Journal of Function Spaces
Volume 2020 (2020), Article ID 5852414, 10 pages
https://doi.org/10.1155/2020/5852414

Research Article

Improvement on Conformable Fractional Derivative and Its Applications in Fractional Differential Equations

Feng Gao¹ and Chunmei Chi²

¹School of Science, Qingdao University of Technology, Qingdao, China, 266033
²School of Information and Control Engineering, Qingdao University of Technology, Qingdao, China, 266033

Correspondence should be addressed to Feng Gao; gaofeng99@sina.com

Received 7 June 2020; Accepted 4 July 2020; Published 1 August 2020

Academic Editor: Xinguang Zhang

Copyright © 2020 Feng Gao and Chunmei Chi. 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.

In this paper, we made improvement on the conformable fractional derivative. Compared to the original one, the improved conformable fractional derivative can be a better replacement of the classical Riemann-Liouville and Caputo fractional derivative in terms of physical meaning. We also gave the definition of the corresponding fractional integral and illustrated the applications of the improved conformable derivative to fractional differential equations by some examples.

1. Introduction

The fractional order derivative has always been an interesting research topic in the theory of functional space for many years [1–11]. Various types of fractional derivatives were introduced, among which the following Riemann-Liouville and Caputo are the most widely used ones.

(1)Riemann-Liouville Definition. For , the derivative of is

(2)Caputo Definition. For , the derivative of is

Both Riemann-Liouville definition and Caputo definition are defined via fractional integrals. Therefore, these two fractional derivatives inherit some nonlocal behaviors including historical memory and future dependence. All definitions including (1) and (2) above satisfy the property that the fractional derivative is linear. This is the only property inherited from the 1st derivative. However, the existing fractional derivatives do not satisfy the following properties which the integral derivatives have.

(1)Most of the fractional derivatives except Caputo-type derivatives do not satisfy , if is not a natural number(2)All fractional derivatives do not obey the familiar product rule for two functions:

(3)All fractional derivatives do not obey the familiar quotient rule for two functions:

(4)All fractional derivatives do not obey the chain rule:

(5)Fractional derivatives do not have corresponding Rolle’s theorem(6)Fractional derivatives do not have a corresponding mean value theorem(7)In general, all fractional derivatives do not obey

(8)The Caputo definition assumes that the function is differentiable

To overcome some of these difficulties, Khalil et al. [12] proposed a new interesting factional derivative definition called conformable derivative that extends the familiar limit definition of the derivative of a function given by the following.

Definition 1. Given a function , then the conformable fractional derivative of of order is defined by

for and . If is differentiable in some , , and exists, then define . It is easy to see that if is differentiable, then . One can find functions which are -differentiable at a point but not differentiable at this point.

As a result of the above definition, the authors in [12] showed that the conformable derivative obeys the product rule and quotient rule and has results similar to Rolle’s theorem and the mean value theorem in classical calculus.

The conformable fractional derivative has two advantages over the classical fractional derivatives. First, the conformable fractional derivative definition is natural and it satisfies most of the properties which the classical integral derivative has such as linearity, product rule, quotient rule, power rule, chain rule, vanishing derivatives for constant functions, Rolle’s theorem, and mean value theorem. Second, the conformable derivative bring us a lot of convenience when it is applied for modelling many physical problems, because the differential equations with conformable fractional derivative are easier to solve numerically than those associated with the Riemann-Liouville or Caputo fractional derivative. In fact, many researchers have already applied conformable fractional derivative to many fields and a lot of corresponding techniques were developed [13–20].

However, there are still shortcomings or disadvantages for the conformable derivative. If we look at the Riemann-Liouville and Caputo fractional derivative definition, we have

for , and

for , whereas

for .

That means the physical meaning of the classical Riemann-Liouville and Caputo fractional derivative is quite different from that of the conformable derivative, especially when and close to . We graph the conformable derivatives of and in Figures 1 and 2 and make comparison with the Riemann-Liouville and Caputo fractional derivatives. We can see if the Riemann-Liouville and Caputo fractional derivatives are replaced by conformable derivative, a large error will occur. To overcome this difficulty, we propose a kind of modified conformable fractional derivative in Section 2. This modified conformable fractional derivative is a local operator on the one hand and approximates the Riemann-Liouville and Caputo fractional derivative better on the other hand. So it is a better choice to replace the classical Riemann-Liouville and Caputo fractional derivative with the improved conformable fractional derivative, especially when and close to .

Figure 1: 0.3th derivative of .

Figure 2: 0.2th derivative of .

2. Improvement on Conformable Fractional Derivative

First, we give the definition for the improved Caputo-type conformable fractional derivative and the improved Riemann-Liouville-type conformable fractional derivative for .

Definition 2. Given a function , the improved Caputo-type conformable fractional derivative of of order is defined by

where , is a given number.

The improved Riemann-Liouville-type conformable fractional derivative of of order is defined by

where , is a given number.

It is easy to see

If , both and coincide with . In Definition 2, we introduce to let and have a kind of historical memory as the Caputo and Riemann-Liouville fractional derivative have.

For , we give the following.

Definition 3. Given a function , the improved Caputo-type conformable fractional derivative of of order is defined by

where .

The improved Riemann-Liouville-type conformable fractional derivative is defined by

where .

It is easy to see

If , both and coincide with .

Remark 4. As a result of Definitions 2 and 3, we can easily show that if is 1-differentiable at , then

for . If is -differentiable at , then

for .

In fact, compared with the conformable fractional derivative, the improved conformable fractional derivative can be a better replacement of the Caputo and Riemann-Liouville fractional derivative. One can see in Figures 3–9 that the improved conformable fractional derivative approximates the Caputo and Riemann-Liouville fractional derivative in a better way for most elementary functions especially when is larger than and close to and even when is close to .

Figure 3: 0.2th fractional derivative of .

Figure 4: 0.5th fractional derivative of .

Figure 5: 0.8th fractional derivative of .

Figure 6: 0.2th fractional derivative of .

Figure 7: 0.5th fractional derivative of .

Figure 8: 0.8th fractional derivative of .

Figure 9: 0.2th fractional derivative of .

Based on the results in [12], one can prove the following theorem.

Theorem 5. We can easily show that the improved conformable fractional derivative satisfies the following properties if .

(1)(2)(3)(4)

Theorem 6. We get the following improved conformable fractional derivatives of certain functions for .

(1)(2)(3)(4)(5)(6)(7)(8)(9)

Proof. It is very easy to verify properties (1)–(8) if we take into account the conclusions in [12]. We only prove property (9) here. From [12], we have

Therefore,

Now, we give the following definition for fractional integral.

Definition 7. For and continuous function , let

When and coincides with the usual Riemann integral.

Theorem 8. Suppose , we have

(1)(2)

Proof. We only prove (1); (2) can be proved in the same way.

Let ; by using a variation of the constant method, we can get

Let

we have

3. Applications of Improved Conformable Fractional Derivative

We solve fractional differential equations by using the improved conformable fractional derivatives.

Example 1. Consider the following fractional differential equation:

If is understood as the Riemann-Liouville or Caputo fractional derivative, the solution to this problem is

Now, we take as the improved conformable fractional derivative; the problem reduces to

We can solve this problem through the variation constant method and get the following solution:

If is understood as the conformable fractional derivative, the problem reduces to

and the solution to this problem is

We compare the three solutions in Figure 10. We can see that the improved conformable fractional derivative solution is much better than the solution obtained by using the conformable fractional derivative. That means if we prefer a replacement of the Riemann-Liouville or Caputo fractional derivative, the improved conformable fractional derivative is a better choice.

Example 2. In general, we consider the problem

We can get the following three solutions:

(i)If is the Riemann-Liouville or Caputo fractional derivative, the solution is , and

(ii)If is the conformable fractional derivative, the problem can be reduced to

The solution is

We can see that

(iii)If is the improved conformable fractional derivative, the problem can be reduced to

Its solution can be obtained through the variation of the constant method.

Figure 10: Solution to Example 1.

Since

we have

Therefore,

The result for the general equation shows that the improved conformable derivative has advantages over other fractional derivatives. First, it is a good approximation to the classical Riemann-Liouville or Caputo fractional derivative so it has a similar physical meaning with the Riemann-Liouville and Caputo fractional derivative. Second, it is a local derivative operator so it is easy for numerical computing.

Example 3. Consider the following fractional differential equation:

If is the Caputo fractional derivative, the solution to this problem is

If is the conformable fractional derivative, the problem reduces to

The solution to this problem is

If is the improved conformable fractional derivative, the problem reduces to

Let , the problem further reduces to

Solve this problem by using the method of constant variation, we get

The solutions to Example 3 can be seen in Figure 11.

Figure 11: Solution of Example 3.

4. Conclusion

We propose a kind of improved conformable fractional derivative in this paper. This improved conformable fractional derivative is also local by its definition, and meanwhile, a kind of historical memory parameter is introduced to its definition. The advantage of the improved conformable derivative is that its physical behavior approximates the Riemann-Liouville and Caputo fractional derivative better than the conformable fractional derivative. So this improved conformable fractional derivative has much potential in modelling many physical problems where the Riemann-Liouville and Caputo fractional derivative is usually used.

Data Availability

Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.

Conflicts of Interest

The authors declare that they have no competing interests.

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