1. Introduction
The theory of fractional calculus has interested many researchers, as a theoretical extension of classical mathematical analysis that has also been successfully applied to solve important problems in different scientific disciplines. In the evolution of the definition of fractional derivatives, two formulations have been proposed: a non-local approach and another based on a local conception. The non-local concept of a fractional derivative, which has played a fundamental role in the development of fractional calculus, includes well-known definitions such as the Riemann–Liouville (RL) and Caputo (C) derivatives. The key properties and applications of these definitions are mentioned in [
1,
2,
3].
Classical fractional derivatives, such as C and RL derivatives, have both advantages and disadvantages. However, in these definitions of derivatives, the linearity property is satisfied, and they do not possess certain essential properties of the ordinary derivative. For instance, the non-zero RL derivative of a constant differs from the behavior of the ordinary derivative. Furthermore, these definitions lack the fundamental properties of the ordinary derivative, including product, quotient, and chain rules. However, the C derivative is only defined for differentiable functions.
The local formulation of non-integer order derivatives arises with the idea of overcoming the disadvantages associated with non-local fractional derivatives. The definitions of local fractional derivatives are established based on incremental ratios. A well-known definition in this category is the conformable derivative, introduced by Khalil et al. [
4]. This definition successfully addresses some of the limitations of non-local fractional derivatives. Furthermore, conformable calculus offers a direct approach for obtaining analytical solutions for various applications of fractional calculus problems. However, according to [
5], the conformable derivative has a disadvantage, and its results cannot be considered satisfactory in comparison to the C definition for certain functions. Recently, in [
6], a new generalized derivative of non-integer order, the so-called Abu-Shady–Kaabar derivative, allows fractional differential equations to be solved analytically in a simple way, whose results are in exact agreement with those obtained through the RL or C derivative. Furthermore, the study of this new derivative has been extended to important fields of classical analysis, such as special functions or the fixed-point theorem [
7,
8].
On the other hand, another notable non-integer local differential operator is the so-called fractal derivative introduced in [
9]. This derivative is established from the fractal definition, and it has been used to model various scientific phenomena concerning power law scaling, such as quantum mechanics and turbulence [
9,
10].
Some recent research studies have proven the internal connection between fractional calculus and fractal calculus in the context of geometry where the fractal dimension of the function with a fractional order can be modified via the change in fractional calculus formulation [
11] (see also [
12] for more recent research about the connection between them via the approximation of continuous functions).
In [
13], a hybrid approach of differentiation that combines both fractional and fractal differentiations is introduced. In this new differential operator, various properties (memory effect, heterogeneity, elastic viscosity, and fractal geometry) of the dynamic system are considered.
Based on the existing literature and the limitations of non-local fractional derivatives, our research aims to introduce a new local FF derivative and explore its fundamental properties. To accomplish this objective, we structured our study into several stages, which are outlined as follows:
The definition of the generalized FF derivative of order α is introduced in this study, which produces results consistent with the outcomes obtained using the FF derivative of order
in the C sense with power law, as mentioned in [
13].
Furthermore, we establish the fundamental elements of generalized FF calculus, including operations with generalized FF -differentiable functions, the chain rule, mean value theorems, and the inverse function theorem.
Then, we define the generalized fractal -integral and present two significant results of integral calculus, namely the fundamental theorem of calculus and Barrow’s rule, in this context.
Finally, we give some interesting applications of the proposed derivative to FF ordinary differential equations.
Our results are original and novel because they provide a simple mathematical tool that can be applied efficiently in modeling various systems and phenomena, proposed with fractional order and fractal dimension, in sciences, engineering, economics, and medicine, where the connection between both fractional calculus and fractal geometry can play an important role in studying those systems.
3. Generalized Fractal–Fractional Derivative and Its Properties
A new local type of FF derivative is discussed in this section based on Definition 1 and Theorems 1 and 2, established in the previous section. Likewise, we present and prove the main properties of this proposed derivative.
Definition 2. For function
, the generalized FF (GFF) derivative of order
, of
at
is written as:where
with
and
If
is GFF
-differentiable in some
,
, and
exists, then it is expressed as:
Remark 6. It is interesting to highlight some special cases of Definition 2:
- (i)
If , then Equation (9) becomes the classical definition of the derivative.
- (ii)
If , then Equation (9) becomes the definition of Abu-Shady–Kaabar fractional derivative of order proposed in [6].
- (iii)
If , then Equation (9) becomes the definition of fractal derivative of order introduced in [9,10].
Theorem 3. Let
and let
be generalized fractal
-differentiable at a point
. If, additionally,
is differentiable function, thenwhere
with
.
Proof. Let
in Equation (9), and then
. Therefore,
□
Remark 7. Note that if ,
, and
, then Equation (9) can be written as:
where
is the conformable derivative of order
introduced in [4].
Remark 8. Consider a function
,
. Using Theorem 3, the following result is easily obtained,Note that the above result is compatible with the result of the FF derivative of order
in the C sense with power law expressed in Equation (1). Theorem 4. If a function
is GFF
-differentiable at
then
is continuous at
Proof. Since
Then,
Let
. Then,
which implies that
Hence,
is continuous at
□
Theorem 5. Let
and let
be GFF
-differentiable at a point
. Then, we have:
- (i)
,
.
- (ii)
constant functions
.
- (iii)
.
- (iv)
.
Proof. Parts
and
are followed directly from the mentioned definition. Let us only show
since it is important. Now, for fixed
Since
is continuous at
this completes the proof of parts
and
, which can be proven in a similar way. □
Remark 9. Now, we show that the results obtained by applying this derivative proposed for certain elementary functions are compatible with the results of the FF derivative of order in the C sense with power law as in Equation (1).
- (i)
Exponential function
From Remark 8, we get:Finally, the following result is obtained, - (ii)
Sine function .
Using the fact that
, we have: Finally, the following result is obtained: - (iii)
Cosine function .
Using the fact that
, we have: Finally, the following result is obtained, Now, we establish a fundamental result of classical mathematical analysis, the chain rule, in the context of FF calculus. Note that this extension is possible due to the local character of the proposed GFF derivative.
Theorem 6 (Chain Rule)
. Let , GFF
-differentiable at
and
is differentiable at
, then Proof. We show the result via the standard limit approach. If the function
is constant in a neighbourhood of
, then
If
is not constant in a neighbourhood of
we can find a
, such that
for any
Now, since
is continuous at
for a sufficiently small
, we have:
Taking
In the first factor, we have:
And from here
The proof is completed. □
Remark 10. From the result above, it is easy to obtain the GFF derivative of order
of the following elementary functions:
- (i)
- (ii)
- (iii)
- (iv)
Remark 11. Using the fact that differentiability implies GFF
-differentiability and assuming
, Equation (16) can be re-written as:where
with
The extension of the mean value theorems of classical mathematical analysis was also the subject of our research. Thus, we establish these theorems for GFF differentiable functions and discuss some interesting consequences.
Theorem 7 (Roll’s theorem for GFF
-differentiable functions)
. Let , and
be a given function that satisfies:
- (i)
is continuous on
,
- (ii)
is GFF
-differentiable on
,
- (iii)
.
Then, there exists
such that
Proof. Since
is continuous on
and
, there is
which is a point of local extrema. With no loss of generality, assume
is a point of local minimum. So
But, the first limit is non-negative, and the second limit is non-positive. Hence,
□
Theorem 8 (mean value theorem for GFF
-differentiable functions)
. Let , and
be a given function that satisfies:
- (i)
is continuous on
,
- (ii)
is local GFF
-differentiable on
,
Then, there exists
such that
Proof. Consider the function,
Then
From Remark 10, we obtain:
At
And the auxiliary function
satisfies all conditions of Theorem 7. Hence, there exists
such that
Then, we obtain:
□
Theorem 9. Let
, and
be a given function that satisfies:
- (i)
is continuous on
,
- (ii)
is GFF
-differentiable on
,
If
for all
then,
is constant on
Proof. Suppose
for all
Let
with
So, the closed interval
is contained in
and the open interval
is contained in
Hence,
is continuous on
and generalized fractal
-differentiable on
So, from Theorem 8, there exists
with
Therefore,
and
Since
and
are arbitrary numbers in
with
is constant on
□
Corollary 1. Let
, and
be functions such that
for all
Then, there exists a constant
such that Proof. By simply applying the above theorem to it can be proven easily. □
Theorem 10. Let
, and
be a given function that satisfies:
- (i)
is continuous on
,
- (ii)
is GFF
-differentiable on
If
for all
then
is increasing on
If
for all
then
is decreasing on
Proof. Similarly, using Theorem 9’s proof, there exists
with
If then for Therefore, is strictly increasing on since and are arbitrary numbers of
If then for Therefore, is strictly decreasing on since and are arbitrary numbers of □
Theorem 11 (racetrack-type principal)
. Let , and
be the given functions that satisfies:
- (i)
and
are continuous on
- (ii)
and
are GFF
-differentiables on
- (iii)
for all
If
then
for all
If
then
for all
Proof. Consider Then, is continuous on and GFF -differentiable on Also, using the linearity of and the fact that for all we obtain for all So, through Theorem 10, is increasing (non-decreasing). Hence, for any we have Since by the assumption, the result follows. Similarly, for part 2 of Theorem 11, since for any we have and the result follows. □
Theorem 12 (extended mean value theorem for GFF
-differentiable functions)
. Let , and
be the given functions that satisfies:
- (i)
are continuous on
- (ii)
are GFF
-differentiable on
Then, there exists
such that
Then, the function satisfies the conditions of Theorem 7. Hence, there exists such that Using the linearity of the GFF -derivative and the fact that with a constant, the result follows. □
Remark 12. Another interesting result in the context of the proposed GFF calculus is a modified version of the mean value theorem for GFF -differentiable functions. Next, we will establish and prove this result.
Theorem 13 (modified value theorem for generalized fractal
-differentiable functions)
. Let , and
be a given function that satisfies:
- (i)
is continuous on
,
- (ii)
is GFF
-differentiable on
.
Proof. Consider the function:
Then, the function
satisfies the conditions of Theorem 7. Hence, there exists
such that
Therefore,
Hence,
□
Remark 13. From the above theorem, we can easily establish similar consequences as those obtained in Theorem 8 (see Theorems 9–11 and Corollary 1).
Definition 3. Let
an open interval,
, and
, we will say that f is of class
on the interval
, which we write as
if
is GFF
-differentiable on
and GFF
-derivative is continuous on
Theorem 14. Let
an open interval,
, and
be a function of class
on the interval
. Suppose
for some
, and
. Then, there is an open neighborhood
of
in which f admits an inverse function
of class
on the open neighborhood
of b, and its GFF
-derivative is: Proof. Since
is continuous in the open interval
, it is a known fact that there exists an open neighborhood
of
in which
has a constant sign (the sign of
). From Remark 13, it follows
that is strictly monotonic on
(increasing if
, decreasing if
). Therefore,
is continuous and strictly monotonic on
, so there is the inverse function of the one-to-one function
, with
. This inverse
is of class
and strictly monotonic (in the same sense that
is) on
. Equation (18) can be easily obtained from the identity
for all
, in which the GFF
-derivative (with respect to
) is calculated, applying the chain rule as follows:
□
Finally, we present the following definition for the GFF -integral of a function starting at :
Definition 4. , where this integral is basically the usual Riemann improper integral, , and
From the definition above, we can establish two important results:
Theorem 15. , for , where is any continuous function in the domain of
.
Proof. Since
is continuous, then
is differentiable. Hence,
□
Theorem 16. Let
,
,
, and
be a continuous real-valued function on interval
. Let
any real-valued function with the property
for all
. Then Proof. First, let be a function on defined as , which can be called GFF -integral function of .
By using Theorem 15, for all .
Since and have the same GFF -derivative, then by corollary 1, there exists a real constant , such that for all .
Finally, is computed as follows:
4. Applications
In this section, we will solve several interesting FF ordinary differential equations in the sense of the proposed GFF derivative.
Example 1. Consider the initial value problem involving a GFF ordinary differential equation of order
as follows:To find the solution to the differential equation in Equation (20), we use the fact that
and apply Equation (11) to obtain:If we rearrange the above equation and integrate on both sides, it follows:By taking
, we have:Finally, using the initial condition
, we obtain:Note that this solution is consistent with the solution to the corresponding FF (in the sense of C) Initial Value Problem (IVP). Example 2. Consider the IVP involving a GFF ordinary differential equation of order
as follows:To find the solution to the differential equation in Equation (21), we use the fact that
and apply Equation (11) to obtain:If we rearrange the above equation and integrate on both sides, it follows:By taking
, we have:Finally, using the initial condition
, we obtain:Note that this solution is consistent with the solution to the corresponding FF (in the sense of C) IVP. Remark 14. In the following example, we will solve a generalized linear fractal–fractional differential equation, but, also, we will define this type of differential equation and prove a result in which its general solution is established.
Definition 5. The generalized linear FF differential equation of order
is defined aswhere ,
, and
are real-valued continuous functions on an interval
.
Theorem 17. The general solution of the GFF differential Equation (22) is expressed by:where is a real constant. Proof. Using Theorem 3, Equation (22) can be expressed as:
Since the above equation is a classical first-order linear differential equation, its general solution is written as:
where
is a real constant. Finally, using Definition 5 and substituting into Equation (25), our result follows directly. □
Now, we can solve an example that involves a generalized linear FF differential equation of order using the proposed method.
Example 3. Consider the generalized linear FF differential equation of order
as follows:Taking in Equation (26)
,
,
, and
, we have:where
is a real constant. Finally, the initial condition
implies that
. Hence,
.
We finish this section by discussing another interesting differential equation in the sense of the GFF derivative, specifically, the generalized Bernoulli FF differential equation. As in the classic case, we propose to solve this equation by reducing it to a generalized linear FF differential equation. Thus, consider the generalized Bernoulli FF differential equation of non-integer order
, given by
where
,
, , and
are real-valued continuous functions on an interval
.
The above equation, through the change of variable
, can be reduced to the following linear ordinary differential equation:
According to Theorem 15, the general solution of Equation (27) is given by:
In the following example, we apply this proposed method to solve a generalized Bernoulli FF differential equation.
Example 4. Consider the generalized Bernoulli FF differential equation of order
as follows:Taking in Equation (29)
,
, ,
, and
, we have:where
is a real constant.