given by (39). Let be an open set in and let be a function such that, for any . If , then satisfies the relation (76)-(77) if and only if satisfies the Volterra integral equation (78).
Proof. First we prove the necessity. Let be the solution to the Cauchy problem (76)-(77). Applying the operator to (76) and taking into account and (77), we arrive at the integral equation (78) since .
Inversely, if satisfies (78), for , applying the operator to both sides of (78) and taking into account (51) and (55), we have In addition, by term-by-term differentiation of (78) and using (51), we havefor . Thus we obtain relations in (77) by letting in (81).
Theorem 28. Let and let be given by (39). Let be an open set in and a function such that, for any . Let satisfies the Lipschitzian condition (82), and . Then there exists a unique solution to initial value problem (76)-(77).
Proof. Since the Cauchy type problem (76)-(77) and the nonlinear Volterra integral equation (78) are equivalent, we only need to prove there exists a unique solution to (78).
We define function sequences: where To simplify our proof, without loss of generality, we assume that is large enough such that .
We obtain by inductive method that In fact, for , since , we have If then According to and by Weierstrass discriminance, we obtain convergent uniformly and the limit is the solution. Thus we prove the existence of solution.
Next we will show the uniqueness. Assume is another solution to (78); that is, Since we have If then By mathematical induction, we have and then we get that Thus, , and then we have owing to the uniqueness of the limit. The result follows.
In the following, we consider generalized Cauchy type problems:
Theorem 29. Let be an open set and let be a function such that, for any , . If , then satisfies (97) if and only if satisfies the integral equation
Suppose that satisfies generalized Lipschitzian condition:
According to the theorem above and the proof of Theorem 28, we have the following theorem.
4.2. The Dependency of the Solution upon the Initial Value
We consider Caputo fractional differential initial value problem again:where .
Using Theorem 27, we have where Suppose is the solution to the initial value problem:We denote . We can derive the dependency of the solution upon the initial value.
Proof. By the proof of Theorem 28, we know that , where Using the Lipschitz condition, we have