Abstract
In this paper, the definitions of the fractional derivative and integrals are given by the neutrix limit. Not only is it consistent with the classical results but also the representations of the fractional derivative and integrals are obtained for , , , and , where is the analytic function.
1. Introduction
Fractional calculus has been used to model physics and engineering processes widely since standard mathematical models of integer-order derivatives, including nonlinear models. In fact, in many fields such as mechanics, electronics, chemistry, biology, economics, notably control theory, and signal and image processing, fractional calculus has been playing more and more important roles in recent years [1–5]. There are several definitions for fractional derivatives and integrals.
The Riemann–Liouville fractional integral and derivatives [6] are defined as.andfor and , respectively. The Grünwald–Letnikov fractional derivative [7] is defined aswhere is the integer part of . It has been proved that this definition is equivalent towhere . The Caputo fractional derivative, from [8], is defined aswhere . The fractional derivative defined above is called the left fractional integral and derivative. It is completely analogous to introducing the corresponding right fractional derivative. For example, the right Riemann–Liouville fractional derivatives is defined as
From the definitions (4) and (5), we have when .
For the fractional integral and derivative of the power function , there areandfor and . When , (7) and (8) are clearly not valid. However,where the symbol represents the Pochhammer symbol, i.e.,and it requires a proper definition of the fractional derivative of . To remedy this problem, the following definition is given by Mauro Bologna [9]:
It is clear that (8) and (11) there is no essential connection, to be more reasonable definition. In fact, if (11) is true, then we have by (7)which is impossible.
Therefore, the traditional definition of fractional derivatives for is unreasonable. In this paper, our goal is to give a reasonable definition and representations of fractional derivatives for , , , , where is an analytic function. The properties of the fractional derivatives of the correlation function and the solution of the correlation fractional differential equation are discussed. For simplicity, we take.
We use neutrix limit [9–16] to define the frational derivatives for (1)–(6).
Definition 1. 1 For complex number , letorandFor convenience of writing, let .
Remark 1. If , then .
The so-called neutrix limit is defined as follows.
Definition 2. Iffor , thenFor example, byfor , we haveBy exchanging the order of integration and summing, we haveFrom Definitions 2 and (23), we obtain thatfor andfor .
It should be pointed out that we can consider the fractional order derivatives of based on definitions (14)–(18), where is the analytic function and , and present the following two examples to illustrate our conclusions.
2. Lemmas
In this section, we shall present some important lemmas that will be frequently used and their proof may be found in [15, 16].
Lemma 1. For Beta function , there isfor where andfor .
Lemma 2. For the complex number and , there iswhere . In particular,
Lemma 3. For , one haswhere .Here, is the digamma function defined byand is the Hurwitz zeta function defined bywhere is the Riemann zeta function.
Lemma 4. For complex number , there is
Lemma 5. For complex number , there is.for and
Lemma 6. For , there areand
Lemma 7. For the complex number , there is
The numerical calculation shows that (40) is always true, but its theory is a little harder to prove.
3. Fractional Derivatives of and
Through the analytical continuation of , we see that (7) and (8) still hold for and . In this section, we shall consider the fractional order derivatives of and for all complex and and expect to get similar results according to the definition of neutrix limit. Our main results read as follows.
Theorem 1. According to definition (14)–(16), (7), and (8) still hold for the complex .
Proof. For , using (21), Definitions 2 and (35) yields thatBy (16) and (41), we havewhich implies that (7) holds.
Employing (16) and (7), we have.which concludes that (8) holds.
Corollary 1. (1)For the complex , there is .(2).
Proof. (1)For , Remark 1. indicates that corollary naturally holds. For , , , we have Due to for we have by Definition 2. which implies that Hence, the first part of the corollary holds.(2)From (41) and we have Then, Therefore, the second part of the corollary holds.
Theorem 2. For complex and , there isorfor andor
Proof. By (21) and exchanging the order of integration and then summing, we haveFrom (36), we haveBy (16), (35), (53), (54), Lemma 3, and Definition 2, we getwhich implies that (49) and (50) hold.
It’s worth noting that if we use the derivative method with parameter, thenwhich is exactly the same as the derivation above.
By (16) and (49), we haveIn fact, can be obtained from (29) and (57), but the expression is more complex. Here, we still use the following derivative method with parameter.so, we get (51) and (52).
Corollary 2. For the complex , there is.
Proof. For , by similar to the proof of Corollary 1, we find that (45) still holds. Hence, .
By (29), we getMoreover, from (49), we haveSubstituting (61) into (60), we getBy (40) and (62), we haveThis completes the proof of the corollary.
Theorem 3. For the complex and , there isfor andfor .
Proof. For , by (21) and (26), we havewhich implies that (64) holds.
By (16), (64), and (30), we getwhere . Usingand (67) yields that (65) holds.
Remark 2. Due tofor , we haveUnder the definition given in this paper, whether , or , there is always . So is just the usual derivative , which implies that the definition in this paper is reasonable. For example, taking there is
Corollary 3. For , there isandBy (72) and (73), we see that and are not equal for .
Remark 3. When , then . However, when , the relationship is broken. For example, by (64) and (30), we havefor .
Obviously, .
Theorem 4. For , there isfor , whereandwith
Proof. By (21), we haveFrom (36), (37), and (79), we getBy and exchanging the sum order, we see that (75) holds.
Using (16) and (29) yields thatHence, we conclude that (77) holds.
Remark 4. (1)For , noting that and employing (38), (39), and (77), we have is just the usual derivative , which implies that the definitions in this paper are reasonable.(2)From (7), (8), (49), (51), (64), and (65), we can still get where , . For example, where and are used. Again, this shows that the definitions in this paper are reasonable.
4. Fractional Derivatives of Some Elementary Functions
In this section, we shall study the fractional order derivatives of some elementary functions. First of all, we give the following general theorem.
Theorem 5. If is the analytic function, i.e.,then the following formulas are true,
In particular,andwhere we have used the following equations
For the fractional derivative, there is the following Leibniz derivative formulawhere is any of or , and are continuous in . Since , by (90), we have
Using (92), we obtain that , which is not consistent with Corollary 3 so that (92) does not hold for . Then, we cannot calculate , by (92). Therefore, (, , and ) cannot be completed by (92).
If is the analytic function, the Leibniz derivative formula (92) is still valid for the Riemann–Liouville fractional integral and derivatives [5]. For example, if is an analytic function in Theorem 5, the following formula still holds:
By (94), we haveor
The numerical calculation shows that (95) and (96) are exactly same as (89), but (89) is the calculated fastest and (95) is the calculated slowest.
For we have by (94)where is one of the . The numerical calculation shows that (97) is exactly same as (90), but (90) is calculated faster.
Remark 5. Based on the derivative formula of elementary function, (94) is more convenient to calculate , but the calculation efficiency is not as fast as (87).
5. Applications to the Fractional Differential and Integral Equations
At last, as an application of the definitions of the fractional derivative and integrals given by the neutrix limit, this section is devoted to study the solution of fractional differential and integral equations.
Theorem 6 (see [1]). If , then the solution of fractional integral equation.is given bywhere is the generalized Mittag-Leffer function and it is defined as
Through the analytical continuation, when , the above conclusion is also established. For (99) is not a solution of the fractional integral equation (98), but we have the following result.
Theorem 7. If , then the solution of the following fractional integral equationis given bywhere
Proof. Applying the operator to both sides of (101), we have by (64)Summing up the above expression with respect to from 0 to gives rise toHence, we deduce that (102).
Theorem 8. If , then the solution of the following fractional differential equationis given by
Proof. Applying the operator to both sides of (106), using (64) andwe get the following integral equation.whereApplying the operator to both sides of (109), and then using (7), (58), and Lemma 3 yield thatwhereSumming up the above expression with respect to from 0 to yields that
Theorem 9. If , then the solution of the following fractional differential equationis given by
Proof. Applying the operator to both sides of (114), using (64) imply thatthus, we get the following integral equation.wherethen, according to Theorem 8, we get (115).
Theorem 10. If , then the solution of the following fractional differential equationis given bywhere
Proof. Applying the operator to both sides of (118), using (75) and (108), we getwherefor .
Exactly according to the discussion of Theorems 8 and 9, we obtainBy Lemma 3 and (124) we obtain (120).
Let us substitute the solutions of (102), (107), (115), and (120) into their respective equations (101), (106), (114), and (119), and make their error curves (see (1)–(4) in Figure 1). Without loss of generality.
Here, let . In (1)–(4) of Figure 1, the figure above is the solution curve and the figure below is the error curve. The numerical results show that the solutions of the fractional differential and integral equations obtained above satisfy their respective equations. It can be seen from (2) and (3) of Figure 1 that although the expressions of (107) and (115) are different, their graphs are almost the same. This shows that equations (106) and (114) do not describe a fundamentally different problem.
Finally, we verify the correctness of the above results by numerical solutions of fractional differential equations. According to the method in literature [5], for (106) and (119), the following iterative algorithms are, respectively, as follows:andfor , whereWithout loss of generality, let , where is the analytic solution. The analytical solutions of (107) or (106) and the numerical solution obtained by (125) are shown in (1) in Figure 2. The analytical solutions of (120) or (119) and the numerical solution obtained by (126) are shown in (2) in Figure 2.
As can be seen from Figure 2, the analytical solution obtained in this paper is basically consistent with the numerical solution.
6. Conclusions
As defined in this paper, the following function is consistent with traditional fractional integrals and derivatives:where is the analytic function, . The fractional integrals and derivatives of the following function can be obtained:where is the analytic function, . The following Leibniz derivative formula of the Riemann–Liouville fractional derivative still holds:and the propertiesare true.
Therefore, we present the definitions of the fractional derivative and integrals given by the neutrix limit in our paper. Based on it, the solutions to some of the differential and integral equations given in the previous section cannot be expressed by traditional methods, and the solutions to these differential and integral equations are also expressed by the definitions in this paper.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (61379009 and 61771010) and the Natural Science Foundation of Shandong Province (ZR2021MA017).