Advanced Theoretical and Applied Studies of Fractional Differential Equations 2013View this Special Issue
Research Article | Open Access
Fractional Cauchy Problem with Riemann-Liouville Derivative on Time Scales
-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.
The subject of fractional calculus (see ) 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  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 . In , 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 , 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., ) is defined by while for discrete case, the fractional sum (see, e.g., ) 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 , 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.
First, we present some preliminaries about time scales in .
Definition 1 (see ). A time scale is an nonempty closed subset of the real numbers.
Definition 2 (see ). 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 ). 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 ). 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 , Taylor’s Formula). Let . Suppose that the function is such that is ld-continuous on . Let , . Then one has
Definition 8 (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 . We denote . If , then we denote .
According to Theorem 4.13 in , we have the following lemma.
Lemma 9. Let be a measurable set. If is integrable on , then
Proof. From Theorem 4.13 in , we have that where denotes the indices set of right-scattered points of .
Definition 10 (see ). 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 ). Let . The fractional sum whose lower limit is is defined as
Definition 12 (see ). 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 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 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
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
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 , 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 , 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,
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,
Corollary 41 (see ). 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 .
Corollary 42 (see ). Let , . Then,
Property 2. Let and , . If , then the fractional -derivative which exists almost everywhere on can be represented in the following forms:
When , there is the following corollary.
Corollary 43 (see ). 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 ). 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 .
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