On the Definitions of Nabla Fractional Operators

Abstract

We show that two recent definitions of discrete nabla fractional sum operators are related. Obtaining such a relation between two operators allows one to prove basic properties of the one operator by using the known properties of the other. We illustrate this idea with proving power rule and commutative property of discrete fractional sum operators. We also introduce and prove summation by parts formulas for the right and left fractional sum and difference operators, where we employ the Riemann-Liouville definition of the fractional difference. We formalize initial value problems for nonlinear fractional difference equations as an application of our findings. An alternative definition for the nabla right fractional difference operator is also introduced.

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Thabet Abdeljawad 1 and Ferhan M. Atici 2

Recommended by Dumitru B...leanu

1, Department of Mathematics, Çankaya University, 06530 Ankara, Turkey
2, Department of Mathematics, Western Kentucky University, Bowling Green, KY 42101, USA

Received 12 April 2012; Accepted 4 September 2012

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.

1. Introduction

The following definitions of the backward (nabla) discrete fractional sum operators were given in [ 1, 2], respectively. For any given positive real number α , we have [figure omitted; refer to PDF] where t ∈ {a +1 ,a +2 , ... } , and [figure omitted; refer to PDF] where t ∈ {a ,a +1 ,a +2 , ... } .

One significant difference between these two operators is that the sum in ( 1.1) starts at a +1 and the sum in ( 1.2) starts at a . In this paper we aim to answer the following question:

Do these two definitions lead the development of the theory of the nabla fractional difference equations in two directions?

In order to answer this question, we first obtain a relation between the operators in ( 1.1) and ( 1.2). Then we illustrate how such a relation helps one to prove basic properties of the one operator if similar properties of the other are already known.

In recent years, discrete fractional calculus gains a great deal of interest by several mathematicians. First Miller and Ross [ 3] and then Gray and Zhang [ 1] introduced discrete versions of the Riemann-Liouville left fractional integrals and derivatives, called the fractional sums and differences with the delta and nabla operators, respectively. For recent developments of the theory, we refer the reader to the papers [ 2, 4- 19]. For further reading in this area, we refer the reader to the books on fractional differential equations [ 20- 23].

The paper is organized as follows. In Section 2, we summarize some of basic notations and definitions in discrete nabla calculus. We employ the Riemann-Liouville definition of the fractional difference. In Section 3, we obtain two relations between the operators ^{∇ a - α} and ^{[white triangle down] a - α} . So by the use of these relations we prove some properties for ^{∇ a - α} -operator. Section 4is devoted to summation by parts formulas. In Section 5, we formalize initial value problems and obtain corresponding summation equation with ^{∇ a - α} -operator. This section can be considered as an application of the results in Section 3. Finally, in Section 6, a definition of the nabla right fractional difference resembling the nabla right fractional sum is formulated. This definition can be used to prove continuity of the nabla right fractional differences with respect to the order α .

2. Notations and Basic Definitions

Definition 2.1.

(i) For a natural number m , the m rising (ascending) factorial of t is defined by [figure omitted; refer to PDF]

(ii) For any real number α , the rising function is defined by [figure omitted; refer to PDF]

Throughout this paper, we will use the following notations.

(i) For real numbers a and b , we denote _{... a} = {a ,a +1 , ... } and _b ... = { ... ,b -1 , b } .

(ii) For n ∈ ... , we define [figure omitted; refer to PDF]

Definition 2.2.

Let ρ (t ) =t -1 be the backward jump operator. Then

(i) the (nabla) left fractional sum of order α >0 (starting from a ) is defined by [figure omitted; refer to PDF]

(ii) the (nabla) right fractional sum of order α >0 (ending at b ) is defined by [figure omitted; refer to PDF]

We want to point out that the nabla left fractional sum operator has the following characteristics.

(i) ^{∇ a - α} maps functions defined on _{... a} to functions defined on _{... a} .

(ii) ^{∇ a -n} f (t ) satisfies the n th order discrete initial value problem [figure omitted; refer to PDF]

(iii): The Cauchy function (t - ρ (s ) ^{) n -1 ¯} / Γ (n ) satisfies ^{∇ n} y (t ) =0 .

In the same manner, it is worth noting that the nabla right fractional sum operator has the following characteristics.

(i) _b ^{∇ - α} maps functions defined on _b ... to functions defined on _b ... .

(ii) _b ^{∇ -n} f (t ) satisfies the n th order discrete initial value problem [figure omitted; refer to PDF]

(iii): The Cauchy function (s - ρ (t ) ^{) n -1 ¯} / Γ (n ) satisfies _[ecedil]6; ^{Δ n} y (t ) =0 .

Definition 2.3.

(i) The (nabla) left fractional difference of order α >0 is defined by [figure omitted; refer to PDF]

(ii) The (nabla) right fractional difference of order α >0 is defined by [figure omitted; refer to PDF]

Here and throughout the paper n = [ α ] +1 , where [ α ] is the greatest integer less than or equal α .

Regarding the domains of the fractional difference operators we observe the following.

(i) The nabla left fractional difference ^{∇ a α} maps functions defined on _{... a} to functions defined on _{... a +n} (on _{... a} if we think f =0 before a ).

(ii) The nabla right fractional difference ^{∇ b α} maps functions defined on _b ... to functions defined on _{b -n} ... (on _b ... if we think f =0 after b ).

3. A Relation between the Operators ^{∇ a - α} and ^{[white triangle down] a - α}

In this section we illustrate how two operators, ^{∇ a - α} and ^{[white triangle down] a - α} are related.

Lemma 3.1.

The following holds:

(i) ^{[white triangle down] a +1 - α} f (t ) = ^{∇ a - α} f (t ) ,

(ii) ^{[white triangle down] a - α} f (t ) = ( 1 / Γ ( α ) ) (t -a +1 ^{) α -1 ¯} f (a ) + ^{∇ a - α} f (t ) .

Proof.

The proof of (i) follows immediately from the above definitions ( 1.1) and ( 1.2). For the proof of (ii), we have [figure omitted; refer to PDF]

Next three lemmas show that the above relations on the operators ( 1.1) and ( 1.2) help us to prove some identities and properties for the operator ^{∇ a - α} by the use of known identities for the operator ^{[white triangle down] a - α} .

Lemma 3.2.

The following holds: [figure omitted; refer to PDF]

Proof.

It follows from Lemma 3.1and Theorem 2.1 in [ 13] [figure omitted; refer to PDF]

Lemma 3.3.

Let α >0 and β > -1 . Then for t ∈ _{... a} , the following equality holds [figure omitted; refer to PDF]

Proof.

It follows from Theorem 2.1 in [ 13] [figure omitted; refer to PDF]

Lemma 3.4.

Let f be a real-valued function defined on _{... a} , and let α , β >0 . Then [figure omitted; refer to PDF]

Proof.

It follows from Lemma 3.1and Theorem 2.1 in [ 2] [figure omitted; refer to PDF]

Remark 3.5.

Let α >0 and n = [ α ] +1 . Then, by Lemma 3.2we have [figure omitted; refer to PDF] or [figure omitted; refer to PDF]

Then, using the identity [figure omitted; refer to PDF]

we verified that ( 3.2) is valid for any real α .

By using Lemma 3.1, Remark 3.5, and the identity ∇ ^{( t -a ) α -1 ¯} = ( α -1 ) ^{( t -a ) α -2 ¯} , we arrive inductively at the following generalization.

Theorem 3.6.

For any real number α and any positive integer p , the following equality holds: [figure omitted; refer to PDF] where f is defined on _{... a} .

Lemma 3.7.

For any α >0 , the following equality holds: [figure omitted; refer to PDF]

Proof.

By using of the following summation by parts formula [figure omitted; refer to PDF]

we have [figure omitted; refer to PDF]

On the other hand, [figure omitted; refer to PDF] where the identity [figure omitted; refer to PDF] and the convention that (0 ^{) α -1 ¯} =0 are used.

Remark 3.8.

Let α >0 and n = [ α ] +1 . Then, by the help of Lemma 3.7we have [figure omitted; refer to PDF] or [figure omitted; refer to PDF]

Then, using the identity [figure omitted; refer to PDF]

we verified that ( 3.12) is valid for any real α .

By using Lemma 3.7, Remark 3.8, and the identity Δ ^{( b -t ) α -1 ¯} = - ( α -1 ) ^{( b -t ) α -2 ¯} , we arrive inductively at the following generalization.

Theorem 3.9.

For any real number α and any positive integer p , the following equality holds: [figure omitted; refer to PDF] where f is defined on _b N .

We finish this section by stating the commutative property for the right fractional sum operators without giving its proof.

Lemma 3.10.

Let f be a real valued function defined on _b ... , and let α , β >0 . Then [figure omitted; refer to PDF]

4. Summation by Parts Formulas for Fractional Sums and Differences

We first state summation by parts formula for nabla fractional sum operators.

Theorem 4.1.

For α >0 , a , b ∈ ... , f defined on _{... a} and g defined on _b ... , the following equality holds [figure omitted; refer to PDF]

Proof.

By the definition of the nabla left fractional sum we have [figure omitted; refer to PDF] If we interchange the order of summation we reach ( 4.1).

By using Theorem 3.6, Lemma 3.4, and ^{∇ a - (n - α )} f (a ) =0 , we prove the following result.

Theorem 4.2.

For α >0 , and f defined in a suitable domain _{... a} , the following are valid [figure omitted; refer to PDF]

We recall that ^{D - α D α} f (t ) =f (t ) , where ^{D - α} is the Riemann-Liouville fractional integral, is valid for sufficiently smooth functions such as continuous functions. As a result of this it is possible to obtain integration by parts formula for a certain class of functions (see [ 23] page 76, and for more details see [ 22]). Since discrete functions are continuous we see that the term ^{∇ a - (1 - α )} f (t ) _{| t =a} , for 0 < α <1 disappears in ( 4.4), with the application of the convention that ^{∑ s =a +1 a} ...f (s ) =0 .

By using Theorem 3.9, Lemma 3.10, and _b ^{∇ - (n - α )} f (b ) =0 , we obtain the following.

Theorem 4.3.

For α >0 , and f defined in a suitable domain _b ... , we have [figure omitted; refer to PDF]

Theorem 4.4.

Let α >0 be noninteger. If f is defined on _b N and g is defined on _{N a} , then [figure omitted; refer to PDF]

Proof.

Equation ( 4.7) implies that [figure omitted; refer to PDF] And by Theorem 4.1we have [figure omitted; refer to PDF]

Then the result follows by ( 4.4).

5. Initial Value Problems

Let us consider the following initial value problem for a nonlinear fractional difference equation [figure omitted; refer to PDF] where 0 < α <1 and a is any real number.

Apply the operator ^{∇ a - α} to each side of ( 5.1) to obtain [figure omitted; refer to PDF]

Then using the definition of the fractional difference and sum operators we obtain [figure omitted; refer to PDF]

It follows from Lemma 3.2that [figure omitted; refer to PDF]

Note that [figure omitted; refer to PDF]

Hence we obtain [figure omitted; refer to PDF] which follows from the power rule in Lemma 3.3.

Let us put this expression back in ( 5.5) and use ( 4.4), we have [figure omitted; refer to PDF]

Thus, we have proved the following lemma.

Lemma 5.1.

y is a solution of the initial value problem, ( 5.1), ( 5.2), if, and only if, y has the representation ( 5.9).

Remark 5.2.

A similar result has been obtained in the paper [ 13] with the operator ^{[white triangle down] a α} . And the initial value problem has been defined in the following form [figure omitted; refer to PDF] where 0 < α ...4;1 and a is any real number. The subscript a of the term ^{[white triangle down] a α} y (t ) on the left hand side of ( 5.10) indicates directly that the solution has a domain starts at a . The nature of this notation helps us to use the nabla transform easily as one can see in the papers [ 2, 12]. On the other hand, the subscript a -1 in ( 5.1) indicates that the solution has a domain starts at a .

6. An Alternative Definition of Nabla Fractional Differences

Recently, the authors in [ 18], by the help of a nabla Leibniz's Rule, have rewritten the nabla left fractional difference in a form similar to the definition of the nabla left fractional sum. In this section, we do this for the nabla right fractional differences.

The following delta Leibniz's Rule will be used: [figure omitted; refer to PDF]

Using the following identity [figure omitted; refer to PDF] and the definition of the nabla right fractional difference (ii) of Definition 2.3, for α >0 , α ∉ ... we have [figure omitted; refer to PDF]

By applying the Leibniz's Rule ( 6.1), n -1 number of times we get [figure omitted; refer to PDF]

In the above, it is to be insisted that α ∉ ... is required due the fact that the term 1 / Γ ( - α ) is undefined for negative integers. Therefore we can proceed and unify the definitions of nabla right fractional sums and differences similar to Definition 5.3 in [ 18]. Also, the alternative formula ( 6.4) can be employed, similar to Theorem 5.4 in [ 18], to show that the nabla right fractional difference ^{∇ α} f b is continuous with respect to α ...5;0 .

7. Conclusions

In fractional calculus there are two approaches to obtain fractional derivatives. The first approach is by iterating the integral and then defining a fractional order by using Cauchy formula to obtain Riemann fractional integrals and derivatives. The second approach is by iterating the derivative and then defining a fractional order by making use of the binomial theorem to obtain Grünwald-Letnikov fractional derivatives. In this paper we followed the discrete form of the first approach via the nabla difference operator. However, we noticed that in the right fractional difference case we used both the nabla and delta difference operators. This setting enables us to obtain reasonable summation by parts formulas for nabla fractional sums and differences in Section 4and to obtain an alternative definition for nabla right fractional differences through the delta Leibniz's Rule in Section 6.

While following the discrete form of the first approach, two types of fractional sums and hence fractional differences appeared; one type by starting from a and the other type, which obeys the general theory of nabla time scale calculus, by starting from a +1 in the left case and ending at b -1 in the right case. Section 3discussed the relation between the two types of operators, where certain properties of one operator are obtained by using the second operator.

An initial value problem discussed in Section 5is an important application exposing the derived properties of the two types of operators discussed throughout the paper, where the solution representation was obtained explicitly for order 0 < α <1 . Regarding this example we remark the following. In fractional calculus, Initial value problems usually make sense for functions not necessarily continuous at a (left case) so that the initial conditions are given by means of ^{a +} . Since sequences are nice continuous functions then in Theorem 4.2, which is the tool in solving our example, the identity ( 4.4) appears without any initial condition. To create an initial condition in our example we shifted the fractional difference operator so that it started at a -1 .

Acknowledgment

The second author is grateful to the Kentucky Science and Engineering Foundation for financial support through research Grant KSEF-2488-RDE-014.

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