1 Introduction

Fractional calculus or non-integer order calculus is an old topic from Leibniz times [1] and has attracted much attention from mathematicians, physicists and engineers in recent decades [2,3,4,5,6,7,8,9,10,11,12,13]. It is regarded as a powerful and effective tool for modelling nonlinear systems [14,15,16,17]. Various types of fractional derivative definitions were introduced in history, such as Riemann–Liouville, Caputo, Grunwald–Letnikov, Riesz and Weyl, etc. Most of them are defined via fractional integrals, thus they inherit non-local properties from integral. Heredity and nonlocality are typical properties of these definitions [18], which are important in many application fields and are different from classical Newton–Leibniz calculus.

However, some inconsistencies arise when we compare these fractional derivative definitions with Newton’s derivative. For example, as stated in Refs. [19] and [20], these derivatives do not obey product rule, quotient rule, chain rule for derivative operations. Those inconsistencies come with difficulties and inconvenience in mathematical handle. To overcome these difficulties, some scholars introduced the concept of local fractional derivative (LFD). Kolwankar and Gangal [21] gave a type of LFD by the means of letting lower or upper limit of Riemann–Liouville derivative approach to the defined point in 1996, several followed works [22,23,24,25,26] further extended its properties and applications. In 2014, Khalil et al. [19] proposed another type of LFD named “conformable fractional derivative” (CFD) whose most properties coincide with Newton derivative and can be used to solve fractional differential equations more easily [27]. The CFD is defined as

Definition 1

Given a function \(f:[0,\infty )\rightarrow \mathbf {R}\). Then the “conformable fractional derivative” of order \(\alpha \) is defined by

$$\begin{aligned} T_{\alpha }(f)(t) = \lim _{\epsilon \rightarrow 0}\frac{f(t + \epsilon t^{1-\alpha })-f(t)}{\epsilon } \end{aligned}$$
(1)

for all \(t>0,\alpha \in (0,1)\). If f is \(\alpha \)-differentiable in some \((0,a), a > 0\), and \( \lim _{t \rightarrow 0^{+}}f^{\alpha }(t)\) exists, then define \(f^{\alpha }(0) = \lim _{t \rightarrow 0^{+}}f^{\alpha }(t)\).

Some scholars followed this work and explored potential applications in Newton mechanics [28], quantum mechanics [29] and so on [30]. However, physical interpretation of this derivative is unknown thus basis of those potential applications seems not solid. At the same time, some readers may wonder what is the role the term \(t^{1-\alpha }\) acts in Definition (1). We will discuss these issues in the following context and point out that the term \(t^{1-\alpha }\) is not essential and it is only a kind of “fractional conformable function”.

It is noticed that the term LFD has two meanings in this paper. Firstly, it refers to the derivative proposed by Kolwankar and Gangal. Secondly, it has a more wide scope that it includes all the opposite derivatives of “nonlocal” fractional derivatives. In order to avoid confusion, LFD refers to the second meaning by default in this paper and we will call the first one a full name—Kolwankar and Gangal LFD.

In the present work, we extend the concept of Gâteaux derivative and discuss principles for the definition of LFD. As an example, we generalize the definition of CFD by means of Linear Extended Gâteaux derivative (LEGD) to general conformable fractional derivative (GCFD). We also give physical and geometrical interpretations of this new derivative which thus indicate potential applications in physics and engineering. It is easy to demonstrate that Khalil’s CFD is the special case, then to the authors’ knowledge, so far we first give the physical and geometrical interpretations of CFD.

The presentation in the rest of paper is organized as follows. We introduce the concepts of Gâteaux derivative, Extended Gâteaux derivative (EGD) and LEGD in Sect. 2 and give the physical interpretation of EGD and LEGD in Sect. 3. Those two sections prepare a new framework for the definition of GCFD. In Sect. 4, we discuss principles for the definition of LFD. As an example, we define GCFD in the framework of LEGD in this section. Various properties of GCFD are given in Sect. 5. Integral of GCFD and the “fundamental theorem” of GCFD are provided in Sect. 6. As an application, we give a scheme for solving the GCFD fractional differential equations in Sect. 7. Finally, we conclude and give some discussions in Sect. 8.

2 Gâteaux derivative, Extended Gâteaux derivative and Linear Extended Gâteaux derivative

Definition 2

(Gâteaux derivative). [31] Suppose X and Y are locally convex topological vector spaces, \(U \subset X\) is open, and \(f: X \rightarrow Y\). The Gâteaux differential \(df(u;\psi )\) of f at \(u \in U\) in the direction \(\psi \in X\) is defined as

$$\begin{aligned} df(u;\psi ) = \lim _{\epsilon \rightarrow 0}\frac{f(u+\epsilon \psi )-f(u)}{\epsilon }, \end{aligned}$$
(2)

if the limit exists.

The limit in Eq. (2) is taken relative to the topology of Y. If X and Y are Euclidean space \({\mathbb {R}}^n\), the Gâteaux derivative is a one-dimensional calculation along a specified direction \(\psi \). We say that f is Gâteaux differentiable at u if the limit in Eq. (2) exists for all \(\psi \in X\).

Readers can refer to textbook [32] to find many concrete examples of Gâteaux differential. We will list some properties of Gâteaux derivative here [32].

Property 1

Differential of a constant. The Gâteaux differential of a constant is zero: \(d(c)(u;\psi ) = 0\).

Property 2

Sum rule. Gâteaux differentiation distributes over sums: \(d(f+g)(u;\psi ) = df(u;\psi ) + dg(u;\psi )\).

Property 3

Product rule and quotient rule. The Gâteaux differential of an elementwise product fg is \(d(fg)(u;\psi ) = df(u;\psi )g + dg(u;\psi )f\) while differential of quotient f / g is \(d(\frac{f}{g})(u;\psi ) = \frac{df(u;\psi )g - dg(u;\psi )f}{g^2}\).

Property 4

Chain rule. Assume \(g: U \rightarrow V\) is Gâteaux differential at \(u \in U\), and \(f: V \rightarrow W\) is Gâteaux differential at \(g(u) \in V\), then

$$\begin{aligned} d(f\circ {g})(u;\psi ) = df(g(u); dg(u;\psi )). \end{aligned}$$
(3)

Note that in Definition 2, \(\psi \) is a constant point in X and has no relationship with u at any given \(u \in U\). We extend this concept and give a new extended definition as follows.

Definition 3

(Extended Gâteaux derivative). Suppose X and Y are locally convex topological vector spaces, \(U \subset X\) is open, \(f: X \rightarrow Y\) and \(\psi (u,\epsilon , p): X \times {\mathbb {R}} \times {\mathbb {R}} \rightarrow X\), where \(p \in {\mathbb {R}}\) is a parameter. The Extended Gâteaux differential (EGD) \(df(u;\psi )\) of f at \(u \in U\) is defined as

$$\begin{aligned} df^{EG}(u;\psi ) = \lim _{\epsilon \rightarrow 0}\frac{f(u+\psi (u,\epsilon ,p))-f(u)}{\epsilon }, \end{aligned}$$
(4)

if the limit exists.

Remark 1

When \(\psi (u,\epsilon , p) = \epsilon \phi \), where \(\phi \in X\), Definition 3 degenerates to Definition 2. In other words, Extended Gâteaux derivative is a natural extension of Gâteaux derivative.

Remark 2

When \(\psi (u,\epsilon , p) = \epsilon ^{\frac{1}{p}}\), Definition 3 is equivalent to Kolwankar and Gangal’s LFD definition. Actually, Kolwankar and Gangal’s LFD definition can be written as [22]

$$\begin{aligned} d_{+}^{p}f(u) = \varGamma (1+p)\lim _{\epsilon \rightarrow 0^{+}}\frac{f(u+\epsilon )-f(u)}{\epsilon ^{p}}. \end{aligned}$$
(5)

If we ignore the coefficient \(\varGamma (1+p)\), Eq. (5) is equivalent to (4) with \(\psi (u,\epsilon , p) = \epsilon ^{\frac{1}{p}}\).

In order to inherit properties from Gâteaux derivative, we discuss a special and useful case of EGD—\(\psi (u,\epsilon , p) = \epsilon \phi (u,p)\), where \(\epsilon \) is separable from u and p, we name it Linear Extended Gâteaux derivative.

Definition 4

(Linear Extended Gâteaux derivative). Suppose X and Y are locally convex topological vector spaces, \(U \subset X\) is open, \(f: X \rightarrow Y\) and \(\psi (u, p): X \times {\mathbb {R}} \rightarrow X\), where \(p \in {\mathbb {R}}\) is a parameter. The Linear Extended Gâteaux differential (LEGD) \(df(u;\psi )\) of f at \(u \in U\) is defined as

$$\begin{aligned} df^{LEG}(u;\psi ) = \lim _{\epsilon \rightarrow 0}\frac{f(u+\epsilon \psi (u,p))-f(u)}{\epsilon }, \end{aligned}$$
(6)

if the limit exists.

3 Physical interpretation of EGD and properties of LEGD

Physical and geometrical interpretations of Newton’s derivative are clear. It means velocity of particle or slope of a tangent respectively. When the independent variable u changes \(\epsilon \), function value changes accordingly. Their ratio limit reflects the strength and direction of the velocity. As for EGD, when the independent variable u changes \(\epsilon \), the map value changes according to \(\psi (u,\epsilon , p)\), and the ratio limit of change of map value and \(\epsilon \) defines the EGD. So physical interpretation of EGD can be regarded as a special velocity, its direction and strength rely on \(\psi (u,\epsilon , p)\). In particular, the LEGD has a more clear interpretation. We first give a result in Euclidean space \({\mathbb {R}}^n\). For simplicity, we write \(df^{LEG}(u;\psi )\) as \(df(u;\psi )\) in the rest of paper if there is no special claim.

Theorem 1

Let X, Y be Euclidean space \({\mathbb {R}}^n\) and \({\mathbb {R}}^m\) respectively, \(f: X \rightarrow Y\) and f be Linear Extended Gâteaux differentiable at \(u=(u_{1},u_{2},\ldots ,u_{n}) \in U \subset X, f=(f_{1},f_{2},\ldots ,f_{m}) \in Y, \psi (u, p): X \times {\mathbb {R}} \rightarrow X\), where \(p \in {\mathbb {R}}\) is a parameter. If \(\frac{\partial {f_{j}}}{\partial {u_{i}}}, \forall i \in {1,2,\ldots ,n}, \forall j \in {1,2,\ldots ,m}\) exists, then,

$$\begin{aligned} df(u;\psi )= & {} (<\nabla f_{1}, \psi (u, p)>,<\nabla f_{2}, \psi (u, p)>,\nonumber \\&\ldots , <\nabla f_{m}, \psi (u, p)>). \end{aligned}$$
(7)

Proof

Firstly, \(df(u;\psi ) = (df_{1}(u;\psi ),df_{2}(u;\psi ),\ldots ,df_{m}(u;\psi ))\).

We rewrite \(\psi (u, p) \in X\) as \((\psi _{1}(p), \psi _{2}(p), \ldots , \psi _{n}(p))\). Then, \(\forall j \in {1,2,\ldots ,m}\),

$$\begin{aligned}&df_{j}(u;\psi ) \\&\quad = \lim _{\epsilon \rightarrow 0}\frac{f_{j}(u+\epsilon \psi (u,p))-f_{j}(u)}{\epsilon } \\&\quad = \lim _{\epsilon \rightarrow 0}\left( \frac{f_{j}(u_{1}+\epsilon \psi _{1}(p), u_{2}+\epsilon \psi _{2}(p), \ldots , u_{n}+\epsilon \psi _{n}(p))}{\epsilon }\right. \\&\qquad \left. -\frac{f_{j}(u_{1},u_{2},\ldots ,u_{n})}{\epsilon }\right) . \end{aligned}$$

Since \(\frac{\partial {f_{j}}}{\partial {u_{i}}}, \forall i \in {1,2,\ldots ,n}, \forall j \in {1,2,\ldots ,m}\) exists, and when \(\epsilon \rightarrow 0, f_{j}(u+\epsilon \psi (u,p))-f_{j}(u) \rightarrow 0\), according to L’Hôpital’s rule of variable \(\epsilon \), we get that

$$\begin{aligned} df_{j}(u;\psi )= & {} \frac{\partial {f_{j}}}{\partial {u_{1}}}(u)\psi _{1}(p) + \frac{\partial {f_{j}}}{\partial {u_{2}}}(u)\psi _{2}(p) + \ldots + \frac{\partial {f_{j}}}{\partial {u_{n}}}(u)\psi _{n}(p) \\= & {} <\nabla f_{j}, \psi (u, p)>. \end{aligned}$$

Then it holds

$$\begin{aligned} df(u;\psi )= & {} (<\nabla f_{1}, \psi (u, p)>,<\nabla f_{2}, \psi (u, p)>,\\&\ldots , <\nabla f_{m}, \psi (u, p)>). \end{aligned}$$

\(\square \)

Remark 3

When \(m = 1\) and \(\psi (u, p) = (0,\ldots ,1,\ldots ,0), df(u;\psi ) = \frac{\partial {f}}{\partial {u_{i}}}(u)\) coincides with classical partial derivative in \({\mathbb {R}}^n\).

Remark 4

When \(m = 1, df(u;\psi ) = <\nabla f, \psi (u, p)>\), then geometrical interpretations of the LEGD is that the gradient of a function f projects onto a function \(\psi (u, p)\) or a special function \(\psi (u, p)\) projects onto the gradient of a function f from a different viewpoint.

Remark 5

When \(n = 1, df(u;\psi ) = (\frac{df_{1}}{du}\psi (u, p), \frac{df_{2}}{du}\psi (u, p), \ldots , \frac{df_{m}}{du}\psi (u, p)) = \frac{df}{du}\psi (u, p)\), it means that velocity in the sense of LEGD is a multiple of the classical velocity.

From Theorem (1) and remarks above, we can find that physical interpretation of the LEGD is a modification of classical velocity in direction and magnitude.

Next, we give a result of the LEGD similar to Gâteaux derivative.

Theorem 2

Let f, g be Linear Extended Gâteaux differentiable at \(u \in U \subset X, \psi (u, p): X \times {\mathbb {R}} \rightarrow X\), where \(p \in {\mathbb {R}}\) is a parameter. Then,

  1. (1)

    \(dc(u;\psi ) = 0\), for all constant functions.

  2. (2)

    \(d(af + bg)(u;\psi ) = adf(u;\psi ) + bdg(u;\psi )\), for all \(a,b \in {\mathbb {R}}\).

  3. (3)

    \(d(fg)(u;\psi ) = fdg(u;\psi ) + gdf(u;\psi )\).

  4. (4)

    \(d(\frac{f}{g})(u;\psi ) = \frac{gdf(u;\psi ) - fdg(u;\psi )}{g^2}\).

  5. (5)

    Chain rule: \(d(f\circ {g})(u;\psi ) = df(g(u); dg(u;\psi ))\).

Proof

Parts (1) and (2) follow immediately from the definition of LEGD.

$$\begin{aligned}&d(fg)(u;\psi ) \\&\quad = \lim _{\epsilon \rightarrow 0}\frac{f(u+\epsilon \psi (u,p))g(u+\epsilon \psi (u,p))-f(u)g(u)}{\epsilon }\\&\quad = \lim _{\epsilon \rightarrow 0}\left( \frac{f(u+\epsilon \psi )g(u+\epsilon \psi )-f(u+\epsilon \psi )g(u)}{\epsilon }\right. \\&\qquad \left. +\frac{f(u+\epsilon \psi )g(u)-f(u)g(u)}{\epsilon }\right) \\&\quad = \lim _{\epsilon \rightarrow 0}\frac{f(u+\epsilon \psi )(g(u+\epsilon \psi )-g(u))}{\epsilon } \\&\qquad + \lim _{\epsilon \rightarrow 0}\frac{((f(u+\epsilon \psi )-f(u))g(u)}{\epsilon }\\&\quad = fdg(u;\psi ) + gdf(u;\psi ). \end{aligned}$$

This completes the proof of (3).

$$\begin{aligned} d\left( \frac{1}{g}\right) (u;\psi )= & {} \lim _{\epsilon \rightarrow 0}\frac{\frac{1}{g(u+\epsilon \psi (u,p))}-\frac{1}{g(u)}}{\epsilon } \\= & {} \lim _{\epsilon \rightarrow 0}\frac{\frac{g(u)-g(u+\epsilon \psi (u,p))}{\epsilon }}{g(u+\epsilon \psi (u,p)g(u))} \\= & {} \frac{-dg(u;\psi )}{g^{2}(u)}. \end{aligned}$$

Then \(d(\frac{f}{g})(u;\psi ) = \frac{gdf(u;\psi ) - fdg(u;\psi )}{g^2}\) holds according to (3).

Next, we prove (5).

First, if f is Linear Extended Gâteaux differentiable at \(u, f(u+\epsilon \psi (u,p)) = f(u) + \epsilon df(u;\psi (u,p) ) + o(\epsilon )\). Then,

$$\begin{aligned}&d(f\circ {g})(u;\psi )\\&\quad = \lim _{\epsilon \rightarrow 0}\frac{f(g(u+\epsilon \psi (u,p)))-f(g(u))}{\epsilon } \\&\quad = \lim _{\epsilon \rightarrow 0}\frac{f(g(u)+\epsilon dg(u;\psi (u,p) ) + o(\epsilon ))-f(g(u))}{\epsilon } \\&\quad = \lim _{\epsilon \rightarrow 0}\frac{f(g(u)+\epsilon (dg(u;\psi (u,p) ) + \epsilon ^{-1}o(\epsilon )))-f(g(u))}{\epsilon } \\&\quad = \lim _{\epsilon \rightarrow 0}\left( \frac{f(g(u))+\epsilon df(g(u); dg(u;\psi (u,p) ) + \epsilon ^{-1}o(\epsilon ))}{\epsilon }\right. \\&\qquad \left. -\frac{f(g(u))}{\epsilon }\right) \\&\quad = df(g(u); dg(u;\psi )). \end{aligned}$$

This completes the proof of (5). \(\square \)

4 Principles for the definition of LFD and GCFD

When we consider the concept of local fractional derivative (LFD), which principles are needed to take into account? Clearly, the first principle is that when fractional order \(p = 1\), the LFD should degenerate to the usual first-order derivative. Even more, we hope that when \(p = 0\), the LFD of a function is the function itself. However, it may be hard to satisfy, because neither Kolwankar–Gangal definition nor Khalil definition meets this demand. The second one is the LFD should have local properties, for example, the derivative of a constant is zero. Meanwhile, its properties should be consistent with the classical derivative as much as possible, thus it is more convenient to do mathematical treatment. Furthermore, we hope that LFD has clear physical or geometric interpretations, as it relates to its value and potential applications. We put these principles together as:

Principle 1

LFD should degenerate to the usual first-order derivative when fractional order equals one.

Principle 2

LFD should have properties consistent with the classical derivative as much as possible.

Principle 3

LFD should have clear physical or geometrical interpretations.

As mentioned in last sections, the LEGD is the generalization of classical Newton’s derivative and has properties similar to it (see Theorem 2), in other words, LEGD satisfies the above Principle 2. Meanwhile, the LEGD has clear physical or geometrical interpretations (see Theorem 1), so Principle 3 is satisfied. As an example, we focus our attention on generalizing the definition of CFD by means of LEGD.

In order to meet Principle 1, \(\psi (u, 1)\) should be equal to one. Meanwhile, for the unique meaning of each order \(p \in (0,1], \psi (u, p)\) should differ from different p. That is:

$$\begin{aligned} \psi (u, 1)= & {} 1,\end{aligned}$$
(8)
$$\begin{aligned} \psi (\cdot , p)\ne & {} \psi (\cdot , q), \text {where } p \ne q \text { and } p,q \in (0,1]. \end{aligned}$$
(9)

Whenever we determine a function \(\psi (u, p)\), we get a CFD definition form. From elementary simple functions, we get various \(\psi (u, p)\) which satisfy Eqs. (8) and (9)

$$\begin{aligned} \text {linear : } \psi (u, p)= & {} pk + b, \text { specially, } \psi (u, p) = p,\end{aligned}$$
(10)
$$\begin{aligned} \text {power : } \psi (u, p)= & {} p^{\alpha }, \text { specially, } \psi (u, p) = p^2,\end{aligned}$$
(11)
$$\begin{aligned} \text {exponent : } \psi (u, p)= & {} a^{(1-p)h(p)},\nonumber \\ \text { specially, }\psi (u, p)= & {} u^{(1-p)h(p)},\end{aligned}$$
(12)
$$\begin{aligned} \text {logarithm : } \psi (u, p)= & {} log_{a}\phi (u,p),\end{aligned}$$
(13)
$$\begin{aligned} \text {trigonometric : } \psi (u, p)= & {} \sin \left( \frac{\pi }{2}p\right) , \text { or, } \psi (u, p) = \tan \left( \frac{\pi }{4}p\right) , \end{aligned}$$
(14)

where k, b are constant number, \(k + b = 1, \alpha > 0, a\) is a non-zero number, h(p) is a polynomial function such that \(h(p) \ne 0\) when \(p \in (0,1), \phi (u,p)\) is a monotonous function of u such that \(\phi (u,1) = a\). It is easy to verify that all the \(\psi (u, p)\) listed above satisfy Eqs. (8) and (9) and readers can find more in elementary simple functions or their combinations.

Definition 5

(Fractional conformable function). We call continuous real functions satisfying the above Eqs. (8), (9) and constant value function \(\psi (u, \alpha ) = 1\) fractional conformable functions.

Basically, we restrict space X to \(\mathbb {R^{+}}\) for simplicity to define the concept of GCFD as follows.

Definition 6

(GCFD). Let \(\psi (u, p)\) be a fractional conformable function and \(p \in (0,1]\). The GCFD is defined as:

$$\begin{aligned} D_{\psi }^{p}f(u) = \lim _{\epsilon \rightarrow 0}\frac{f(u+\epsilon \psi (u,p))-f(u)}{\epsilon }. \end{aligned}$$
(15)

Apparently, this definition is local and its operation on any constant number is zero.

Remark 6

When \(\psi (u, p) = 1, D_{\psi }^{p}f(u)\) degenerates to the usual first-order derivative and has no relationship with fractional order p.

Remark 7

When \(h(p)=1\) such that \(\psi (u, p) = u^{1-p}, D_{\psi }^{p}f(u)\) coincides with Khalil’s definition [19]. In other words, CFD is a special case of GCFD.

Remark 8

As for the definition of Katugampola [20], it is a special case of the EGD, where \(\psi (u,\epsilon , p) = ue^{\epsilon u^{-p}} - u\). However, since \(ue^{\epsilon u^{-p}} - u = \epsilon u^{1-p} + o(\epsilon ^{2})\), it has the same form as the Khalil’s definition if we omit \(o(\epsilon ^{2})\). So we regard Katugampola’s definition and Khalil’s definition as the same definition type.

It is natural to define the GCFD of arbitrary order as follows.

Definition 7

(GCFD of arbitrary order). Let \(p \in (n, n+1]\), for some \(n \in {\mathbb {N}}\) and f be n-differentiable at \(u>0\). Then the p-fractional derivative of f is defined as:

$$\begin{aligned} D_{\psi }^{p}f(u)= & {} D_{\psi }^{p-n}D^{n}f(u) \nonumber \\= & {} \lim _{\epsilon \rightarrow 0}\frac{f^{(n)}(u+\epsilon \psi (u,p-n))-f^{(n)}(u)}{\epsilon }, \end{aligned}$$
(16)

if the limit exists.

5 Properties of GCFD

We now give some properties of the GCFD.

Theorem 3

If a function \(f : {\mathbb {R}}^{+} \rightarrow {\mathbb {R}}\) is p-differentiable at \(u > 0, p \in (0,1]\), then f is continuous at u.

Proof

Since \(f(u+\epsilon \psi (u,p)) - f(u) = \frac{f(u+\epsilon \psi (u,p))-f(u)}{\epsilon }\epsilon \). Then,

$$\begin{aligned}&\lim _{\epsilon \rightarrow 0} (f(u+\epsilon \psi (u,p)) - f(u)) \\&\quad = \lim _{\epsilon \rightarrow 0}\frac{f(u+\epsilon \psi (u,p))-f(u)}{\epsilon }\cdot \lim _{\epsilon \rightarrow 0}\epsilon . \end{aligned}$$

Let \(h = \epsilon \psi (u,p)\), then,

$$\begin{aligned}&\lim _{\epsilon \rightarrow 0} (f(u+\epsilon \psi (u,p)) - f(u)) \\&\quad = \lim _{h \rightarrow 0} (f(u+h) - f(u)) = D_{\psi }^{p}f(u)\cdot 0 = 0, \end{aligned}$$

which implies \(\lim \limits _{h \rightarrow 0} f(u+h) = f(u)\). In other words, f is continuous at u. \(\square \)

Theorem 4

If a function f is p-differentiable at \(u > 0, p \in (0,1]\), then

$$\begin{aligned} f(u+\epsilon \psi (u,p)) = f(u) + \epsilon D_{\psi }^{p}f(u) + o(\epsilon ). \end{aligned}$$
(17)

Proof

It is a direct consequence of the definition of GCFD in Eq. (15). \(\square \)

Theorem 5

Let \(p \in (0,1]\), and f, g be p-differentiable at \(u > 0\). Then,

  1. (1)

    \(D_{\psi }^{p}(af + bg) = aD_{\psi }^{p}(f) + bD_{\psi }^{p}(g)\), for all \(a,b \in {\mathbb {R}}\).

  2. (2)

    \(D_{\psi }^{p}(c) = 0\), for all constant functions.

  3. (3)

    \(D_{\psi }^{p}(fg) = fD_{\psi }^{p}(g) + gD_{\psi }^{p}(f)\).

  4. (4)

    \(D_{\psi }^{p}(\frac{f}{g}) = \frac{gD_{\psi }^{p}(f) - fD_{\psi }^{p}(g)}{g^2}\).

  5. (5)

    If f is differentiable, then \(D_{\psi }^{p}(f) = \frac{df}{du}(u)\psi (u, p) \).

  6. (6)

    Chain rule: if f is differentiable, then \(D_{\psi }^{p}(f\circ {g}(u)) = f^{'}(g(u))D_{\psi }^{p}(g(u))\).

Proof

Parts (1) to (4) and (6) are direct corollaries of Theorem 2. Part (5) is the corollary of Theorem 1. \(\square \)

Remark 9

When \(\psi (u, p) = u^{1-p}\), result (5) comes out to be \(D^{p}(f) = \frac{df}{du}(u)u^{1-p} \). It coincides with Khalil’s result in Ref. [19].

Remark 10

If our Definition (6) holds for \(p = 0\), then we have \(D_{\psi }^{0}(f) = f\). Meanwhile, by result (5), we get that \(D_{\psi }^{0}(f) = f'\psi (u,0)\). Therefore, \(f'\psi (u,0) = f\), in other words, \(\psi (u,0) = f/f'\), which indicates \(\psi (u,0)\) relies on some function. It is unreasonable. So we never require Definition (6) holds for \(p = 0\). That is why Khalil does not require his definition hold for zero too.

According to (5) in Theorem (5), one can easily and conveniently compute the GCFD of some useful functions.

Theorem 6

Let \(a, n \in {\mathbb {R}}\) and \(p \in (0,1]\), then we have the following results.

  1. (1)

    \(D_{\psi }^{p}(1) = 0\).

  2. (2)

    \(D_{\psi }^{p}(u^{n}) = nu^{n-1}\psi (u,p)\).

  3. (3)

    \(D_{\psi }^{p}(e^{au}) = ae^{au}\psi (u,p)\).

  4. (4)

    \(D_{\psi }^{p}(\sin {ax}) = a\cos {ax}\psi (u,p)\).

  5. (5)

    \(D_{\psi }^{p}(\cos {ax}) = -a\sin {ax}\psi (u,p)\).

One can also find the fixed point of operator \(D_{\psi }^{p}\).

Theorem 7

If f is differentiable, and \(D_{\psi }^{p}(f) = f\), then

$$\begin{aligned} f(u) = ce^{\int {\frac{1}{\psi (u,p)}du}}, \end{aligned}$$
(18)

where c is any positive constant.

Proof

According to (5) in Theorem (5), \(D_{\psi }^{p}(f) = f\) is equivalent to \(\psi (u,p)f^{'} = f\). To solve this simple ODE, one can get the result: \(f(u) = ce^{\int {\frac{1}{\psi (u,p)}du}} \), where c is any positive constant. \(\square \)

Remark 11

When \(\psi (u, p) = u^{1-p}\), it is easy to get that \(f(u) = ce^{\frac{u^p}{p}}\). It coincides with Khalil’s result in Ref. [19].

Next, we give Rolle’s theorem and the Mean Value theorem of GCFD.

Theorem 8

(Rolle’s theorem for GCFD). Let \(a > 0\) and \(f: [a,b] \rightarrow {\mathbb {R}}\) be a given function that satisfies

  1. (i)

    f is p-differentiable for some \(p \in (0,1]\) at any \(u \in [a,b]\),

  2. (ii)

    \(f(a) = f(b)\).

Then, there exists \(c \in (a,b)\), such that \(D_{\psi }^{p}(f(c)) = 0\).

Proof

Since f is p-differentiable at any \(u \in [a,b]\), we know that f is continuous on [a, b] by Theorem (3). Because \(f(a) = f(b)\), there must be \(c \in (a,b)\), which is a point of local extrema. Then,

$$\begin{aligned} D_{\psi }^{p}(c)= & {} \lim _{\epsilon \rightarrow 0^{+}}\frac{f(c+\epsilon \psi (c,p))-f(c)}{\epsilon } \\= & {} \lim _{\epsilon \rightarrow 0^{-}}\frac{f(c+\epsilon \psi (c,p))-f(c)}{\epsilon }. \end{aligned}$$

If \(\psi (c,p) = 0\), then \(D_{\psi }^{p}(f(c)) = 0\). Otherwise, the two limits have opposite signs. Hence, \(D_{\psi }^{p}(f(c)) = 0\). \(\square \)

Theorem 9

(Mean Value theorem for GCFD). Let \(a > 0\) and \(f: [a,b] \rightarrow {\mathbb {R}}\) be a p-differentiable function for some \(p \in (0,1]\) at any \(u \in [a,b]\). \(h(u) = \int {\frac{1}{\psi (u,p)}du}\). Then, there exists \(c \in (a,b)\), such that \(D_{\psi }^{p}(f(c)) = \frac{f(b)-f(a)}{h(b)-h(a)}\).

Proof

First, we notice that \(D_{\psi }^{p}(h(u)) = 1\). It is apparent from direct computation.

Now consider the function

$$\begin{aligned} g(u) = f(u) - f(a) - \frac{f(b)-f(a)}{h(b)-h(a)}(h(u)-h(a)). \end{aligned}$$

Then the function g(u) satisfies the conditions of Rolle’s theorem (8).

So there exists \(c \in (a,b)\), such that \(D_{\psi }^{p}(g(c)) = 0\). That is

$$\begin{aligned} D_{\psi }^{p}(g(c)) = D_{\psi }^{p}(f(c)) - \frac{f(b)-f(a)}{h(b)-h(a)}D_{\psi }^{p}(h(c)) = 0. \end{aligned}$$

Because \(D_{\psi }^{p}(h(c)) = 1\), then we get \(D_{\psi }^{p}(f(c)) = \frac{f(b)-f(a)}{h(b)-h(a)}\). \(\square \)

Remark 12

When \(\psi (u, p) = 1, h(u) = u\). It coincides with classical Mean Value theorem.

Remark 13

When \(\psi (u, p) = u^{1-p}\), it is easy to get that \(h(u) = \frac{u^{p}}{p}\). It coincides with Khalil’s result in Refs. [19] and [20].

6 Inverse operator of GCFD

Inverse operator of GCFD, or corresponding form of integration, clearly should make the formula \(I_{p}D_{\psi }^{p} = I\) holds, where \(I_{p}\) is a symbol of integral and I is the identity operator. Inspired by relation (5) in Theorem (5), we can define it as follows.

Definition 8

(Integral of GCFD). Let \(a \ge 0\) and \(u \ge a, f\) be a function defined on (a, u]. Then the p-fractional integral of f is defined as:

$$\begin{aligned} I_{a}^{p,\psi } f(u) = \int _{a}^{u}\frac{f(t)}{\psi (t,p)}dt, \end{aligned}$$
(19)

if the Riemann integral exists.

Remark 14

When \(\psi (u, p) = 1\), it coincides with classical Riemann integral.

Remark 15

When \(\psi (u, p) = u^{1-p}\), it coincides with Khalil’s integral in Refs. [19] and [20].

The “fundamental theorem” of GCFD calculus is as follows.

Theorem 10

(Fundamental theorem 1 for GCFD). Let \(a \ge 0\) and \(u \ge a, p \in (0,1]\). Let f be a continuous function such that \(I_{a}^{p,\psi } f(u)\) exists. Then,

$$\begin{aligned} D_{\psi }^{p}{I_{a}^{p,\psi } f}(u) = f(u). \end{aligned}$$

Proof

Since f is continuous, then \(I_{a}^{p,\psi } f(u)\) is differentiable. According to (5) in Theorem (5), we have

$$\begin{aligned} D_{\psi }^{p}{I_{a}^{p,\psi } f}(u)= & {} \psi (u,p)({I_{a}^{p,\psi } f}(u))' \\= & {} \psi (u,p)\left( \int _{a}^{u}\frac{f(t)}{\psi (t,p)}dt\right) ' \\= & {} \psi (u,p)\frac{f(u)}{\psi (u,p)} \\= & {} f(u). \end{aligned}$$

\(\square \)

Theorem 11

(Fundamental theorem 2 for GCFD). Let \(a \ge 0\) and \(u \ge a, p \in (0,1]\). Let \(f: [a, +\infty )\) be differentiable. Then,

$$\begin{aligned} I_{a}^{p,\psi }D_{\psi }^{p} f(u) = f(u) - f(a). \end{aligned}$$
(20)

Proof

Since f is differentiable, according to (5) in Theorem (5), we have

$$\begin{aligned}&I_{a}^{p,\psi }D_{\psi }^{p} f(u) \\&\quad = I_{a}^{p,\psi }(\psi (t, p)f')(u) \\&\quad = \int _{a}^{u}\frac{\psi (t, p)f'(t)}{\psi (t,p)}dt \\&\quad = \int _{a}^{u}f'(t)dt \\&\quad = f(u) - f(a). \end{aligned}$$

\(\square \)

7 Application in fractional differential equations

In this section, we will give a simple scheme for solving the GCFD fractional differential equations (FDE). We consider the FDE in the form with

$$\begin{aligned} y^{(p)} + f(x)y = g(x), \end{aligned}$$
(21)

where \(0 < p \le 1\) and \(y^{(p)}\) denotes the fractional derivative of y. We assume that we are looking for a differentiable y.

According to (5) in Theorem (5), Eq. (21) is transformed to

$$\begin{aligned} \psi (x,p)y^{\prime } + f(x)y = g(x), \end{aligned}$$
(22)

which is a first-order differential equation and easy to solve.

Example 1

\(y^{(p)} + y = 0, 0 < p \le 1\).

We follow the above scheme and transform the original FDE to

$$\begin{aligned} \psi (x,p)y^{\prime } + y = 0. \end{aligned}$$
(23)

It is easy to find the solution of Eq. (23) is \(ce^{-\int {\frac{1}{\psi (x,p)}dx}}, c\) is any positive constant. If we take \(\psi (x,p) = x^{1-p}\), the solution equals \(ce^{-\frac{1}{p}x^{p}}\), which is derived in Ref. [19].

8 Conclusions and discussions

In the present paper, a class of new fractional derivative named GCFD is introduced to describe the physical world. We give physical and geometrical interpretations of GCFD which thus indicate potential applications in physics and engineering. It is easy to demonstrate that CFD is a special case of GCFD, then to the authors’ knowledge, so far we first give the physical and geometrical interpretations of CFD. The above work is done by a new framework named EGD and LEGD which are natural extensions of Gâteaux derivative. LEGD includes Gâteaux derivative as a special case, EGD includes both of LEGD and Kolwankar–Gangal’s LFD definition as special cases.

We discuss principles for the definition of LFD, and claim that the LEGD with a special class of functions (fractional conformable functions) satisfies those principles. As an example, we then define GCFD by means of LEGD. We do not follow the routine—define CFD and then guess what are the physical and geometrical interpretations of it? However, we try to find the definition from the point of view “How can it be closer to the nature and practical applications” and follow another routine—the LEGD has clear physical and geometrical interpretations, so we can define GCFD in that framework.

To discuss concrete applicable examples of GCFD is an interesting topic and we will present its application in fluid dynamics in the future works.