Abstract and Applied Analysis

Volume 2013 (2013), Article ID 401596, 19 pages

http://dx.doi.org/10.1155/2013/401596

Research Article

Fractional Cauchy Problem with Riemann-Liouville Fractional Delta Derivative on Time Scales

Jiang Zhu and Ying Zhu

School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China

Received 9 August 2013; Accepted 6 October 2013

Academic Editor: Ali H. Bhrawy

Copyright © 2013 Jiang Zhu and Ying Zhu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The -power function and fractional -integrals and fractional -differential are defined, and then the definitions and properties of -Mittag-Leffler function are given. The properties of fractional -integrals and fractional -differential on time scales are discussed in detail. After that, the existence of the solution and the dependency of the solution upon the initial value for Cauchy type problem with fractional -derivative are studied. Also the explicit solutions to homogeneous fractional -differential equations and nonhomogeneous fractional -differential equations are derived by using Laplace transform method.

1. Introduction

The fractional differential equation theory is an important subject of mathematics, which includes continuous fractional differential equations and discrete fractional difference equations. The theory of fractional differential equations has gained considerable popularity and importance during the past three decades or so. Many applications in numerous seemingly diverse and widespread fields of science and engineering have been gained. It does indeed provide several potentially useful tools for solving differential and integral equations and various other problems involving special functions of mathematical physics as well as their extensions and generalizations in one and more variables. About these advances, one can refer to [1, 2], the books [3, 4], and the references of them. For the recent developments about continuous fractional differential equations and discrete fractional difference equations, one can refer to [5–11]. To unify differential equations and difference equations, Higer proposed firstly the time scale and built the relevant basic theories (see [12–15]). Recently, some authors studied fractional calculus on time scales (see [16, 17]), where Williams [16] gives a definition of fractional -integral and -derivative on time scales to unify three cases of specific time scales. Bastos gives definitions of fractional -integral and -derivative on time scales by the inverse of Laplace transform in [17].

Inspired by these works, the aim of this paper is to give a new definition of fractional -integral and -derivative on general time scales and then study some fractional differential equations on time scales. To define the fractional -integral and fractional -derivative, we would need to obtain a definition of fractional order power functions on time scales to generalize the monomials. Different from definition of -power functions by axiomatization method in [16], we define fractional -power functions on general time scales by using inversion of time scale Laplace transform and shift transform in Section 3, and Riemann-Liouville fractional -integral and Riemann-Liouville fractional -derivative on general time scales are also given. In Section 4, we present the properties of fractional -integrals and fractional -differential on time scales. Then in Section 5, Cauchy type problem with Riemann-Liouville fractional -derivative is discussed. In Section 6, for the Riemann-Liouville fractional -differential initial value problem, we discuss the dependency of the solution upon the initial value. In Section 7, by applying the Laplace transform method, we derive explicit solutions to homogeneous fractional -differential equations with constant coefficients. In Section 8, we also use the Laplace transform method to find particular solutions and general solutions of the corresponding fractional -differential nonhomogeneous equations.

2. Preliminaries

First, we present some preliminaries about time scales in [12].

Definition 1 (see [12]). A time scale is a nonempty closed subset of the real numbers. Throughout this paper, or denotes a time scale.

Definition 2 (see [12]). Let be a time scale. For one defines the forward jump operator by , while the backward jump operator is defined by . If , one says that is right-scattered, while if , one says 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.

Definition 3 (see [12]). A function is called regulated provided that 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 .

Definition 4 (see [12]). Let , be defined by

and then recursively by

Definition 5 (see [12]). One defines the Cauchy function for the linear dynamic equation

to be for each fixed the solution of the initial value problem

Remark 6 (see [12]). Note that

is the Cauchy function of .

Theorem 7 (variation of constants [12]). Let ; then the solution of the initial value problem

is given by

where is the Cauchy function for

Definition 8 (see [2]). The factorial polynomial is defined as

For arbitrary , define

where denotes gamma function (see [3]).

Definition 9 (see [12]). One says that a function is regressive provided that

for all holds. The set of all regressive and rd-continuous functions will be denoted by

Theorem 10 (see [12]). If , then the function defined by

is also an element of .

Definition 11 (see [12]). If , then one defines the -exponential function by

Definition 12 (see [12]). If , then the first order linear dynamic equation

is called regressive.

Theorem 13 (see [12]). Suppose that (15) is regressive and fix . Then is a solution of the initial value problem

on .

Theorem 14 (see [12]). If , then

Definition 15 (see [12]). Assume that is regulated. Then the -Laplace transform of is defined by

for , where consists of all complex numbers for which the improper integral exists.

Definition 16 (uniqueness of the inverse [12]). If the functions and have the same Laplace transform, then .

In order to give fractional integral and derivative on a time scale, we need to define fractional power function which is derived by the inverse of Laplace transform and is introduced in the following section. Before this, we need definitions of shift and convolution and some properties about convolution, such as convolution theorem and associativity, which are introduced in [18].

Definition 17 (see [18]). Let be a time scale that and fix . For a given , the solution of the shifting problem

is denoted by and is called the shift (or delay) of .

Example 18 (see [18]). Consider for all , independent of .

Definition 19 (see [18]). For given functions , their convolution is defined by

where is the shift of introduced in Definition 17.

Theorem 20 (associativity of the convolution [18]). The convolution is associative; that is,

Theorem 21 (see [18]). If is delta differentiable, then

and if is delta differentiable, then

Theorem 22 (see [18]). If and are infinitely often -differentiable, then for all

Theorem 23 (convolution theorem [18]). Suppose that are locally -integrable functions on and their convolution is defined by (20). Then,

Theorem 24 (see [12]). Assume that is a mapping, such that is regulated. Then

for those regressive satisfying

Theorem 25 (see [12]). Assume that , are defined as in Definition 4. Then

for those regressive satisfying

3. -Power Function and Fractional Integral and Derivative on Time Scales

In this section, inspired by property of in Theorem 25 for , we define fractional -power functions for by using inversion of -Laplace transform and give definitions of fractional integral and derivative on time scales.

Definition 26. One defines fractional generalized -power function on time scales

to those suitable regressive such that exist for , . Fractional generalized -power function on time scales is defined as the shift of ; that is,

Applying the initial value theorem of Laplace transform (see, e.g., [15, Theorem 1.3]), for , we have

Theorem 27. For , one has

Proof. According to convolution theorem,

By the uniqueness of inverse transform for Laplace transform, we obtain

Moreover, if we take , then

That is,

Now, we will give the definitions of fractional -integral and -derivative which are the main context in this section.

Definition 28. Let be a finite interval on a time scale , . For and for a function , the Riemann-Liouville fractional -integral of order is defined by and

for , .

When , , according to Definition 17,

satisfy

When , , according to Definition 17,

satisfy

As an especial case of Definition 28, we have the following examples.

Example 29 (see [3]). When , the fractional -integral of order is defined by

Example 30. When , Consider the following.

(1)The th integral of is defined by

 Here , . Note that power function vanishes at . So

(2)The th fractional sum of is defined by

Definition 31. Let , , and . For with , the Riemann-Liouville fractional -derivative of order is defined by the expression

if it exists.

Throughout this paper, we denote , , and, for ,   means and means .

Finally, we present the definition of -Mittag-Leffler function which is an important tool for solving fractional difference equation.

Definition 32. -Mittag-Leffler function is defined by

provided that the right series is convergent, where , .

Example 33. When , for any ,  , we can obtain that the series is convergent.

Example 34. When , Since is convergent for any , is convergent. That is, is defined as .

As to the Laplace transform of -Mittag-Leffler function, we have the following theorem.

Theorem 35. The Laplace transform of -Mittag-Leffler function is

Proof. According to the definition of Laplace transform, it is obtained that

By differentiating times with respect to on both sides of the formula in Theorem 35, we get the following result:

4. Properties of Fractional -Integral and -Derivative on Time Scales

In this section, we mainly give the properties of fractional -integral and -derivative on time scales which are needed in the following sections.

Theorem 36. Let , , and . Then

Proof. (1) According to Definition 28 and Theorem 27, we have

(2) By Definition 31, it is obtained that

Then

In particular, if , , then the Riemann-Liouville fractional -derivatives of a constant are, in general, not equal to zero:

On the other hand, for ,

In fact,

From Theorem 36, we derive the following result in [3] when .

Corollary 37 (see [3]). If and , then

In particular, if and , then the Riemann-Liouville fractional derivatives of a constant are, in general, not equal to zero:

On the other hand, for ,

As to the fractional sum and difference, we have the following result, which is an improvement of Lemma 3.1 in [1].

Corollary 38. Let . Then

Lemma 39 (Taylor’s formula). Let . Suppose that the function is times differentiable on . Let , , and . Then one has

Proof. Let . Then solves the initial value problem

Note that the Cauchy function for is . By the variation of constants formula in Theorem 7,

where solves the initial value problem

To validate the claim that , set

By the properties of , . We have moreover that

so that

for . We consequently have that also solves (66), whence by uniqueness.

Lemma 40. (1) For , , let be a function which is times -differentiable on with rd-continuous over , and it is valid that

(2) For , , let be a function which is times -differentiable on with rd-continuous over and exists almost on , and it is valid that

Proof. By Taylor’s formula

we have

Besides,

where

When , there is the following corollary.

Corollary 41 (see [3]). Let and . If , then the fractional derivative exists almost everywhere on and can be represented in the form

Theorem 42. For and , then .

Proof. According to Definition 28, Theorems 20 and 27,

Theorem 43. For , is a positive integer; if is -differentiable and the highest order derivative is rd-continuous over , then it is valid that

Proof. (1) Suppose that is a function which is times -differentiable on with rd-continuous over . By Lemma 40(2),

By a similar way, we can get .

When , we have the following corollary.

Corollary 44. Let be given. For any and with , one has

Theorem 45. For , is a positive integer; if is -differentiable and the highest order derivative is rd-continuous over , then it is valid that

Proof. (1) In the proof of Theorem 43, if we take , then we have

As is times -differentiable on , we have

By (82) and (83), if is at least times -differentiable with the highest order derivative rd-continuous over , then we have

Thus

is valid if and only if

(2) Similarly, we have

Therefore

Provided that is at least times -differentiable with the highest order derivative rd-continuous over .

Thus

is valid if and only if

In particular, there are corollaries for and for .

Corollary 46 (see [3]). Let and be such that , and , and let and . Then one has the following index rule:

Corollary 47. Let be given. For any and ,

Theorem 48. Let be -differentiable and let its highest order derivative be rd-continuous over . When , , one has the following:

Proof. According to Theorem 42 and (24), we have

In addition,

As a direct corollary of Theorem 48, we get Lemma 2.5 in [3].

Corollary 49 (see [3]). Let , , and let . If and , then

holds almost everywhere on .

For fractional sum and difference, there is also the following theorem in [1].

It is different from Theorem 3.3 in [1], and from Theorem 48, we can get the following corollary.

Corollary 50. Let be given. For any and with ,

Next, we will give the Laplace transform of fractional integral and derivative on time scales.

Theorem 51. (1) Let and be locally -integrable. For with , one has

(2) Let and be locally -integrable. For with , one has

Proof. (1) According to Definition 28 and convolution theorem, we have

(2) By Definition 31 and (26) and taking the Laplace transform of fractional integral into account, we get

5. Cauchy Type Problem with Riemann-Liouville Fractional Derivative

In this section, we consider Cauchy type problem with Riemann-Liouville fractional derivative

In the space defined for by

Here is the space of -Lebesgue summable functions in a finite interval .

In the following, we prove that Cauchy type problem and nonlinear Volterra integral equation are equivalent in the sense that if satisfies one of these relations, then it also satisfies the other.

Theorem 52. Let , , , . Let be an open set in and let be a function such that for any . If , then Cauchy type problem (102) and (103) is equivalent to

Proof. First we prove the necessity. We apply to both sides of (102) and get by Theorem 48

Thus

Now we prove the sufficiency. Applying the operator to both sides of (105) and by (57) and Theorem 48(1), we have

Now we show that the relations in (103) also hold. For this, we apply the operators    to both sides of (105):

Since

In the following, we bring in Lipschitzian-type condition:

where does not depend on . We will derive a unique solution to the Cauchy problem (102)-(103).

Theorem 53. Let the condition of Theorem 52 be valid, let satisfy the Lipschitzian condition (111), and , is defined on , where is the Lipschitzian constant in (111). Then there exists a unique solution to initial value problem (102)-(103) in the space .

Proof. Since the Cauchy type problem (102)-(103) and the nonlinear Volterra integral equation (105) are equivalent, we only need to prove that there exists a unique solution to (105).

We define function sequences:

where

We obtain, by induction that,

In fact, for , as , we have

If

then

Let

and we have

By Weierstrass discriminance, we obtain convergent uniformly. Next we will show the uniqueness. Assume that is another solution to (105); that is,

As

If

then

By mathematical induction, we have

and then get owing to the uniqueness of the limit. This completes the proof of the theorem.

Next we consider the generalized Cauchy type problem:

Theorem 54. Let be a function such that for any . If , then satisfies a.e. the relations (126) and (125) if and only if it satisfies a.e. the integral equation

Assume that satisfies generalized Lipschitzian condition

According to Theorem 54 and by a similar proof to that of Theorem 53, we have the following theorem.

Theorem 55. Let the condition of Theorem 54 be valid and let satisfy the Lipschitzian condition (128). Then there exists a unique solution to the generalized Cauchy type problem.

6. The Dependency of the Solution upon the Initial Value

We consider fractional differential initial value problem again:

where .

Using Theorem 52, we have

Suppose that is the solution to the initial value problem:

We can derive the dependency of the solution upon the initial value.

Theorem 56. Let , and suppose that satisfy the Lipschitz condition; that is,

Then one has

Proof. By the proof of Theorem 53, we know that , , where

Using the Lipschitz condition, we have

Suppose that

Then

According to mathematical induction, we have

Taking the limit , we obtain that

As a special case, when fractional equation is linear, we can obtain its explicit solutions and we will explain it in next section.

7. Homogeneous Equations with Constant Coefficients

In this section, we apply the Laplace transform method to derive explicit solutions to homogeneous equations of the form

with the Liouville fractional derivatives . Here are real constants, and, generally speaking, we can take .

In order to solve the equation, we need the following Laplace transform formula:

First, we derive explicit solutions to (140) with :

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

Definition 57. The function is defined by

Lemma 58. The solutions are linearly independent if and only if   at some point .

Proof. We first prove sufficiency. If, to the contrary, are linearly dependent on , 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 ,  . Consider

where , . As , the equations have nontrivial solution . Now we construct a function using these constants:

and we get that is a solution. From (146), 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 .

There hold the following statements.

Theorem 59. Let and . Then the functions

yield the fundamental system of solutions to (143). Moreover, , , satisfy

Proof. Applying the Laplace transform to (143) and taking (141) into account, we have

where are given by (142).

Formula (49) with yields

Thus, from (152), we derive the following solution to (143):

It is easily verified that the functions are solutions to (143):

In fact,

Moreover,

It follows from (157) that

If , then

and since for any , the following relations hold:

By (158) and (160), . Then , , yield the fundamental system of solutions to (143).

Corollary 60. Consider that

has its solution given by

while

has the fundamental system of solutions given by

Next we derive the explicit solutions to (140) with of the form

with .

Theorem 61. Let , , and . Then the functions

yield the fundamental system of solutions to (165) provided that the series in (166) are convergent. Moreover, if , then , , in (166) satisfy (151).

Proof. Let . Applying the Laplace transform to (165) and using (141) as in (152), we obtain

where , .

For and , we have

and hence (167) has the following representation:

By (51), for and , we have

From (169) and (170), we derive the solution to (165) as

where are given by (166). For , the direct evaluation yields

For , , and for , . Thus we have . It follows from Lemma 58 that the functions , in (166) are linearly independent solutions, and then they yield the fundamental system of solutions to (165). Furthermore, if , then we rewrite (172) as follows:

If , then for , , , and for, , . Besides, we also have for . These imply that . Thus the relations in (151) are valid. The proof is finished.

Corollary 62. Consider that

has its fundamental system of solution given by

Finally, we find the explicit solutions to (140) with any . It is convenient to rewrite (140) in the form

Theorem 63. Let , , let and be such that , and let . Then the functions

yield the fundamental system of solutions to (176) provided that the series in (177) are convergent. The inner sum is taken over all such that . Moreover, if , then , , in (177) satisfy (151).

Proof. Let , . Applying the Laplace transform to (176) and using (141) as in (167), we obtain

where

Here 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 (178), (180), and (182), we derive the solution to (176), as

which shows that arbitrary solution can be expressed by , , where are given by (177). For , , the direct evaluation yields

For , , and for , . Thus we have . It follows from Lemma 58 that the functions , in (177) are linearly independent solutions and then they yield the fundamental system of solutions to (176). Furthermore, if , then we rewrite (184) as follows:

If , then for , , and for , , . Besides, we also have       for . These imply that . Thus the relations in (151) are valid. The result follows.

8. Nonhomogeneous Equations with Constant Coefficients

In Section 7, we have applied the Laplace transform method to derive explicit solutions to the homogeneous equations (140) with the Liouville fractional derivatives. Here we use this approach to find particular solutions to the corresponding nonhomogeneous equations

with real .

By (141)-(142), for suitable functions , the Laplace transform of is given by

Applying the Laplace transform to (186) and taking (187) into account, we have

Using the inverse Laplace transform from here, we obtain a particular solution to (186) in the form

Using the Laplace convolution formula

we can introduce the Laplace fractional analog of the Green function as follows:

and we can express a particular solution of (152) in the form of the Laplace convolution and :

Generally speaking, we can consider (186) with . First we derive a particular solution to (186) with in the form

Theorem 64. Let , . Then (193) is solvable, and its particular solution has the form

provided that the integral in the right-hand side of (194) is convergent.

Proof. Equation (193) is the same as (186) with , , and , and (191) takes the form

Thus (192), with , yields (194). Theorem is proved.

Next we derive a particular solution to (186) with of the form

Theorem 65. Let , . Then (196) is solvable, and its particular solution has the form

provided that the series in (198) and the integral in (197) are convergent.

Proof. Equation (196) is the same as (186) with , , , , , and , and (191) is given by

According to (168) for and , we have

In addition, for and , we have

and hence (200) takes the following form:

Thus the result in (197) follows from (192) with .

Finally, we find a particular solution to (186) with any . It is convenient to rewrite (186) just as (176) in the form

with , , and .

Theorem 66. Let , , and let . Then (203) is solvable, and its particular solution has the form

provided that the series (205) and integral in (204) are convergent. The inner sum is taken over all such that .

Proof. Equation (203) is the same equation as (186) with , , , and with instead of for . Since , (191) takes the form

For and , in accordance with (180), we have

For and , we have

The proof is finished.

As in the case of ordinary differential equations, a general solution to the nonhomogeneous equation (186) is a sum of a particular solution to this equation and of the general solution to the corresponding homogeneous equation (140). Therefore, the results established in Section 7 and in Section 8 can be used to derive general solutions to the nonhomogeneous equations (193), (196), and (203). The following statements can thus be derived from Theorems 59, 64, 61, 65 and Theorems 63 and 66, respectively.

Theorem 67. Let , . Then (193) is solvable, and its general solution is given by

where are arbitrary real constants.

Theorem 68. Let , , . Then (196) is solvable, and its general solution has the form

where is given by (198) and are arbitrary real constants.

Theorem 69. Let , , let and be such that and , and let . Then (203) is solvable, and its general solution is given by

where is given by (205) and are arbitrary real constant.

9. Conclusions

In this paper, we first give a generalized definition of fractional -power function on general time scale by inversion of Laplace transform and shift transform. The fractional -power function on a time scale is an important basis of fractional -integral and fractional -differential on time scales. Then, based on the fractional -power function, we give a new definition of Riemann-Liouville fractional -integral and Riemann-Liouville fractional -derivative on time scales. Some of properties of Riemann-Liouville fractional -integral and Riemann-Liouville fractional -derivative on time scales are studied in detail. On this basis, equivalencies of Cauchy type problem with Riemann-Liouville fractional -derivative and nonlinear Volterra integral equation are obtained. By employing Laplace transform, we derive explicit solutions to homogeneous and nonhomogeneous equations of Riemann-Liouville fractional -derivative with constant coefficient. We give the conditions when the solutions of linear fractional differential equation will be linearly independent and when these linearly independent solutions form the fundamental system of solutions. On the other hand, we know that continuous fractional differential equation theory and discrete fractional difference equation theory have been studied by many authors, the existence and uniqueness of the solution to the boundary value problems for fractional differential equations have been studied a lot by many methods involving partial order method, fixed point method, lower and upper solutions method, transform method, and so on. For example, for the recent developments about continuous fractional differential equations and discrete fractional difference equations, one can refer to [3, 5–11] and the references therein. However, the fractional differential equation theory on time scales is still an open problem. Therefore, the authors believe that the present work will potentiate further research in the study of fractional differential equation theory on time scales in the future.

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 to the writing of this paper. All authors read and approved the final paper.

Acknowledgments

First, the authors are very grateful to the referees for their careful reading of the paper, and lots of valuable comments and suggestions, which greatly improve this manuscript. Next, this work was supported by the National Natural Science Foundation of China (11171286) and by Jiangsu Province Colleges and Universities Graduate Scientific Research Innovative Program (CXZZ12-0974).

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