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Analysis and applications of the proportional Caputo derivative


In this paper, we investigate the analysis of the proportional Caputo derivative that recently has been constructed. We create some useful relations between this new derivative and beta function. We discretize the new derivative. We investigate the stability and obtain a stability condition for the new derivative.


Fractional calculus is an emerging field of mathematics [1] having important contributions in modeling the dynamics of complex systems [2, 3] from various fields of science and engineering [4, 5]. Nowadays a huge debate was opened by asking the simple” question: can we classify the fractional operators?” Curiously the answer of this question is not simple and, so far, several answers seemed to be possible [611]. A new non-singular fractional operator was proposed by Caputo and Fabrizio [12] and their result was generalized by Atangana and Baleanu [13] and applied successfully to a lot of complex phenomena including biological ones.

Khalid et al. [14] have studied the computational research of the Caputo time fractional Allen–Cahn equation. Owolabi [15] has studied by analysis and numerical simulation a multicomponent system with the Atangana–Baleanu fractional derivative. Akgül [16] has presented a novel method for a fractional derivative with non-local and non-singular kernel. Akgül [17] has investigated the solutions of differential equations with the generalized fractional derivatives. Atangana et al. [18] have investigated the analysis of the fractal fractional derivatives in detail. Fernandez et al. [19] investigated the series representations for fractional-calculus operators involving generalized Mittag-Leffler functions. Wu et al. [20] have investigated the fractional impulsive differential equations including the exact solutions, integral equations and short memory case. Some inequalities were investigated within the proportional fractional operators [21, 22] and in [23] was investigated the proportional derivatives of a function with respect to another function. Very recently, a new fractional operator has been constructed in [24]:

$$ {}^{\mathit{PC}}_{0} D_{t}^{\alpha }f(t)= \frac{1}{\Gamma (1-\alpha )} \int _{0}^{t} \biggl( k_{1}(\alpha ,\tau ) f(\tau )+ k_{0}(\alpha ,\tau ) \frac{df(\tau )}{d\tau } \biggr) (t-\tau )^{-\alpha }\,d\tau . $$

In this paper, we aim to analyze the above derivative in detail for \(k_{0}(\alpha ,t)= (\alpha t^{1-\alpha } ) c^{2 \alpha }\) and \(k_{1}(\alpha ,t)=(1-\alpha ) t^{\alpha }\). Here c is as a constant of the time dimension t for the two terms involved in the new derivative (1.1).

The new fractional operator in the Caputo sense is a generalization of the classical proportional derivative introduced by [24] which has deep applications in control theory. The new fractional operator will provide better applications in control theory. Due to the physical meaning of the initial conditions we concentrate here on the Caputo fractional generalization. For more details see [2528].

We construct the paper as follows. We give some scientific theorems for the new derivative in Sect. 2. We present the discretization and the applications of the proportional Caputo derivative in Sect. 3. We show the stability analysis in Sect. 4. We demonstrate the numerical results in Sect. 5. We discuss the conclusion in the last section.

Analysis of the proportional Caputo derivative

We present the following scientific results for the new derivative.

Lemma 2.1

We have the following relation for the new derivative given by (1.1):

$$\begin{aligned} \bigl\vert {}^{\mathit{PC}}_{0} D_{t}^{\alpha }f(t) \bigr\vert < & \frac{t (1-\alpha )}{\Gamma (1-\alpha )} \bigl\Vert f(\tau ) \bigr\Vert _{\infty }B(\alpha +1,1-\alpha ) \\ & {}+\frac{\alpha c^{2 \alpha }}{\Gamma (1-\alpha )} t^{2-2 \alpha } \biggl\Vert \frac{df(\tau )}{d\tau } \biggr\Vert _{\infty }B(2-\alpha ,1- \alpha ). \end{aligned}$$


We have

$$\begin{aligned} \bigl\vert {}^{\mathit{PC}}_{0} D_{t}^{\alpha }f(t) \bigr\vert =& \frac{1}{\Gamma (1-\alpha )} \biggl\vert \int _{0}^{t} \biggl((1- \alpha ) \tau ^{\alpha } f(\tau )+ c^{2 \alpha } \alpha \tau ^{1-\alpha } \frac{df(\tau )}{d\tau } \biggr) (t-\tau )^{-\alpha }\,d\tau \biggr\vert \\ \leq & \frac{1}{\Gamma (1-\alpha )} \biggl\vert \int _{0}^{t} \bigl((1-\alpha ) \tau ^{\alpha } f(\tau ) \bigr) (t-\tau )^{-\alpha }\,d\tau \biggr\vert \\ &{} +\frac{c^{2 \alpha }}{\Gamma (1-\alpha )} \biggl\vert \int _{0}^{t} \biggl( \alpha \tau ^{1-\alpha } \frac{df(\tau )}{d\tau } \biggr) (t- \tau )^{-\alpha }\,d\tau \biggr\vert \\ < & \frac{1}{\Gamma (1-\alpha )} \int _{0}^{t} \bigl\vert f(\tau ) \bigr\vert (1-\alpha ) \tau ^{\alpha } (t-\tau )^{-\alpha }\,d\tau \\ &{} +\frac{c^{2 \alpha }}{\Gamma (1-\alpha )} \int _{0}^{t} \biggl\vert \frac{df(\tau )}{d\tau } \biggr\vert \alpha \tau ^{1-\alpha } (t- \tau )^{-\alpha }\,d\tau \\ < & \frac{1}{\Gamma (1-\alpha )} \int _{0}^{t} \sup_{\tau \in [0,t]} \bigl\vert f(\tau ) \bigr\vert (1-\alpha ) \tau ^{\alpha } (t-\tau )^{- \alpha }\,d\tau \\ &{} +\frac{c^{2 \alpha }}{\Gamma (1-\alpha )} \int _{0}^{t} \sup_{ \tau \in [0,t]} \biggl\vert \frac{df(\tau )}{d\tau } \biggr\vert \alpha \tau ^{1-\alpha } (t- \tau )^{-\alpha }\,d\tau . \end{aligned}$$

Then we obtain

$$\begin{aligned} \bigl\vert ^{PC}_{0} D_{t}^{\alpha }f(t) \bigr\vert < & \frac{(1-\alpha )}{\Gamma (1-\alpha )} \bigl\Vert f(\tau ) \bigr\Vert _{\infty } \int _{0}^{t} \tau ^{\alpha } (t-\tau )^{-\alpha }\,d\tau \\ &{} +\frac{\alpha c^{2 \alpha }}{\Gamma (1-\alpha )} \biggl\Vert \frac{df(\tau )}{d\tau } \biggr\Vert _{\infty } \int _{0}^{t} \tau ^{1- \alpha } (t-\tau )^{-\alpha }\,d\tau . \end{aligned}$$

Let \(\tau = th\). Then we obtain

$$\begin{aligned} \bigl\vert {}^{\mathit{PC}}_{0} D_{t}^{\alpha }f(t) \bigr\vert < & \frac{(1-\alpha )}{\Gamma (1-\alpha )} \bigl\Vert f(\tau ) \bigr\Vert _{\infty } \int _{0}^{1} (th)^{\alpha } (t-th)^{-\alpha } t \,dh \\ & {}+\frac{\alpha c^{2 \alpha }}{\Gamma (1-\alpha )} \biggl\Vert \frac{df(\tau )}{d\tau } \biggr\Vert _{\infty } \int _{0}^{1} (th)^{1- \alpha } (t-th)^{-\alpha } t \,dh \\ < & \frac{t (1-\alpha )}{\Gamma (1-\alpha )} \bigl\Vert f(\tau ) \bigr\Vert _{\infty } \int _{0}^{1} h^{\alpha } (1-h)^{-\alpha }\,dh \\ & {}+\frac{\alpha c^{2 \alpha }}{\Gamma (1-\alpha )} t^{2-2\alpha } \biggl\Vert \frac{df(\tau )}{d\tau } \biggr\Vert _{\infty } \int _{0}^{1} h^{1-\alpha } (1-h)^{-\alpha }\,dh. \end{aligned}$$

Then we get the desired result:

$$\begin{aligned} \bigl\vert {}^{\mathit{PC}}_{0} D_{t}^{\alpha }f(t) \bigr\vert < & \frac{t (1-\alpha )}{\Gamma (1-\alpha )} \bigl\Vert f(\tau ) \bigr\Vert _{\infty }B(\alpha +1,1-\alpha ) \\ & {}+\frac{\alpha c^{2 \alpha }}{\Gamma (1-\alpha )} t^{2-2 \alpha } \biggl\Vert \frac{df(\tau )}{d\tau } \biggr\Vert _{\infty }B(2-\alpha ,1- \alpha ). \end{aligned}$$

This completes the proof. □

Remark 2.2

We consider

$$\begin{aligned}& {}^{\mathit{PC}}_{0} D_{x}^{\gamma } \bigl( u(x) v(x) \bigr) \\& \quad = \frac{1}{\Gamma (1-\gamma )} \int _{0}^{x} \biggl((1-\gamma ) t^{ \gamma } u(t) v(t)+ \gamma c^{2 \gamma } t^{1-\gamma } \frac{du(t)v(t)}{dt} \biggr) (x-t)^{-\gamma }\,dt. \end{aligned}$$

If u and v are continuous and bounded, then we get

$$\begin{aligned} {}^{\mathit{PC}}_{0} D_{x}^{\alpha } \bigl( u(x) v(x) \bigr) =& \frac{1}{\Gamma (1-\gamma )} \int _{0}^{x} \bigl((1-\gamma ) t^{ \gamma } u(t) v(t) \bigr) (x-t)^{-\gamma }\,dt \\ &{} + \frac{1}{\Gamma (1-\gamma )} \int _{0}^{x} \gamma c^{2 \gamma } t^{1- \gamma } \biggl( \frac{du(t)}{dt}v(t)+ \frac{dv(t)}{dt}u(t) \biggr) (x-t)^{- \gamma }\,dt. \end{aligned}$$

Lemma 2.3

Assume that f and g are differentiable and bounded. Then we obtain

$$\begin{aligned} \bigl\vert {}^{\mathit{PC}}_{0} D_{t}^{\alpha } \bigl( f(t)g(t) \bigr) \bigr\vert < & \frac{t (1-\alpha )}{\Gamma (1-\alpha )} \bigl\Vert f(t) \bigr\Vert _{\infty } \bigl\Vert g(t) \bigr\Vert _{\infty }B( \alpha +1,1- \alpha ) \\ & {}+ \frac{\alpha t^{2-2 \alpha }}{\Gamma (1-\alpha )} \biggl\Vert \frac{df(t)}{dt} \biggr\Vert _{\infty } \bigl\Vert g(t) \bigr\Vert _{\infty }B(2-\alpha ,1- \alpha ) \\ & {}+ \frac{\alpha t^{2-2 \alpha }}{\Gamma (1-\alpha )} \biggl\Vert \frac{dg(t)}{dt} \biggr\Vert _{\infty } \bigl\Vert f(t) \bigr\Vert _{\infty }B(2-\alpha ,1- \alpha ). \end{aligned}$$


We have

$$\begin{aligned} \bigl\vert {}^{\mathit{PC}}_{0} D_{t}^{\alpha } \bigl( f(t)g(t) \bigr) \bigr\vert < & \frac{t (1-\alpha )}{\Gamma (1-\alpha )} \bigl\Vert f(t) \bigr\Vert _{\infty } \bigl\Vert g(t) \bigr\Vert _{\infty }B( \alpha +1,1-\alpha ) \\ & {}+ \frac{\alpha t^{2-2 \alpha }}{\Gamma (1-\alpha )} \biggl\Vert \frac{df(t)}{dt} \biggr\Vert _{\infty } \bigl\Vert g(t) \bigr\Vert _{\infty }B(2-\alpha ,1- \alpha ) \\ & {}+ \frac{\alpha t^{2-2 \alpha }}{\Gamma (1-\alpha )} \biggl\Vert \frac{dg(t)}{dt} \biggr\Vert _{\infty } \bigl\Vert f(t) \bigr\Vert _{\infty }B(2-\alpha ,1- \alpha ). \end{aligned}$$

Let \(\tau = th\). Then we obtain

$$\begin{aligned} \bigl\vert {}^{\mathit{PC}}_{0} D_{t}^{\alpha } \bigl( f(t)g(t) \bigr) \bigr\vert < & \frac{t (1-\alpha )}{\Gamma (1-\alpha )} \bigl\Vert f(t) \bigr\Vert _{\infty } \bigl\Vert g(t) \bigr\Vert _{\infty }B( \alpha +1,1- \alpha ) \\ & {}+ \frac{\alpha t^{2-2 \alpha }}{\Gamma (1-\alpha )} \biggl\Vert \frac{df(t)}{dt} \biggr\Vert _{\infty } \bigl\Vert g(t) \bigr\Vert _{\infty }B(2-\alpha ,1- \alpha ) \\ & {}+ \frac{\alpha t^{2-2 \alpha }}{\Gamma (1-\alpha )} \biggl\Vert \frac{dg(t)}{dt} \biggr\Vert _{\infty } \bigl\Vert f(t) \bigr\Vert _{\infty }B(2-\alpha ,1- \alpha ). \end{aligned}$$

This completes the proof. □

Lemma 2.4

If f and g are differentiable and satisfy the following condition:

$$ \biggl\Vert \frac{df}{dt}-\frac{dg}{dt} \biggr\Vert _{\infty } < K \Vert f-g \Vert _{\infty }, $$

then we have

$$ \big\Vert {}^{\mathit{PC}}_{0} D_{t}^{\alpha } f(t) - {}^{\mathit{PC}}_{0} D_{t}^{\alpha } g(t) \big\Vert _{\infty } < K \Vert f-g \Vert _{\infty }. $$


We have

$$\begin{aligned} \big\Vert {}^{\mathit{PC}}_{0} D_{t}^{\alpha } f(t) - {}^{\mathit{PC}}_{0} D_{t}^{\alpha } g(t) \big\Vert _{\infty } < & \frac{(1-\alpha )}{\Gamma (1-\alpha )} \bigl\Vert f(t)-g(t) \bigr\Vert _{\infty } \int _{0}^{t} \tau ^{ \alpha } (t-\tau )^{-\alpha }\,d\tau \\ & {}+ \frac{\alpha c^{2 \alpha }}{\Gamma (1-\alpha )} \biggl\Vert \frac{df(t)}{dt}- \frac{dg(t)}{dt} \biggr\Vert _{\infty } \int _{0}^{t} \tau ^{1-\alpha } (t-\tau )^{-\alpha }\,d\tau . \end{aligned}$$

Let \(\tau = th\). Then we obtain

$$\begin{aligned} \big\Vert {}^{\mathit{PC}}_{0} D_{t}^{\alpha } f(t) - {}^{\mathit{PC}}_{0} D_{t}^{\alpha } g(t) \big\Vert _{\infty } < & K \bigl\Vert f(t)-g(t) \bigr\Vert _{ \infty }. \end{aligned}$$

This completes the proof. □

Lemma 2.5

Let f be analytic around 0, then we obtain

$$\begin{aligned} {}^{\mathit{PC}}_{0} D_{t}^{\alpha } f(t) =& t (1-\alpha ) \sum_{j=0}^{\infty } a_{j} t^{j} \frac{\Gamma (j+\alpha +1)}{\Gamma (j+2)} \end{aligned}$$
$$\begin{aligned} & {}+ t^{1-2\alpha } \alpha c^{2 \alpha } \sum _{j=0}^{\infty } j a_{j} t^{j} \frac{\Gamma (j-\alpha +1)}{\Gamma (j-2\alpha +2)}. \end{aligned}$$


We have

$$\begin{aligned} {}^{\mathit{PC}}_{0} D_{t}^{\alpha }f(t) =& \frac{1}{\Gamma (1-\alpha )} \int _{0}^{t} \biggl((1-\alpha ) \tau ^{\alpha } f(\tau )+ \alpha c^{2 \alpha } \tau ^{1+ \alpha } \frac{df(\tau )}{d\tau } \biggr) (t-\tau )^{-\alpha }\,d\tau \\ =& \frac{(1-\alpha )}{\Gamma (1-\alpha )} \sum_{j=0}^{\infty } a_{j} \int _{0}^{t} \tau ^{\alpha +j} (t-\tau )^{-\alpha }\,d\tau \\ & {}+ \frac{\alpha c^{2 \alpha } }{\Gamma (1-\alpha )} \sum_{j=0}^{ \infty } j a_{j} \int _{0}^{t} \tau ^{j+\alpha } (t-\tau )^{-\alpha }\,d\tau . \end{aligned}$$

We let \(\tau =ht\). Then we obtain

$$\begin{aligned} {}^{\mathit{PC}}_{0} D_{t}^{\alpha } f(t) =& t (1-\alpha ) \sum_{j=0}^{\infty } a_{j} t^{j} \frac{\Gamma (j+\alpha +1)}{\Gamma (j+2)} \\ & {}+ t^{1-2\alpha } \alpha c^{2 \alpha } \sum _{j=0}^{\infty } j a_{j} t^{j} \frac{\Gamma (j-\alpha +1)}{\Gamma (j-2\alpha +2)}. \end{aligned}$$

This completes the proof. □

Discretization and applications of the proportional Caputo derivative

We consider the new derivative [24]:

$$ {}^{\mathit{PC}}_{0} D_{t}^{\alpha }f(t)= \frac{1}{\Gamma (1-\alpha )} \int _{0}^{t} \biggl((1-\alpha ) \tau ^{\alpha } f(\tau )+ \alpha c^{2 \alpha } \tau ^{1- \alpha } \frac{df(\tau )}{d\tau } \biggr) (t-\tau )^{-\alpha }\,d\tau . $$

We put \(t_{n}=n\Delta t\), then at \(t_{n+1}\), we have

$$\begin{aligned} {}^{\mathit{PC}}_{0} D_{t}^{\alpha }f(t_{n+1}) =& \frac{1}{\Gamma (1-\alpha )} \int _{0}^{t_{n+1}} \biggl((1-\alpha ) \tau ^{\alpha } f(\tau )+ \alpha c^{2 \alpha } \tau ^{1-\alpha } \frac{df(\tau )}{d\tau } \biggr) (t_{n+1}- \tau )^{-\alpha }\,d\tau \\ =& \frac{1}{\Gamma (1-\alpha )} \sum_{j=0}^{n} \int _{t_{j}}^{t_{j+1}} \biggl((1-\alpha ) t_{j}^{\alpha } f^{j+1}+ \alpha c^{2 \alpha } t_{j}^{1- \alpha } \frac{f^{j+1}-f^{j}}{\Delta t} \biggr) (t_{n+1}-\tau )^{- \alpha }\,d\tau \\ =& \frac{1}{\Gamma (1-\alpha )} \sum_{j=0}^{n} \biggl((1-\alpha ) t_{j}^{ \alpha } f^{j+1}+ \alpha c^{2 \alpha } t_{j}^{1-\alpha } \frac{f^{j+1}-f^{j}}{\Delta t} \biggr) \\ &{}\times \int _{t_{j}}^{t_{j+1}} (t_{n+1}- \tau )^{-\alpha }\,d\tau \\ =& \frac{1}{\Gamma (1-\alpha )} \sum_{j=0}^{n} \biggl((1-\alpha ) t_{j}^{ \alpha } f^{j+1}+ \alpha c^{2 \alpha } t_{j}^{1-\alpha } \frac{f^{j+1}-f^{j}}{\Delta t} \biggr) \\ &{}\times \bigl[(n-j+1)^{1-\alpha }-(n-j)^{1- \alpha } \bigr]. \end{aligned}$$

We take into consideration [18]

$$ {}^{\mathit{PC}}_{0} D_{t}^{\alpha } u(x,t)= f \bigl(x,t,u(x,t) \bigr). $$

Here \(u(x,0)=g(x)\), \(x_{m}-x_{m-1}=\Delta x\), \(t_{n+1}-t_{n}=\Delta t\), \(t_{n}=n \Delta t\), \(x_{m}=m\Delta x\). The above equation can be approximated as

$$\begin{aligned}& \frac{1}{\Gamma (1-\alpha )} \sum_{j=0}^{n} \biggl((1-\alpha ) t_{j}^{ \alpha } u_{m}^{j+1}+ \alpha c^{2 \alpha } t_{j}^{1-\alpha } \frac{u_{m}^{j+1}-u_{m}^{j}}{\Delta t} \biggr) \bigl[(n-j+1)^{1- \alpha }-(n-j)^{1-\alpha } \bigr] \\& \quad = f \bigl(x_{m},t_{n+1},u_{m}^{n+1} \bigr). \end{aligned}$$

Stability analysis

We discretize the following problem and investigate the stability of it. We consider the heat equation,

$$ \frac{\partial u(x,t) }{\partial t} = k \frac{\partial ^{2} u(x,t) }{\partial x^{2}}. $$

We change the left hand side of the above equation with the new derivative and we obtain

$$ {}^{\mathit{PC}}_{0} D_{t}^{\alpha } u(x,t) = k \frac{\partial ^{2} u(x,t) }{\partial x^{2}}. $$

We obtain

$$\begin{aligned}& \frac{1}{\Gamma (1-\alpha )} \sum_{p=0}^{s} \biggl((1-\alpha ) t_{p}^{ \alpha } u_{m}^{p+1}+ \alpha c^{2 \alpha } t_{p}^{1-\alpha } \frac{u_{m}^{p+1}-u_{m}^{p}}{\Delta t} \biggr) \bigl[(s-p+1)^{1- \alpha }-(s-p)^{1-\alpha } \bigr] \\& \quad = k \frac{u_{m+1}^{s+1}-2u_{m}^{s+1}+u_{m-1}^{s+1}}{(\Delta x)^{2}} \end{aligned}$$

at \((t_{s+1},x_{m})\). We put \(u_{m}^{s}= \delta _{s} \exp (ik_{m} x)\). Plugging this into the above equation, we obtain

$$\begin{aligned}& \frac{1}{\Gamma (1-\alpha )} \sum_{p=0}^{s} \biggl((1-\alpha ) t_{p}^{ \alpha } \delta _{p+1} \exp (ik_{m} x)+ \alpha c^{2 \alpha } t_{p}^{1- \alpha } \frac{\delta _{p+1} \exp (ik_{m} x)-\delta _{p} \exp (ik_{m} x)}{\Delta t} \biggr) \\& \qquad {} \times \bigl[(s-p+1)^{1-\alpha }-(s-p)^{1-\alpha } \bigr] \\& \quad = k \frac{\delta _{s+1} \exp (ik_{m} ( x+\Delta x))-2\delta _{s+1} \exp (ik_{m} x)+\delta _{s+1} \exp (ik_{m} (x-\Delta x))}{(\Delta x)^{2}}. \end{aligned}$$

After simplification we get

$$\begin{aligned}& \frac{1}{\Gamma (1-\alpha )} \sum_{p=0}^{s} \biggl((1-\alpha ) t_{p}^{ \alpha } \delta _{p+1} + \alpha c^{2 \alpha } t_{p}^{1-\alpha } \frac{\delta _{p+1}-\delta _{p}}{\Delta t} \biggr) \\& \qquad {} \times \bigl[(s-p+1)^{1-\alpha }-(s-p)^{1-\alpha } \bigr] \\& \quad = k \frac{\delta _{s+1} \exp (ik_{m} (\Delta x))-2\delta _{s+1} +\delta _{s+1} \exp (ik_{m} (-\Delta x))}{(\Delta x)^{2}}. \end{aligned}$$

For simplicity, we take

$$ A_{p,\alpha } = \frac{(1-\alpha ) (p \Delta t)^{\alpha }}{\Gamma (1-\alpha )}, \qquad B_{p, \alpha } = \frac{\alpha c^{2 \alpha } (p \Delta t)^{1-\alpha }}{\Gamma (1-\alpha )\Delta t},\qquad a=\frac{k}{(\Delta x)^{2}}. $$

Then we obtain

$$\begin{aligned}& \sum_{p=0}^{s} \bigl( A_{p,\alpha } \delta _{p+1} + B_{p,\alpha } (\delta _{p+1}-\delta _{p} ) \bigr) \bigl[(s-p+1)^{1- \alpha }-(s-p)^{1-\alpha } \bigr] \\& \quad = a \delta _{s+1} \exp \bigl(ik_{m} (\Delta x) \bigr)-2a\delta _{s+1} +a \delta _{s+1} \exp \bigl(ik_{m} (-\Delta x) \bigr). \end{aligned}$$

Thus, we obtain

$$\begin{aligned}& \sum_{p=0}^{s} \bigl( (A_{p,\alpha } + B_{p,\alpha } ) \delta _{p+1}- B_{p,\alpha } \delta _{p} \bigr) \bigl[(s-p+1)^{1- \alpha }-(s-p)^{1-\alpha } \bigr] \\& \quad = a \delta _{s+1} \bigl( \exp \bigl(ik_{m} (\Delta x) \bigr)-2+ \exp \bigl(-ik_{m} (\Delta x) \bigr) \bigr). \end{aligned}$$

Using the relation between the trigonometric functions and exponential functions gives

$$\begin{aligned} \sum_{p=0}^{s} \bigl( (A_{p,\alpha } + B_{p,\alpha } ) \delta _{p+1}- B_{p,\alpha } \delta _{p} \bigr) \bigl[(s-p+1)^{1- \alpha }-(s-p)^{1-\alpha } \bigr]=-4 a \delta _{s+1} \sin ^{2} \biggl( \frac{k_{m} \Delta x}{2} \biggr). \end{aligned}$$

For \(s=0\), we obtain

$$\begin{aligned} \bigl( (A_{0,\alpha } + B_{0,\alpha } )\delta _{1}- B_{0, \alpha } \delta _{0} \bigr) =-4 a \delta _{1} \sin ^{2} \biggl( \frac{k_{m} \Delta x}{2} \biggr). \end{aligned}$$

Here \(\vert \frac{\delta _{1}}{\delta _{0}} \vert <1\) implies

$$ \biggl\vert \frac{B_{0,\alpha }}{A_{0,\alpha }+B_{0,\alpha }+4 a \sin ^{2} ( \frac{k_{m} \Delta x}{2} )} \biggr\vert < 1. $$

This is true for m. Thus, we get

$$ \biggl\vert \frac{B_{0,\alpha }}{A_{0,\alpha }+B_{0,\alpha }+4a} \biggr\vert < 1. $$

We assume that \(\vert \frac{\delta _{s}}{\delta _{0}} \vert <1\). We need to show that \(\vert \frac{\delta _{s+1}}{\delta _{0}} \vert <1\). We know that

$$\begin{aligned} \sum_{p=0}^{s} \bigl( (A_{p,\alpha } + B_{p,\alpha } ) \delta _{p+1}- B_{p,\alpha } \delta _{p} \bigr) \bigl[(s-p+1)^{1- \alpha }-(s-p)^{1-\alpha } \bigr]=-4 a \delta _{s+1} \sin ^{2} \biggl( \frac{k_{m} \Delta x}{2} \biggr). \end{aligned}$$

Then we get

$$\begin{aligned} \biggl\vert -4 a \delta _{s+1} \sin ^{2} \biggl( \frac{k_{m} \Delta x}{2} \biggr) \biggr\vert = \Biggl\vert \sum _{p=0}^{s} \bigl( (A_{p,\alpha } + B_{p,\alpha } )\delta _{p+1}- B_{p, \alpha } \delta _{p} \bigr) \bigl[(s-p+1)^{1-\alpha }-(s-p)^{1- \alpha } \bigr] \Biggr\vert . \end{aligned}$$

Thus, we reach

$$\begin{aligned} \delta _{s+1} \biggl\vert -4 a \sin ^{2} \biggl( \frac{k_{m} \Delta x}{2} \biggr) \biggr\vert < \delta _{0} \Biggl\vert \sum_{p=0}^{s} \bigl( (A_{p,\alpha } + B_{p,\alpha } )- B_{p,\alpha } \bigr) \bigl[(s-p+1)^{1-\alpha }-(s-p)^{1- \alpha } \bigr] \Biggr\vert . \end{aligned}$$

Here \(\vert \frac{\delta _{s+1}}{\delta _{0}} \vert <1\) implies

$$\begin{aligned} \frac{\sum_{p=0}^{s} \vert ( (A_{p,\alpha } + B_{p,\alpha } )- B_{p,\alpha } ) [(s-p+1)^{1-\alpha }-(s-p)^{1-\alpha } ] \vert }{ \vert -4 a \sin ^{2} ( \frac{k_{m} \Delta x}{2} ) \vert } < 1. \end{aligned}$$

This is true for m. Thus, we get

$$\begin{aligned} \frac{\sum_{p=0}^{s} \vert ( (A_{p,\alpha } + B_{p,\alpha } )- B_{p,\alpha } ) [(s-p+1)^{1-\alpha }-(s-p)^{1-\alpha } ] \vert }{ \vert -4 a \vert } < 1. \end{aligned}$$

Therefore, the method is stable if

$$\begin{aligned} \min \biggl( \biggl\vert \frac{B_{0,\alpha }}{A_{0,\alpha }+B_{0,\alpha }+4 a } \biggr\vert , \frac{\sum_{p=0}^{s} \vert ( (A_{p,\alpha } + B_{p,\alpha } )- B_{p,\alpha } ) [(s-p+1)^{1-\alpha }-(s-p)^{1-\alpha } ] \vert }{ \vert -4 a \vert } \biggr)< 1. \end{aligned}$$

Numerical results

We consider the following problem:

$$ {}^{\mathit{CPC}}_{0} D_{x}^{\alpha }u(x)= \sin (x) $$


$$ {}^{\mathit{CPC}}_{0} D_{x}^{\alpha }u(x)= \frac{1}{\Gamma (1-\alpha )} \int _{0}^{x} \biggl( k_{1}(\alpha ) u(t)+ k_{0}(\alpha ) \frac{du(t)}{dt} \biggr) (x-t)^{- \alpha }\,dt. $$

We apply the Laplace transform to Eq. (5.1):

$$ L \bigl( {}^{\mathit{CPC}}_{0} D_{x}^{\alpha }u(x) \bigr) = L \bigl( \sin (x) \bigr). $$

Then we obtain

$$ \biggl[\frac{K_{1}(\alpha )}{s}+K_{0}(\alpha ) \biggr] s^{\alpha }L \bigl(u(t) \bigr)-K_{0}( \alpha ) s^{\alpha -1} u(0) = \frac{1}{1+s^{2}}. $$

After simplification, we get

$$ L \bigl(u(x) \bigr)= \frac{(1+s^{2}) K_{0}(\alpha ) s^{\alpha -1} u(0)+1}{(1+s^{2}) (s^{\alpha -1} K_{1}(\alpha )+s^{\alpha }K_{0}(\alpha ))}. $$

If we apply the inverse Laplace transform to the above equation, we will obtain

$$\begin{aligned} u(x) =u(0) \exp \biggl( \frac{-K_{1}(\alpha )}{K_{0}(\alpha )} x \biggr)+ \frac{ x^{\alpha }A(x,\alpha )}{ (K_{1}(\alpha )^{2} + K_{2}(\alpha )^{2} ) \Gamma (\alpha )} \end{aligned}$$


$$\begin{aligned} A(x,\alpha ) =& K_{1}(\alpha ) \exp \biggl( \frac{-K_{1}(\alpha )}{K_{0}(\alpha )} x \biggr) \biggl(- \frac{K_{1}(\alpha )}{K_{0}(\alpha )} \biggr)^{-\alpha } \biggl(- \Gamma (\alpha ) +\Gamma \biggl(\alpha , \frac{-K_{1}(\alpha )}{K_{1}(\alpha )}x \biggr) \biggr) \\ & {}+ \frac{1}{\alpha } \text{HypergeometricPFQ} \biggl[ \biggl\{ \frac{\alpha }{2} \biggr\} , \biggl\{ \frac{1}{2}, 1+ \frac{\alpha }{2} \biggr\} ,- \frac{x^{2}}{4} \biggr] \\ & {}\times \bigl(K_{1}(\alpha ) \cos (x)+K_{0}(\alpha ) \sin (x) \bigr) + \frac{1}{1+\alpha } x \bigl(-K_{0}(\alpha ) \cos (x)+K_{1}( \alpha ) \sin (x) \bigr) \\ & {}\times \text{HypergeometricPFQ} \biggl[ \biggl\{ \frac{1}{2}+ \frac{\alpha }{2} \biggr\} , \biggl\{ \frac{3}{2}, \frac{3}{2}, \frac{\alpha }{2} \biggr\} ,- \frac{x^{2}}{4} \biggr]. \end{aligned}$$

We demonstrate the above solution by the following figures for different values of α. We choose \(K_{1}(\alpha )=(1-\alpha ) w^{\alpha } \), \(K_{0}(\alpha )=\alpha c^{2 \alpha } w^{1-\alpha } \), \(c=1\), \(w=0.5\) and \(u(0)=1\) in Figs. 16. In Fig. 7, we choose \(c=w=\alpha =0.8\). In these figures, we can see the effect of the fractional order.

Figure 1

Solution of the problem for \(\alpha =0.1\)

Figure 2

Solution of the problem for \(\alpha =0.3\)

Figure 3

Solution of the problem for \(\alpha =0.5\)

Figure 4

Solution of the problem for \(\alpha =0.7\)

Figure 5

Solution of the problem for \(\alpha =0.9\)

Figure 6

Solution of the problem for \(\alpha =0.99\)

Figure 7

Solution of the problem for \(\alpha =w=c=0.8\)


We presented the analysis of the proportional Caputo derivative in this paper. We presented some scientific theorems for this new derivative. We discretized the new derivative. We presented the stability analysis and experiments. We obtained the stability condition for a problem using the new derivative. We considered a problem with the constant proportional Caputo derivative. We solved the problem by the Laplace transform. We demonstrated the numerical simulations by some figures.

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  1. 1.

    Kilbas, A.A., Srivastava, H.H., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, New York (2006)

    Google Scholar 

  2. 2.

    Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)

    Google Scholar 

  3. 3.

    Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos, Frist edn. 2012/Second edn. 2016. World Scientific, Singapore (2016)

    Google Scholar 

  4. 4.

    Baleanu, D., Etemad, S., Rezapour, S.: A hybrid Caputo fractional modeling for thermostat with hybrid boundary value conditions. Bound. Value Probl. 2020, 64 (2020)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Kumar, S., Baleanu, D.: A new numerical method for time fractional non-linear Sharma–Tasso–Oliver equation and Klein–Gordon equation with exponential kernel law. Front. Phys. 8, 136 (2020)

    Article  Google Scholar 

  6. 6.

    Ross, B.: A brief history and exposition of the fundamental theory of fractional calculus. In: Fractional Calculus and Its Applications. Lect. Notes Math., pp. 1–36. Springer, Berlin (1975)

    Google Scholar 

  7. 7.

    Ortigueira, M., Tenreiro Machado, J.A.: What is a fractional derivative? J. Comput. Phys. 293, 4–13 (2015)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Caputo, M., Fabrizio, M.: On the notion of fractional derivative and applications to the hysteresis phenomena. Meccanica 52(13), 3043–3052 (2017)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Zhao, D., Luo, M.: Representations of acting processes and memory effects: general fractional derivative and its application to theory of heat conduction with finite wave speeds. Appl. Math. Comput. 346, 531–544 (2019)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Hilfer, R., Luchko, Y.: Desiderata for fractional derivatives and integrals. Mathematics 7, 149 (2019).

    Article  Google Scholar 

  11. 11.

    Baleanu, D., Fernandez, A.: On fractional operators and their classifications. Mathematics 7, 830 (2019)

    Article  Google Scholar 

  12. 12.

    Caputo, M., Fabrizio, M.: A new defifinition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1(2), 73–85 (2015)

    Google Scholar 

  13. 13.

    Atangana, A., Baleanu, D.: New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Therm. Sci. 20(2), 763–769 (2016)

    Article  Google Scholar 

  14. 14.

    Khalid, N., Abbas, M., Iqbal, M.K., Baleanu, D.: A numerical investigation of Caputo time fractional Allen–Cahn equation using redefined cubic B-spline functions. Adv. Differ. Equ. 2020, 158 (2020)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Owolabi, K.M.: Analysis and numerical simulation of multicomponent system with Atangana–Baleanu fractional derivative. Chaos Solitons Fractals 115, 127–134 (2018)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Akgül, A.: A novel method for a fractional derivative with non-local and non-singular kernel. Chaos Solitons Fractals 114, 478–482 (2018)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Akgül, E.K.: Solutions of the linear and nonlinear differential equations within the generalized fractional derivatives. Chaos 29, 023108 (2019)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Atangana, A., Akgül, A.: Analysis and Applications of the New Derivative, Chapter of Fractional Order Analysis: Theory, Methods and Applications. Wiley, New York (2020)

    Google Scholar 

  19. 19.

    Fernandez, A., Baleanu, D., Srivastava, H.M.: Series representations for fractional-calculus operators involving generalised Mittag-Leffler functions. Commun. Nonlinear Sci. Numer. Simul. 67, 517–527 (2019)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Wu, G.C., Zeng, D.Q., Baleanu, D.: Fractional impulsive differential equations: exact solutions, integral equations and short memory case. Fract. Calc. Appl. Anal. 22, 180–192 (2019)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Rahman, G., Nisar, K.S., Abdeljawad, T.: Certain Hadamard proportional fractional integral inequalities. Mathematics 8(4), 504 (2020)

    Article  Google Scholar 

  22. 22.

    Jarad, F., Alqudah, M.A., Abdeljawad, T.: On more general forms of proportional fractional operators. Open Math. 18(1), 167–176 (2020)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Alzabut, J., Abdeljawad, T., Jarad, F., Sudsutad, W.: A Gronwall inequality via the generalized proportional fractional derivative with applications. J. Inequal. Appl. 2019, 101 (2019)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Baleanu, D., Fernandez, A., Akgül, A.: On a fractional operator combining proportional and classical differintegrals. Mathematics 8, 360 (2020)

    Article  Google Scholar 

  25. 25.

    Agarwal, P., Choi, J., Paris, R.B.: Extended Riemann–Liouville type fractional derivative operator with applications. Open Math. 15(1), 1667–1681 (2017)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Al-Sayed, A.A., Agarwal, P.: Numerical solution of multiterm variable-order fractional differential equations via shifted Legendre polynomials. Math. Methods Appl. Sci. 42(11), 3978–3991 (2019)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Agarwal, P., Rogosin, S.V., Trujillo, J.J.: Certain fractional integral operators and the generalized multi-index Mittag-Leffler functions. Proc. Math. Sci. 125(3), 291–306 (2015)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Cetinkaya, A., Kiymaz, I.O., Agarwal, P., Agarwal, R.: A comparative study on generating function relations for generalized hypergeometric functions via generalized fractional operators. Adv. Differ. Equ. 2018(1), 1 (2018)

    MathSciNet  Article  Google Scholar 

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We thank the reviewers for their comments to greatly improve the quality of our paper.


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Akgül, A., Baleanu, D. Analysis and applications of the proportional Caputo derivative. Adv Differ Equ 2021, 136 (2021).

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