Fractional Cauchy Problem with Caputo Nabla Derivative on Time Scales

Abstract

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.

1. Introduction

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 [15], 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 [16] 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 [24] gives a definition of fractional integral and derivative on time scales to unify three cases of specific time scales, which improved the results in [23]. Bastos gives definition of fractional -integral and -derivative on time scales in [25]. 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 [29], 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.

2. Preliminaries

First, we present some preliminaries about time scales in [17].

Definition 1 (see [17]). A time scale is a nonempty closed subset of the real numbers.

Definition 2 (see [17]). 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 [17]). 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 [17]). 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

Definition 7 (see [17]). One defines the Cauchy function for the linear dynamic equation (9) to be for each fixed the solution of the initial value problem

Remark 8 (see [17]). Note that is the Cauchy function for .

Theorem 9 (see [17] (variation of constants)). Let and . If , then the solution of the initial value problem is given by where is the Cauchy function for (9).

Theorem 10 (see [17] (Taylor’s Formula)). Let . Suppose the function is such that is ld-continuous on . Let , . Then one has

Definition 11 (see [24]). 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 [4]). 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 [29].

Lemma 13 (see [29]). 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 [29]). 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 [29]). Assume that is such that is regulated. Then for those regressive satisfying , .

Definition 16 (see [29]). 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 [29]). For a given , the solution of the shifting problem is denoted by and is called the shift of .

Definition 18 (see [29]). For given functions , their convolution is defined by where is the shift of , which is introduced in Definition 17.

Definition 19 (see [29]). 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 [29]). Let . The Riemann-Liouville fractional -integral of order is defined by

Definition 21 (see [29]). Let . The Riemann-Liouville fractional -derivative of order is defined by

Throughout this paper, we denote , .

Property 1 (see [29]). Let . Then

Property 2 (see [29]). If and , then the equation is satisfied at almost every point for .

Property 3 (see [29]). If and , then the following equality holds almost everywhere on .

Property 4 (see [29]). If , then, for , the relation holds almost everywhere on . In particular, when and , then

Property 5 (see [29]). Let , and let .  If and , then   If and , then the equality  holds almost everywhere on , where .

Lemma 22 (see [29]). Let , and . For with . Then one has the following.
If , then If , then for those regressive satisfying , .

Definition 23 (see [29]). -Mittag-Leffler function is defined as provided the right-hand series is convergent, where , .

Theorem 24 (see [29]). 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 by where

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 by where . Thus when , , where the notation denote the limit of   as .
In particular, when and , (b)If , then is represented by (43). In particular,

Proof. (a) By Taylor’s formula on time scales and using (29), we have (b) If , then (38) takes the form and, from Taylor’s formula and (28), we derive .

Property 7. Let and let be given by (39), , . Then In particular,

Proof. From Property 6 and (26), it is obtained that for , while for ,

Property 8. Let and let or . Then

Proof. Let , and let and . Since , then for a.e. (for any) , we get for any , and hence Thus using (41) for with replaced by and (28), we derive For ,

Property 9. Let and let be given by (39). If , then In particular, if and , then

Proof. Let . If , then using Property 6, (27) and (32), we have For , the result is obvious from Property 6 and (32).

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 have In fact, from Property 6, we can get . Since , then by (41), (29), and Property 6 Similarly, 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 . Then for those regressive satisfying .
In particular, if , then for those regressive satisfying .

Proof. By Property 6, (33), and (17), for , we have and for , we have The result follows.

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 conditions We 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 equation First 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 have for . Thus we obtain relations in (77) by letting in (81).

In the following, we bring into Lipschitzian-type condition: where does not depend on . We will derive a unique solution to the Cauchy problem (76)-(77).

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.

Theorem 30. Let the condition of Theorem 29 be valid. If satisfies Lipschitzian condition (99) and holds, then there exists a unique solution to initial value problem (97).

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.

Theorem 31. Let be the solutions to (100) and (103), respectively, and let , . Suppose satisfies the Lipschitz condition; that is, Then we have

Proof. By the proof of Theorem 28, we know that , where Using the Lipschitz condition, we have Suppose then According to mathematical induction, we have Taking the limit , we obtain that and the proof is completed.

4.3. Initial Value Problems for Nonlinear Term Containing Fractional Derivative

In this section, we are interested in the nonlinear differential equation of fractional order , where , , and , with the initial conditions We obtain the existence of at least one solution for integral equations using the Leray-Schauder Nonlinear Alternative for several types of initial value problems and establish sufficient conditions for unique solutions using the Banach contraction principle.

Our objective is to find solutions to the initial value problem (112) and (113) in the space . There are two cases to investigate: and .

Throughout this section, we suppose that the following are satisfied: is a ld-continuously and nabla differentiable function;there exist nonnegative functions such that ; and on a compact subinterval of .

The following shows that the solvability of the initial value problem (112) and (113) is equivalent to that of the Volterra-type integral equation (115) in the space .

Lemma 32. Let and assume that and hold. A function is a solution of the initial value problem (112) and (113) if and only if where is a solution of the integral equation

Proof. By (63), we have By using Property 6, , thus we have Let , by using Theorem 27, we obtain As and by (113), the above equation transforms into (115). An application of Definition 7 and Theorem 9 yields (114) in view of and (113).
To prove the converse, let be a solution of (115). Since , the function is ld-continuous on and so is We have Since , by so is a solution of (112) in view of . By absolute continuity of the integral, differentiating (115), we obtain for each . Thus, , ; that is, , . On the other hand, from (114), and thus . Also it is easy to see that .

For the sake of brevity, by , we denote the first term in the right-hand side of (115).

Theorem 33. Suppose that hold. Then the integral equation (115) has a solution in provided

Proof. In the normed space with the sup-norm , we define the mapping by for all . Indeed, one can easily verify that the mapping is well defined and .
Let with Let be defined by .
Let ; that is, . Then which shows that .
In addition, where , , .
Thus, is equicontinuous on . This shows that is a compact mapping.
Consider the eigenvalue problem Under the assumption that is a solution of (131) for a , one obtains which shows that . By Theorem 12, has a fixed point in , which we denote by , such that .

It follows from Lemma 32 that is a solution of (112) and (113).

In the following, we will discuss another case: .

Lemma 34. Let and suppose that and hold. A function is a solution of the initial value problem (112) and (113) if and only if where is a solution of

Proof. Let be a solution of the which, after the substitution , becomes Next, by Property 9 and (113) By Property 6, we have that , and thus the above equation becomes (135).
Conversely, let be a solution of the integral equation (135); that is, Then, by , and . Hence and we obtain (112). Also, it follows from (140) that and (113) are satisfied since, for ,

Our next existence result corresponds to the case .

Theorem 35. Suppose that are satisfied. Then the integral equation (135) has a solution in provided

Proof. We endow with the sup-norm and define, for , the mapping by The mapping is well defined and . Let with Let be defined by .
If , then that is, . Certainly, is continuous and compact. Consider The rest of the proof is the same as the corresponding part of the proof of Theorem 30.

The uniqueness results are based on applications of the Banach contraction principle.

The main assumption in the existence theorems below is that  for each , there exists a nonnegative function such that , , .

The first uniqueness result is for the case .

Theorem 36. Suppose that , and hold. Assume that Then the integral equation (135) has a unique solution.

Proof. In the Banach space we define by where We define the mapping as in the proof of Theorem 31.
If , then that is, .
Let . Then that is, is a contraction since .
By the Banach contraction principle, has a unique fixed point, which is a solution of the integral equation (135).

For the case , the uniqueness result is given without proof.

Theorem 37. Suppose that , and hold and assume that Assume further that Then the integral equation (115) has a unique solution.

5. Laplace Transform Method for Solving Ordinary Differential Equations with Caputo Fractional Derivatives

5.1. Homogeneous Equations with Constant Coefficients

In this section we apply the Laplace transform method to derive explicit solutions to homogeneous equations of the form involving the Caputo fractional derivatives , with real constants and .

The Laplace transform method is based on the formula:

First, we derive explicit solutions to (157) with :

In order to prove our result, we also need the following definition and lemma.

Definition 38. The function is defined by

Lemma 39. The solutions are linearly independent if and only if at some point .

Proof. We first prove sufficiency. If, to the contrary, are linearly dependent in , then there exist constants , not all zero, such that holds, and thus, which leads to a contradiction. Therefore, if at some point , then are linearly independent. Now we prove the necessity. Suppose, to the contrary, for any , . Consider the equations where , . As , the equations have nontrivial solution . Now we construct a function using these constants: and we get as a solution. From (163), we obtain that satisfies initial value condition However, is also a solution to equation satisfying the initial value condition. By the uniqueness of solution, we have and thus, are linearly dependant which leads to a contradiction. Thus, if the solutions are linearly independent, then at some point . The result follows.

The following statements hold.

Theorem 40. Let and . Then the functions yield the fundamental system of solutions to (160).

Proof. Applying the Laplace transform to (160) and taking (158) into account, we have where are given by (159).
Formula (36) with yields Thus, from (168), we derive the following solution to (160): It is easily verified that the functions are solutions to (160): In fact, Moreover, It follows from (173) and (20) that If , then and, since for any , the following relations hold: By (174) and (176), the Wronskian function at is equal to 1: . Then yield the fundamental system of solutions to (160).

Corollary 41. The equation has its solution given by while the equation has the fundamental system of solutions given by

Next we derive the explicit solutions to (157) with : with .

Theorem 42. Let , , and . Then the functions yield the fundamental system of solutions to (182), provided that the series in (183) and (184) are convergent.

Proof. Let . Applying the Laplace transform to (182) and using (158), we obtain where are given by (159).
For and , we have In addition, for and , we have From (185) and (187), we derive the solution to (182): where are given by (183) for and by (184) for . For , the direct evaluation yields For , and for . Thus we have . Thus the functions in (183) and (184) are linearly independent solutions to (182). The result follows.

Corollary 43. The equation has its fundamental system of solutions given by

Corollary 44. The equation has its solution by In particular, is a solution to the equation

Corollary 45. The equation where ; , ; , has one solution , given by (193), and a second solution given by for , while, for , by In particular, the equation has one solution given by (194), and a second given by for , while for , by

Finally, we find explicit solutions to (157) with any . It is convenient to rewrite (157) in the form (202)

Theorem 46. Let and be such that and let . Then the fundamental system of solutions to (202) is given by the formulas for ; by for ; and by for .

Proof. Applying the Laplace transform to (202) and using (158), we obtain where are given by (159).
For and , we have if we also take into account the following relation: where the summation is taken over all such that .
In addition, for and , we have From (210) to (212), we derive the solution to (202): where are given by (204) for , by (205) for , and by (206) for . For , the direct evaluation yields for , for , and for . For , and for . Thus we have . Thus the functions in (204)–(206) are linearly independent solutions to (202). The result follows.

5.2. Nonhomogeneous Equations with Constant Coefficients

In this section, we still use Laplace transform method to find general solutions to the corresponding nonhomogeneous equations with real coefficients and a given function . The general solution to (217) is a sum of its particular solution and of the general solution to the corresponding homogeneous equation (157). It is sufficient to construct a particular solution to (217).

By (158) and (159), for suitable functions , the Laplace transform of is given by Applying the Laplace transform to (217) and taking (218) into account, we have Using the inverse Laplace transform from here we obtain a particular solution to (217) in the form

Using the Laplace convolution formula we can introduce the Laplace fractional analog of the Green function as follows: and express a particular solution of (217) in the form of the Laplace convolution and

Theorem 47. Let , , and be a given function. Then the equation is solvable, and its general solution is given by where are arbitrary real constants.
In particular, the general solutions to (224) with and have the forms respectively, where and are arbitrary real constants.

Proof. Equation (224) is (217) with , , , and (222) takes the form Thus (223), with , and Theorem 40 yield (225). Theorem is proved.

Theorem 48. Let , , , and let be a given function. Then the equation is solvable, and its general solution has the form where are given by (183) and (184) and are arbitrary real constants.

Proof. Equation (229) is the same as (217) with , , , , , , and (222) is given by For and , we have and then we get In addition, for and , we have and hence (233) takes the following form: Thus the result in (230) follows from (223) with and Theorem 42.

Theorem 49. Let , , be such that , and let , and let be a given function. The equation is solvable, and its general solution is given by where are given by (204)–(206) and are arbitrary real constants.

Proof. Equation (236) is the same equation as (217) with , , , , and with instead of for . Since , (222) takes the form For and , we have if we also take into account the following relation: where the summation is taken over all such that , and then we get For and , we have Thus the result in (237) follows from (223) with and Theorem 46.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contribution

All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.

Acknowledgments

The authors would like to thank the referees for their useful comments and suggestions. This work was supported by the National Natural Science Foundation of China (11171286) and by Jiangsu Province Colleges and Universities Undergraduate Scientific Research Innovative Program (CXZZ12-0974).