Abstract and Applied Analysis

Volume 2013 (2013), Article ID 795701, 23 pages

http://dx.doi.org/10.1155/2013/795701

Research Article

Fractional Cauchy Problem with Riemann-Liouville Derivative on Time Scales

Ling Wu and Jiang Zhu

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

Received 19 July 2013; Revised 22 September 2013; Accepted 22 September 2013

Academic Editor: Juan J. Trujillo

Copyright © 2013 Ling Wu and Jiang 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.

-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 the Riemann-Liouville fractional -derivative are studied. Also the explicit solutions to homogeneous equations and nonhomogeneous equations are derived by using Laplace transform method.

1. Introduction

The subject of fractional calculus (see [1]) has gained considerable popularity and importance during the past three decades or so due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering. 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.

On the other hand, in real applications, it is not always a continuous case, but also a discrete case. So, an useful tool as that time scale is considered. In order to unify differential equations and difference equations, Higer proposed firstly the time scale and built the relevant basic theories (see [2–4]). Recently, some authors studied fractional calculus on time scales (see [5–7]). Williams [6] gives a definition of fractional integral and derivative on time scales to unify three cases of specific time scales, which improved the results in [5]. Bastos gives definition of fractional -integral and -derivative on time scales in [7]. In [8], the theory of fractional difference equations has been studied in detail. In the light of the above work, we will further study the theory of fractional integral and derivative on general time scales.

From Theorem in [6], we know that the integer order -integral on time scales is

where is defined in Definition 31. For continuous case, the fractional -integral (see, e.g., [3]) is defined by

while for discrete case, the fractional sum (see, e.g., [8]) is defined by

Thus, we expect that fractional -integral general on time scales can be defined by

To do this, the key problem is that how to define generalized -power function on time scales. In [6], Williams, by using axiomatization method, gives a definition of fractional generalized -power function. However, this definition has no specific form, but it only has an abstract expression. On the other hand, we find that some properties of -power function on time scales under the Laplace transform are important to define fractional generalized -power function on time scales. So, in Section 3, we will give a definition of -Laplace transform, fractional generalized -power function on time scales. Then by using these definitions, we define and study the Riemann-Liouville fractional -integral, Riemann-Liouville fractional -derivative, and -Mittag-Leffler function on time scales. In Section 4, we present some properties of fractional -integral and fractional -differential on time scales. Then, in Section 5, Cauchy-type problem with the 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 equations with constant coefficients. In Section 8, we also use the Laplace transform method to find particular solutions of the corresponding nonhomogeneous equations.

2. Preliminaries

First, we present some preliminaries about time scales in [2].

Definition 1 (see [2]). A time scale is an nonempty closed subset of the real numbers.

Definition 2 (see [2]). For , we define 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 [2]). If has a right-scattered minimum , then we define ; otherwise, . Assume that is a function and let . Then we define 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 [2]). 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 .

Definition 5 (see [2, page 100]). The generalized polynomials are the functions , , defined recursively as follows: The function is

and given for , the function is

Definition 6 (see [3, page 38]). The generalized polynomials are the functions , , defined recursively as follows: The function is

and given for , the function is

Theorem 7 (see [2], Taylor’s Formula). Let . Suppose that the function is such that is ld-continuous on . Let , . Then one has

Definition 8 (see [6]). 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 . We denote . If , then we denote .

According to Theorem  4.13 in [9], we have the following lemma.

Lemma 9. Let be a measurable set. If is integrable on , then

Proof. From Theorem  4.13 in [9], we have that

where denotes the indices set of right-scattered points of .

Definition 10 (see [8]). The increasing factorial function is defined as

where is a positive integer and is a real number, and the symbol is defined as

Definition 11 (see [8]). Let . The fractional sum whose lower limit is is defined as

Definition 12 (see [8]). Let and . The fractional difference whose lower limit is is defined as

In our discussion, we also need some information about -exponential function.

Definition 13 (see [2, Definition 3.4]). The function is -regressive if

for all . Define the -regressive class of functions on to be

For , define circle minus by

Definition 14 (see [2, Definition 3.9]). For , let

Define the -cylinder transformation by

where Log is the principal logarithm function.

Definition 15 (see [2, Definition 3.10]). If , then one defines the nabla exponential function by

where the -cylinder transformation is as in (24).

Definition 16 (see [2, Definition 3.12]). If , then the first-order linear dynamic equation

is called -regressive.

Lemma 17 (see [2, Lemma 3.11]). If , then the semigroup property

is satisfied.

Theorem 18 (see [2, Theorem 3.13]). Suppose that (26) is -regressive and fix . Then is a solution of the initial value problem

on .

Theorem 19 (see [2, Theorem ]). If , then

3. -Laplace Transform, Fractional Generalized -Power Function, Fractional -Integral and Derivative, and -Mittag-Leffler Function

In this section, we first define -Laplace transform and discuss the properties of -Laplace transform. By using the inverse -Laplace transform, we define fractional generalized -power function, which is a basis of our definitions of fractional -integral and fractional -derivative.

From now on, we always assume that , . Note that if we assume that is a constant, then and is well defined. With this in mind we make the following definition.

Definition 20. 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.

The following result is needed frequently.

Lemma 21. If is regressive, then

Proof. By Theorem 19, we have

This proves our claim.

We now will use the Lemma 21 to find the Laplace transform of as follows:

for all complex values of such that holds. The following two results are derived using integration by parts.

Theorem 22. Assume that is such that is regulated. Then,

for those regressive satisfying

Proof. Integration by parts and Lemma 21 directly yield

provided that (35) holds.

By a similar way, we have

provided that and (35) holds and thus we can get the following result by induction:

for those regressive satisfying , .

Theorem 23. Assume that is regulated. If

for ; then

for those regressive satisfying

Proof. By using integration by parts and Lemma 21, we obtain that

provided that (41) holds.

Theorem 24. Assume that , are defined as in Definition 5. Then,

for those regressive satisfying

Proof. It follows from (33) that (43) holds for . Assume that (43) is valid for , and we will show that it is right for . In fact, by using Theorem 23, we have that

The claim follows by the principle of mathematical induction.

It is similar to the proof of Theorems 1.5 and  1.3 in [10], we get the following uniqueness result about the inverse of Laplace transform and initial value theorem.

Theorem 25 (uniqueness of the inverse). If the functions and have the same Laplace transform, then .

Theorem 26 (initial value theorem). Let have generalized Laplace transform . Then, .

By the uniqueness of inverse Laplace transform and fixing , we can define fractional -power function .

Definition 27. We define fractional generalized -power function on time scales as follows:

to those suitable regressive such that exist for , .

Applying the initial value theorem of Laplace transform, for , we have

In particular, when , it follows from Theorems 24 and 25, we can know that is usual power function on time scales for defined in Definition 5.

Example 28. When , the time scale power functions

provided that makes sense. In fact, it follows from Definition 27 that

On the other hand,

Thus, we have that

By using uniqueness of the inverse Laplace transform, we imply that

Next, in order to define fractional generalized -power function for general , we will present some preliminaries about convolution on time scales. In [11], the definitions of shift and convolution, and some properties about convolution, such as convolution theorem and associativity, are presented for delta case, and in the following, we give them similarly for nabla case.

Let be a time scale such that and fix .

Definition 29. For a given , the solution of the following shifting problem:

is denoted by and is called the shift of .

Example 30. Let . Then, for ,

In fact, it is similar to the discussion for the delta case (refer to [3, page 38]), and we can prove that

where is defined by

Thus, according to Definition 29, we can derive the result.

Definition 31. For given functions , their convolution is defined by

where is the shift introduced in Definition 29.

Theorem 32 (associativity of the convolution). The convolution is associative; that is,

Theorem 33. If is nabla differentiable, then

and if is nabla differentiable, then

Theorem 34 (convolution theorem). Suppose that are locally -integrable functions on . Then,

In the following, we will define fractional generalized -power function for general .

Definition 35. Fractional generalized -power function on time scales is defined as the shift of , that is

According to convolution theorem and Definition 27, we have

By the uniqueness of inverse Laplace transform, we obtain

that is,

In particular, if , then

that is,

Thus,

If , then

that is,

According to Theorem 33 and (47), (68), for , we have

Now, we can give definitions of fractional -integral and fractional -derivative on time scales.

From now on, we will always denote a finite interval on a time scale .

Definition 36. Let . The Riemann-Liouville fractional -integral of order is defined by

Definition 37. Let . The Riemann-Liouville fractional -derivative of order is defined by

Throughout this paper, we denote , .

In the following, we will give the Laplace transform of fractional -integral and fractional -derivative.

Lemma 38. Let , and . For with . Then, we have

(1)if , then

(2)if , then

for those regressive satisfying , .

Proof. According to Definition 36, Definition 27 and convolution theorem, we have

By Definition 37, (38), and taking the Laplace transform of fractional -integral into account, we get

where follows from the definition of fractional -derivative, and ( is right-dense); ( is right scattered).

Finally, we present the definition of -Mittag-Leffler function which is an important tool for solving fractional differential equation. We have known that the Mittag-Leffler function is the fractional order case of exponential function , and is generalized from . Inspired by these results and exponential function on time scales (see [7, Remark 124]), we give the following definition.

Definition 39. -Mittag-Leffler function is defined as

provided that the right hand series is convergent, where , .

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

Theorem 40. The Laplace transform of  -Mittag-Leffler function is

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

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

4. Properties of Fractional -Integral and Fractional -Derivative

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

Property 1. Let , , , . Then

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

On the other hand, for ,

In fact,

Proof. (1) According to Definition 36 and (64), we have

(2) From Definition 37 and (82), (68), it is obtained that

From Property 1, we derive the following result in [1] when .

Corollary 41 (see [1]). If and , then

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

On the other hand, for ,

As to the fractional sum and difference, there is also a similar result in [8].

Corollary 42 (see [8]). Let , . Then,

Property 2. Let and , . If , then the fractional -derivative which exists almost everywhere on can be represented in the following forms:

Proof. By Taylor’s formula,

and using (82) and (64), we have

Besides, according to (95) and taking (68) and (71) into account, we have

When , there is the following corollary.

Corollary 43 (see [1]). Let , and . If , then the fractional derivative exists almost everywhere on and can be represented in the following form:

Similarly, for the fractional sum and difference, there is also the following corollary.

Corollary 44 (see [8]). For and , it is valid that

The semigroup property of the fractional -integral operator is given by the following result.

Property 3. If and , then the equation

is satisfied at almost every point for .

Proof. According to Definition 36 and (64), and using associativity of the convolution, we have

The following assertion shows that the fractional differentiation is an operation inverse to the fractional integration from the left.

Property 4. If and , then the following equality

holds almost everywhere on .

Proof. According to the definition of the fractional -derivative and using (99), we get

In the following, we will derive the composition relations between fractional -differentiation and fractional -integration operators.

Property 5. If , then, for , the relation

holds almost everywhere on . In particular, when and , then

Proof. The proof is the same with the proof of Property 4, so we omit it.

Property 6. Let , , . If , then

Thus,

is valid if and only if

Proof. Since , by (93) in Property 2 and (71), (68), we have

On the other hand, from , we know that and thus, we have

Comparing with (108) and (109), we can get that

which proves the result.

Property 7. Let , , . If , then

Proof. Applying Laplace transform to , we have

Using the uniqueness of the inverse Laplace transform, we can derive that

The proof is finished.

To present the next property, we use the space of function defined for and by

The composition of the fractional -integral operator with the fractional -differentiation operator is given by the following result.

Property 8. Let , and let .

(1)If and , then

(2)If and , then the equality

holds almost everywhere on , where .

Proof. From the definition of and (101), it is easy to obtain the result.

Applying Laplace transform to , we can get

By the uniqueness of Laplace transform, we have

Property 9. Let . When , , if and , then we have the following equation:

Proof. Let . According to Property 3 and the definition of fractional -derivative, we have

In addition,

Property 10. Let and , . If and , then we have the following equation:

Proof. According to Property 6 and Property 9, we have

From Property 10, we derive the following result in [1] when .

Corollary 45 (see [1]). Let and be such that , and , and let and . Then, there is the following index rule:

It follows from Property 10, for fractional sum and difference, that there is also the following theorem in [8].

Corollary 46 (see [8]). Let , . Then

5. Cauchy-Type Problem with Riemann-Liouville Fractional Derivative

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

where for and for , and the notation means that if is right-dense, then

if is right-scatter, then .

We discuss this Cauchy-type problem in the space defined for by

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

In particular, if , then the problem in (126) and (127) is reduced to the usual Cauchy problem for the ordinary differential equation of order on the following time scales:

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

Theorem 47. Let , , . Let be an open set in and let be a function such that for any . If , then Cauchy-type problem (126) and (127) is equivalent of the following equation:

Proof. First we prove the necessity. Let satisfy a.e. the relations (126) and (127). Since , (126) means that there exists a.e. on the fractional nabla derivative . According to

we have . Thus, we apply to both sides of (126) and, in accordance with (116), we have

Thus,

Now, we prove the sufficiency. Let satisfy a.e. (131). Applying the operator to both sides of (131), we have

From here, in accordance with the formula (85) and (101), we arrive at (131).

Now, we show that the relations in (127) also hold. For this, applying the operators to both sides of (131) and using (83) and (103), we have

Thus, we obtain the relations in (127).

In the following, we establish the existence of a unique solution to the Cauchy-type problem (126)-(127) in the space defined in (129) under the conditions of Theorem 47, and an additional Lipschitzian-type condition on with respect to the second variable, for all and for all ,

where does not depend on . we will derive a unique solution to the Cauchy-problem (126)-(127).

Theorem 48. Let , . Let be an open set in and let be a function such that for any . Let satisfy the Lipschitzian condition (137) and . Then, there exists a unique solution to the Cauchy-type problem (126)-(127) in the space .

Proof. Since the Cauchy-type problem (126)-(127) and the nonlinear Volterra integral equation (131) are equivalent, we only need to prove there exists a unique solution to (131).

We define function sequences as follows:

where

We obtain by induction

In fact, for , since , we have

If

Then,

Thus, from Lemma 9, we have

By Weierstrass discriminance, we obtain convergent uniformly. Let then is a solution of (131). Next, we will show the uniqueness. Assume that is another solution to (131), that is,

As

If

then

By mathematical induction, we have

and then, from Lemma 9, we get that

Thus, , and then get owing to the uniqueness of the limit. To complete the proof of Theorem 48, we must show that such a unique solution belongs to the space . In accordance with (129), it is sufficient to prove that . By the above proof, the solution is a limit of the sequence as follows

By (126) and (137), we have

Thus, by (151), we get

and hence . This completes the proof of Theorem 48.

Next, we consider the generalized Cauchy-type problem as follows:

Theorem 49. Let be a function such that for any . If , then satisfies a.e. the relations (154) if and only if satisfies a.e. the integral equation as follows

Assume that satisfies generalized Lipschitzian condition as follows:

According to the theorem above and by a similar proof of Theorem 48, we have the following theorem

Theorem 50. Let the condition of Theorem 49 be valid and let satisfy the Lipschitzian condition (156) and Then there exists a unique solution to be the generalized Cauchy-type problem in the space .

6. The Dependency of the Solution upon the Initial Value

We consider fractional differential initial value problem (126)-(127) again as follows

where .

Using Theorem 47, we have

where

Suppose that is the solution to the initial value problem as follows

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

Theorem 51. Let , and suppose satisfy the Lipschitz condition, that is,

Then, we have

Proof. By the proof of Theorem 48, we know that , where

Using the Lipschitz condition, we have

Suppose that

Then,

According to mathematical induction, we have

Taking the limit and from Lemma 9, we obtain that

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

7. Homogeneous Equations with Constant Coefficients

In this section, we apply the Laplace transform method to derive the fundamental system of solutions to homogeneous equations of the following form:

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

The Laplace transform method is based on the relation (75) which is equivalent to the following one:

First, we derive explicit solutions to (169) with as follows:

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

Definition 52. The function is defined by

Lemma 53. 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 exists 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 necessity. Suppose, to the contrary, for , . Consider the following equations:

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

and we get that is a solution. From (175), we obtain that satisfies the following 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 holds the following statements.

Theorem 54. Let and . Then, the following functions:

yield the fundamental system of solutions to (172). Moreover, , satisfy

Proof. Applying the Laplace transform to (172) and taking (170) into account, we have

where are given by (171).

Formula (79) with yields

Thus, from (181), we derive the following solution to (172) as follows:

which shows that an arbitrary solution can be represented by , . It is easily verified that the functions are solutions to (172) as follows:

In fact,

Moreover,

It follows from (186) that

If , then

and since for any , the following relations hold:

By (187) and (189), . Then, , which are linearly independent yield the fundamental system of solutions to (172).

Corollary 55. The following equation:

has its solution given by

while the equation

has the fundamental system of solutions given by

Next, we derive the explicit solutions to (169) with of the following form:

with .

Theorem 56. Let , and . Then, the following functions:

yield the fundamental system of solutions to (194), provided that the series in (195) is convergent. Moreover, if , then, , , in (195) satisfy (180).

Proof. Let . Applying the Laplace transform to (194) and using (170) as in (181), we obtain

where , .

For and , we have

and hence (196) has the following representation:

By (81), for and , we have

From (198) and (199), we derive the solution to (194) as follows

which shows that an arbitrary solution can be represented by , , where are given by (195). For , , the direct evaluation yields

For , , and for , . Thus, we have . It follows from Lemma 53 that the functions , in (195) are linearly independent solutions, and then they yield the fundamental system of solutions to (194). Furthermore, if , then we rewrite (201) as follows:

If , then for , , , and for, , . Besides, we also have for , . These imply that . Thus, the relations in (180) are valid. The proof is finished.

Corollary 57. The following equation form:

has its fundamental system of solution given by

Finally, we find the fundamental system of solutions to (169) with any . It is convenient to rewrite (169) in the form

Theorem 58. Let , and let and be such that , and let . Then, the following functions:

yield the fundamental system of solutions to (205), provided that the series in (206) are convergent. The inner sum is taken over all such that . Moreover, if , then , , in (206) satisfy (180).

Proof. Let , . Applying the Laplace transform to (205) and using (170) as in (196), 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 (207), (209), and (211), we derive the solution to (205),

which shows that arbitrary solution can be expressed by , , where are given by (206). For , the direct evaluation yields

For , , and for , . Thus, we have . It follows from Lemma 53 that the functions , in (206) are linearly independent solutions and then they yield the fundamental system of solutions to (205). Furthermore, if , then we rewrite (213) as follows:

If , then for , , , and for , , . Besides, we also have for , . These imply that . Thus the relations in (180) are valid. The result follows.

8. Nonhomogeneous Equations with Constant Coefficients

In above section, we applied the Laplace transform method to derive explicit solutions to the homogeneous equations (169) with the Riemann-Liouville fractional derivatives. Here, we use this approach to find particular solutions to the corresponding nonhomogeneous equations as follows:

with real .

By (170) and (171), for suitable functions , the Laplace transform of is given by

Applying the Laplace transform to (215) and taking (216) into account, we have

Using the inverse Laplace transform from here we obtain a particular solution to (215) in the following 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 (181) in the form of the Laplace convolution and as follows:

Generally speaking, we can consider (215) with . First, we derive a particular solution to (215) with in the following form:

Theorem 59. Let , . Then, (222) is solvable, and its particular solution has the following form:

Proof. Equation (222) is (215) with , , , and (220) takes the following form:

Thus, (221), with , yields (223). The result is proved.

Next, we derive a particular solution to (215) with of the following form:

Theorem 60. Let , . Then, (225) is solvable, and its particular solution has the following form:

provided that the series in (227) is convergent.

Proof. Equation (225) is the same as (215) with , , , , , , and (220) is given by

According to (197) for and ,we have

In addition, for and , we have

and hence (229) takes the following form:

Thus, the result in (226) follows from (221) with .

Finally, we find a particular solution to (215) with any . It is convenient to rewrite (215), just as (205) in the following form:

with , , and .

Theorem 61. Let , , and let . Then, (232) is solvable, and its particular solution has the following form:

provided that the series (234) is convergent. The inner sum is taken over all such that .

Proof. Equation (232) is the same equation as (215) with , , , and with instead of for . Since , (220) takes the following form:

For and , in accordance with (209), 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 (215) is a sum of a particular solution to this equation and of the general solution to the corresponding homogeneous equation (169). Therefore, the results established in this section and in the previous section can be used to derive general solutions to the nonhomogeneous equation (222), (225), and (232). The following statements can thus be derived from Theorems 54, 59, 56, 60, 58, and 61, respectively.

Theorem 62. Let , . Then, (225) is solvable, and its general solution is given by

where are arbitrary real constants.

Theorem 63. Let , , . Then, (225) is solvable, and its general solution has the following form:

where is given by (227) and are arbitrary real constants.

Theorem 64. Let , and let and be such that and and let . Then, (232) is solvable, and its general solution is given by

where is given by (234) and are arbitrary real constants.

Acknowledgments

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

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