Abstract
This paper is motivated by some papers treating the fractional derivatives. We introduce a new definition of fractional derivative which obeys classical properties including linearity, product rule, quotient rule, power rule, chain rule, Rolle’s theorem, and the mean value theorem. The definition for all , and . If , this definition coincides to the classical definition of the first order of the function .
1. Introduction
The objective of fractional calculus is to generalize traditional derivatives to noninteger orders, see [1–4]. As is well known, many dynamic systems are best characterized by a dynamic fractional-order model, generally based on the notion of differentiation or integration of noninteger order. The study of fractional order systems is more delicate than for their whole order counterparts. Indeed, fractional systems are, on the one hand, considered as memory systems, in particular to take into account the initial conditions, and on the other hand, they present a much more complex dynamic system.
The theory of fractional derivative is a very old theory, which dates back to a conversation on September 30, 1695, between Hôpital and Leibniz concerning the definition of the operator for . Thus, as the time progresses, certain approaches have been given in the literature such as the definition of Riemann–Liouville and that of Caputo.
Recently, the authors in [5] and in [6] define new well-behaved simple fractional derivatives called the conformable fractional derivative depending just on the basic limit definition of the derivative.
Khalil et al. [5] have introduced a new derivative called the conformable fractional derivative of f of order and is defined byfor a function and , and the fractional derivative at 0 is defined as .
They then defined the fractional derivative of higher order (i.e., of order ). They also defined the fractional integral of order only. They then proved the product rule, and the fractional mean value theorem solved some (conformable) fractional differential equations where the fractional exponential function played an important rule.
Katugampola introduced in [6] the new derivative which is defined byfor , . If is differentiable in some , and exists, then define
As a consequence of the above definitions, the authors in [5, 6] showed that the derivatives obey the product rule and quotient rule and have results similar to Rolle’s theorem and the mean value theorem in classical calculus.
The purpose of this work is to further generalize the results obtained in [5] and in [6] and introduce a new conformable fractional derivative as the most natural extension of the familiar limit definition of the derivative of a function at a point.
Remark 1. If and or are differentiable, we have(for the derivative in [5]) and(for the derivative in [6]).
These expressions tends to infinity when is very small, but this brings regularities in several mathematical problems especially when one seeks to bounded as well as these integer derivatives , … (generalized function theory, for example).
This remark is the main motivation for our definition:
2. New Fractional Derivative
Definition 1. Given a function , and then the conformable fractional derivative of order is defined byfor all , and .
If is differentiable in some , and exists, then define
Theorem 1. If a function and differentiable at , then f is continuous at .
Proof. Since , then .
Let . Then, , which implies that .
Hence, f is continuous at .
Theorem 2. Let and be differentiable at a point . Then,(1), for all a, b.(2) for all .(3), for all constant functions .(4).(5).(6)If in addition, is differentiable, then .
Proof. Parts (1) through (3) follow directly from the definition. We choose to prove (4) and (6) only since they are crucial. Now, for fixed ,Since is continuous at then . This completes the proof of part (4).
(5) can be proved in a similar way.
To prove (6), let in Definition 1. and then . Therefore,
Theorem 3. Let and . Then we have the following results:(1) for all .(2).(3), for all .(4), .(5), .
Theorem 4. However, it is worth noting the following conformable fractional derivatives of certain functions:(1).(2).(3).(4).
Remark 2. In the case of the derivative proposed in [5], we have the following remark: a function could be differentiable at a point but not differentiable, for example, take , then where for . But does not exist. This is not the case for the known classical fractional derivatives.
But for our definition, we have the same results of the classic case, and that is one more advantage of our derivative.
Next, we consider the possibility of , for some . We have the following definition.
Definition 2. Let , for some , and function be an differentiable at . Then the fractional derivative of is defined byif the limit exists.
Remark 3. As a direct consequence of Definition 2, we can show thatwhere and is differentiable at .
Remark 4. The previous definitions of fractional derivative Riemann–Liouville and Caputo do not enable us to study the analysis of differentiable functions. However, our definition makes it possible to prove basic analysis theorems such as Rolle’s theorem and the mean value theorem.
Theorem 5. Rolle’s theorem for conformable fractional differentiable functions.
Let and be a given function that satisfies(1)f is continuous on ,(2)f is -differentiable for some ,(3).Then, there exists , such that .
Proof. Since f is continuous on , and , there is which is a point of local extrema. With no loss of generality, assume c is a point of local minimum. SoBut, the first limit is nonnegative, and the second limit is nonpositive.
Hence .
Theorem 6. Mean value theorem for conformable fractional differentiable functions).
Let and , be a given function that satisfies(1)f is continuous on ,(2)f is -differentiable for some .Then, there exists such that
Proof. Consider the functionThen,andThus . By Rolle’s theorem, there exists , such that . Using the fact that , the result follows.
Along the same lines in basic analysis, one can use the present mean value theorem to prove the following proposition.
Proposition 1. Let be differentiable for some .(i)If is bounded on , where , then is uniformly continuous on , and hence, is bounded.(ii)If is bounded on and continuous at , then is uniformly continuous on , and hence, is bounded.
2.1. Fractional Integral
As in the work [5], it is interesting to note that, in spite of the variation of the definitions of the fractional derivatives, we can still adopt the same definition of the fractional integral here due to the fact that we obtained similar results in Theorem 4 as of the results (1)–(6) and (i)–(iii) in [5]. So, we have the following definition.
Definition 3. Let and , let f be a function defined on , Then, the fractional integral of f is defined by
Theorem 7. If is any continuous function in the domain of and . Then, for , we have
Proof. Since f is continuous, then is clearly differentiable. Hence,
Lemma 1. Let be the function differentiable and . Then, for , we have
2.2. Applications
The authors of [5] introduced a new definition of fractional derivative to facilitate the calculations performed to solve the differential equation proposed by Professor S. Momanibut using the following approximation to 1.
Here we consider the following equation:
Let us find a solution of the homogeneous equation . We look for a solution of the form . It is easy to verify that is a particular solution of the nonhomogeneous equation.
Now, the general solution is where is a constant. Finally, the initial condition y(0) = 0 implies that . Hence .
Example 1. Based on the point (3) of Theorem 4, one can easily see that the auxiliary equation for is , so that the solution is given by .
The details of the solution of certain differential equations will be given later.
Remark 5. In the following example, we will show the benefit of the fractional derivative product rule which allows us to use the idea of the integrating factor
Example 2. Consider the following example:By multiplying it by , we obtain and take advantage of the product rule for this fractional derivative (which is not possible with Riemann–Liouville and Caputo fractional derivatives): .
By using the fractional integral (18), we have .
Therefore, .
Example 3. Let us look for a differentiable solution which verifies this equation.
Since , then .
Thus, the fractional differential equation (25) becomeswhich brings back toThis is a differential equations of Bernouilli and can be solved easily.
Example 4. Consider now the fractional differential equation:As before, the fractional differential equation (28) givesFrom whereAnd this is a differential equation of Riccati.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Authors’ Contributions
All the authors contributed equally to the writing of this paper. All the authors read and approved the final manuscript.