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Note on weakly fractional differential equations

Advances in Difference Equations20192019:143

https://doi.org/10.1186/s13662-019-2086-4

  • Received: 7 February 2019
  • Accepted: 8 April 2019
  • Published:

Abstract

In this paper, we compare solutions of q-order fractional differential equations of Caputo type for q near 1 with solutions of the corresponding 1-order ordinary differential equations. By establishing the explicit lower and upper bounds of Mittag-Leffler functions, we obtain the effective convergence results. It is shown that the limit cases \(q\to1_{+}\) and \(q\to1_{-}\) are different. A simple illustrative example is also presented.

Keywords

  • Weakly fractional differential equation
  • Caputo fractional derivative
  • Comparison
  • Mittag-Leffler function

1 Introduction

Fractional differential equations (FDEs) are a rapidly developing area of mathematics with many stimulating applications [14]. Recently, plenty interesting existence and controllability results on the theory of solutions of FDEs or fractional inclusions have been given in [522]. Mathematical modeling approaches using fractional derivatives are presented in [1722] with numerical simulations on various challenging topics.

On the one hand, several properties of ordinary or partial differential equations (DEs) appear in FDEs as well, like asymptotic properties of solutions or equilibria. On the other hand, unlike to DEs, FDEs have no nonconstant periodic solutions and they do not create dynamical systems, which is one of the most obvious characteristics in studying FDEs. So there is a natural question to study the relationship between solutions of FDEs and DEs when the order q of FDEs is near to a natural number \(n\in \mathbb {N}\). Here, we call such FDEs weakly fractional, which can be used to seek numerically the solutions of DEs.

In this paper, we investigate for simplicity the case when q is near to \(n=1\), but our method can be directly extended to any n. We study two cases: \(q\to1_{-}\) in Sect. 2 and \(q\to1_{+}\) in Sect. 3. We derive error estimates in both cases. A simple numerical illustrative example is given to demonstrate theoretical results. Our next step will be to extend this paper for weakly fractional semilinear evolution equations in Banach spaces.

2 The case \(q\to1_{-}\)

Consider a fractional differential equation
$$ \begin{gathered} D_{0}^{q}x(t) = f \bigl(t,x(t)\bigr), \quad t\in \mathbb {R}_{+}=[0,\infty), \\ x(0) = x_{0}, \end{gathered} $$
(1)
where \(D_{0}^{q}\) is the Caputo fractional derivative of order \(q\in (0,1)\) with the lower limit at zero,
$$D_{0}^{q}x(t)=\frac{1}{\varGamma(1-q)}\frac{d}{dt} \int _{0}^{t}(t-s)^{-q}\bigl(x(s)-x(0) \bigr)\,ds, $$
and \(f\in C(\mathbb {R}_{+}\times \mathbb {R}^{n},\mathbb {R}^{n})\) along with an ordinary differential equation
$$ \begin{gathered} y'(t) = f\bigl(t,y(t) \bigr), \quad t\in \mathbb {R}_{+}, \\ y(0) = y_{0}, \end{gathered} $$
(2)
where \(x_{0},y_{0}\in \mathbb {R}^{n}\). We suppose
  1. (H)

    There are nonnegative constants M and L such that \(\Vert f(t,x)\Vert \le M\) and \(\Vert f(t,x)-f(t,y)\Vert \le L\Vert x-y\Vert \) for any \(t\in \mathbb {R}_{+}\) and \(x,y\in \mathbb {R}^{n}\), where \(\Vert \cdot\Vert \) is a norm on \(\mathbb {R}^{n}\).

     
It is well known [4] that problem (1) is equivalent to the following integral equation:
$$x(t)=x_{0}+\frac{1}{\varGamma(q)} \int_{0}^{t} (t-s)^{q-1}f\bigl(s,x(s)\bigr) \,ds. $$
Then we derive
$$\begin{aligned} & \bigl\Vert x(t)-y(t) \bigr\Vert \\ &\quad \le \Vert x_{0}-y_{0} \Vert +\frac{1}{\varGamma(q)} \int _{0}^{t}(t-s)^{q-1} \bigl\Vert f \bigl(s,x(s)\bigr)-f\bigl(s,y(s)\bigr) \bigr\Vert \,ds \\ &\quad \quad {}+ \int_{0}^{t} \biggl\vert 1-\frac{1}{\varGamma(q)}(t-s)^{q-1} \biggr\vert \bigl\Vert f\bigl(s,y(s)\bigr) \bigr\Vert \,ds \\ &\quad \le \Vert x_{0}-y_{0} \Vert +\frac{L}{\varGamma(q)} \int_{0}^{t}(t-s)^{q-1} \bigl\Vert x(s)-y(s) \bigr\Vert \,ds+M \int_{0}^{t} \biggl\vert 1-\frac{1}{\varGamma(q)}(t-s)^{q-1} \biggr\vert \,ds \\ &\quad = \Vert x_{0}-y_{0} \Vert +\frac{L}{\varGamma(q)} \int_{0}^{t}(t-s)^{q-1} \bigl\Vert x(s)-y(s) \bigr\Vert \,ds+M \int_{0}^{t} \biggl\vert 1-\frac{1}{\varGamma(q)}s^{q-1} \biggr\vert \,ds. \end{aligned}$$
Thus by the Henry–Gronwall inequality (see [23, Corollary 2]), we get
$$\bigl\Vert x(t)-y(t) \bigr\Vert \le \biggl( \Vert x_{0}-y_{0} \Vert +M \int_{0}^{t} \biggl\vert 1-\frac{1}{\varGamma (q)}s^{q-1} \biggr\vert \,ds \biggr)E_{q}\bigl(Lt^{q}\bigr) $$
for any \(t\in \mathbb {R}_{+}\), where \(E_{q}\) is the Mittag-Leffler function [24]. We continue with the case \(x_{0}=y_{0}\). Then we get
$$ \bigl\Vert x(t)-y(t) \bigr\Vert \le M\theta_{q}(t), \quad\quad \theta_{q}(t):= \int_{0}^{t} \biggl\vert 1-\frac{1}{\varGamma(q)}s^{q-1} \biggr\vert \,ds E_{q}\bigl(Lt^{q}\bigr) $$
(3)
for any \(t\in \mathbb {R}_{+}\). The equation
$$1-\frac{1}{\varGamma(q)}s^{q-1}=0 $$
has the only solution \(s_{0}>0\) given by
$$ s_{0}=s_{0}(q)=\varGamma(q)^{\frac{1}{q-1}}. $$
(4)
Note that the function \(s_{0}(q)\) is increasing on \((0,1)\) with \(\lim_{q\to0_{+}}s_{0}(q)=0\) and
$$\lim_{q\to1_{-}}s_{0}(q)=e^{\lim_{q\to1_{-}}{\frac{\ln[\varGamma [q]]}{q-1}}}=e^{\lim_{q\to1_{-}}{\frac{\varGamma'[q]}{\varGamma [q]}}}=e^{-\gamma} \doteq0.561459 $$
for the Euler constant γ. Next, clearly, we have
$$1-\frac{1}{\varGamma(q)}s^{q-1} \textstyle\begin{cases} < 0 & \text{for } 0< s< s_{0},\\ =0 & \text{for } s=s_{0},\\ >0 & \text{for } s>s_{0}. \end{cases} $$
Consequently, we obtain
$$ \int_{0}^{t} \biggl\vert 1-\frac{1}{\varGamma(q)}s^{q-1} \biggr\vert \,ds= \textstyle\begin{cases} \frac{t^{q}}{\varGamma(q+1)}-t & \text{for } 0< t< s_{0},\\ \lambda(q):=\frac{\varGamma(q)^{\frac{q}{q-1}}}{\varGamma(q+1)}-\varGamma (q)^{\frac{1}{q-1}} & \text{for } t=s_{0},\\ \frac{-t^{q}+t\varGamma(q+1)+2\varGamma(q)^{\frac{q}{q-1}}}{\varGamma(q+1)}-2 \varGamma(q)^{\frac{1}{q-1}} & \text{for } t>s_{0}. \end{cases} $$
(5)
We can check numerically that \(\lambda''(q)>0\) for \(q\in(0,1)\), then that \(\lambda'(q)\) is increasing from −∞ to \(-e^{-\gamma }\doteq-0.561459\), and then that \(\lambda(q)\) is decreasing from 1 to 0. So we consider \(q\in(1/2,1)\) and then \(-0.751988\le\lambda '(q)\le-0.561459\). This implies that
$$ 0< \lambda(q)\le0.8(1-q) $$
(6)
for \(q\in(1/2,1)\). Next, by [25, Lemma 2], we have the following.

Lemma 2.1

For all \(t\in \mathbb {R}_{+}\), \(q\in(0,1)\), and \(\kappa>0\), it holds
$$1\le E_{q}\bigl(\kappa t^{q}\bigr)\le\frac{e^{\kappa^{\frac{1}{q}}t}}{q}. $$
Furthermore, (5) implies
$$\int_{0}^{t} \biggl\vert 1-\frac{1}{\varGamma(q)}s^{q-1} \biggr\vert \,ds\le \biggl\vert t-\frac{t^{q}}{\varGamma(q+1)} \biggr\vert +2 \lambda(q) $$
for \(t\in \mathbb {R}_{+}\). So if \(q\in(1/2,1)\), then by Lemma 2.1 we get
$$ \theta_{q}(t)\le\frac{e^{L^{\frac{1}{q}}t}}{q} \biggl( \biggl\vert t-\frac {t^{q}}{\varGamma(q+1)} \biggr\vert +2\lambda(q) \biggr)\le2e^{t\tilde{L}} \biggl( \biggl\vert t-\frac{t^{q}}{\varGamma(q+1)} \biggr\vert +2\lambda(q) \biggr) $$
(7)
for \(\tilde{L}=\max\{L,L^{2}\}\).

Now we are ready to deal with (3). First, (3) immediately implies the following expected result.

Theorem 2.2

Under assumption (H), the solution \(x(t)\) of (1) uniformly converges on any finite interval \([0,T]\), \(T>0\), of \(\mathbb {R}_{+}\) to the solution \(y(t)\) of (2) if \(q\to1_{-}\) and \(x_{0}=y_{0}\).

Proof

The proof follows directly from (3), (5) and by
$$\lim_{q\to1_{-}} \biggl\vert t-\frac{t^{q}}{\varGamma(q+1)} \biggr\vert =0 $$
uniformly for \(t\in[0,T]\) and any fixed \(T>0\). □
Next, we take any \(\epsilon>0\) and consider an equation
$$ \theta_{q}(t)=\epsilon. $$
(8)
Clearly, \(\theta_{q}(t)\) is increasing on \(\mathbb {R}_{+}\) from 0 to ∞. Thus (8) has the only solution \(\bar{t}(\epsilon,q)\in \mathbb {R}_{+}\). By the above observations we can easily see that \(\lim_{\epsilon\to 0_{+}}\bar{t}(\epsilon,q)=0\) and \(\lim_{q\to1_{-}}\bar{t}(\epsilon ,q)=\infty\).
Furthermore, the function \(t\mapsto t-\frac{t^{q}}{\varGamma(q+1)}\) is nonpositive on \([0,r_{0}]\) and nonnegative on \([r_{0},\infty)\) for
$$ r_{0}=r_{0}(q)=\varGamma(q+1)^{\frac{1}{q-1}}. $$
(9)
Note that the function \(r_{0}(q)\) is increasing on \((0,1)\) from \(\lim_{q\to0_{+}}r_{0}(q)=1\) to \(\lim_{q\to1_{-}}r_{0}(q)=e^{1-\gamma}\doteq1.526205\).
Next, we study the function \(\phi_{t}(q):=\frac{t^{q}}{\varGamma(q+1)}\) on \((0,1)\) for \(t>0\). We have
$$ \phi_{t}'(q)=\frac{t^{q} \ln t}{\varGamma(q+1)}- \frac{t^{q} \varGamma '(q+1)}{\varGamma^{2}(q+1)}. $$
(10)
For \(t\in(0,1]\) and \(q\in(1/2,1)\), we get
$$\bigl\vert \phi_{t}'(q) \bigr\vert \le- \frac{\sqrt{t}\ln t}{\varGamma(q+1)}+ \biggl\vert \frac {\varGamma'(q+1)}{\varGamma^{2}(q+1)} \biggr\vert \le1.253, $$
while for \(1\le t\le T\) and \(q\in(1/2,1)\), we get
$$\bigl\vert \phi_{t}'(q) \bigr\vert \le1.12838T^{q}\ln T+0.422784T^{q} $$
for \(T>1\). Consequently, we have
$$\bigl\vert \phi_{t}'(q) \bigr\vert \le1.253+1.12838T^{q}\ln T+0.422784T^{q} $$
for \(t\in(0,T]\), \(T>1\), and \(q\in(1/2,1)\). This implies
$$ \biggl\vert t-\frac{t^{q}}{\varGamma(q+1)} \biggr\vert = \bigl\vert \phi_{t}(1)-\phi_{t}(q) \bigr\vert \le \bigl(1.253+1.12838T^{q} \ln T+0.422784T^{q}\bigr) (1-q) $$
(11)
for \(t\in[0,T]\), \(T>1\), and \(q\in(1/2,1)\). Using (6), (7), and (11), we arrive at
$$\theta_{q}(t)\le2e^{T\tilde{L}} \bigl(3+2T^{q}\ln T+T^{q} \bigr) (1-q) $$
for \(t\in[0,T]\), \(T>1\), and \(q\in(1/2,1)\). Now, we consider instead of (8) the following one:
$$ \eta_{L,q}(T):=2e^{T\tilde{L}} \bigl(3+2T^{q} \ln T+T^{q} \bigr)=\frac {1}{\sqrt{1-q}}. $$
(12)
The function \(\eta_{L,q}(T)\) is increasing from \(8e^{\tilde{L}}\) to ∞ on \([1,\infty)\). So, for any
$$ q>1-\frac{1}{64e^{2\tilde{L}}},\quad q\in(1/2,1), $$
(13)
(12) has a unique solution \(T_{L}(q)>1\). Note
$$\lim_{q\to1_{-}}T_{L}(q)=\infty. $$
Summarizing, we have the following result.

Theorem 2.3

Under assumption (H) and for any q fulfilling (13), the solutions \(x(t)\) and \(y(t)\) of (1) and (2) with \(x_{0}=y_{0}\), respectively, satisfy
$$ \bigl\Vert x(t)-y(t) \bigr\Vert \le M\sqrt{1-q} $$
(14)
for any \(t\in[0,T_{L}(q)]\), where \(T_{L}(q)>1\) is the unique solution of (12).

3 The case \(q\to1_{+}\)

Consider a fractional differential equation
$$ \begin{gathered} D_{0}^{q}x(t) = f \bigl(t,x(t)\bigr), \quad t\in \mathbb {R}_{+}, \\ x(0) = x_{0}, \\ x'(0) = x_{1}, \end{gathered} $$
(15)
where \(q\in(1,2)\) and \(f\in C(\mathbb {R}_{+}\times \mathbb {R}^{n},\mathbb {R}^{n})\) along with an ordinary differential equation
$$ \begin{gathered} y'(t) = f\bigl(t,y(t) \bigr)+y_{1}, \quad t\in \mathbb {R}_{+}, \\ y(0) = y_{0}, \end{gathered} $$
(16)
where \(x_{0},x_{1},y_{0},y_{1}\in \mathbb {R}^{n}\). Again, we suppose assumption (H). It is known [2, Theorem 3.24] that initial value problem (15) is equivalent to the integral equation
$$x(t)=x_{0}+x_{1} t+\frac{1}{\varGamma(q)} \int_{0}^{t}(t-s)^{q-1}f\bigl(s,x(s)\bigr) \,ds. $$
Analogously to the previous section, we derive
$$\begin{aligned} \bigl\Vert x(t)-y(t) \bigr\Vert &\leq \Vert x_{0}-y_{0} \Vert + \Vert x_{1}-y_{1} \Vert t \\ &\quad {}+\frac{L}{\varGamma(q)} \int_{0}^{t}(t-s)^{q-1} \bigl\Vert x(s)-y(s) \bigr\Vert \,ds +M \int_{0}^{t} \biggl\vert 1-\frac{1}{\varGamma(q)}s^{q-1} \biggr\vert \,ds, \end{aligned}$$
and the Henry–Gronwall inequality yields
$$\bigl\Vert x(t)-y(t) \bigr\Vert \leq \biggl( \Vert x_{0}-y_{0} \Vert + \Vert x_{1}-y_{1} \Vert t+M \int_{0}^{t} \biggl\vert 1-\frac{1}{\varGamma(q)}s^{q-1} \biggr\vert \,ds \biggr)E_{q}\bigl(Lt^{q}\bigr). $$
Hence, for \(x_{0}=y_{0}\), \(x_{1}=y_{1}\), estimation (3) follows for any \(t\in \mathbb {R}_{+}\). Function \(s_{0}(q)\) of (4) is increasing on \((1,2)\) from
$$\lim_{q\to1_{+}}s_{0}(q)=e^{\lim_{q\to1_{+}}{\frac{\ln[\varGamma [q]]}{q-1}}}=e^{\lim_{q\to1_{+}}{\frac{\varGamma'[q]}{\varGamma [q]}}}=e^{-\gamma} \doteq0.561459 $$
to 1. So this time,
$$1-\frac{1}{\varGamma(q)}s^{q-1} \textstyle\begin{cases} >0 & \text{for } 0< s< s_{0},\\ =0 & \text{for } s=s_{0},\\ < 0 & \text{for } s>s_{0}. \end{cases} $$
Consequently, we have (compare with (5))
$$ \int_{0}^{t} \biggl\vert 1-\frac{1}{\varGamma(q)}s^{q-1} \biggr\vert \,ds= \textstyle\begin{cases} t-\frac{t^{q}}{\varGamma(q+1)} & \text{for } 0< t< s_{0},\\ -\lambda(q) & \text{for } t=s_{0},\\ \frac{t^{q}-t\varGamma(q+1)-2\varGamma(q)^{\frac{q}{q-1}}}{\varGamma(q+1)}+2 \varGamma(q)^{\frac{1}{q-1}} & \text{for } t>s_{0}, \end{cases} $$
(17)
where \(\lambda(q)\) is given by (5). One can check numerically that \(-\lambda''(q)<0\) for \(q\in(1,2)\). So, \(-\lambda'(q)\) is decreasing from \(e^{-\gamma}\doteq0.561459\) to \(\frac{3}{4}-\frac {\gamma}{2}\doteq0.461392\), and \(-\lambda(q)\) is increasing from 0 to \(1/2\). Hence for \(q\in(1,2)\), we can estimate
$$ 0\leq-\lambda(q)\leq0.6(q-1). $$
(18)
Next, we need the following analog to Lemma 2.1.

Lemma 3.1

For all \(t\in \mathbb {R}_{+}\), \(q\in(1,4/3)\), and \(\kappa>0\), it holds
$$1\leq E_{q}\bigl(\kappa t^{q}\bigr)\leq\frac{e^{\kappa^{\frac{1}{q}}t}}{q} + \frac{4\sqrt{3}\sin\frac{\pi q}{2}}{9q}. $$

Proof

Using Dzherbashyan’s recursion formula [26],
$$E_{\alpha,\beta}(z)=\frac{1}{m}\sum_{h=0}^{m-1}E_{\frac{\alpha }{m},\beta} \bigl(z^{\frac{1}{m}}e^{\frac{2\pi\imath h}{m}}\bigr) $$
for \(\alpha,\beta>0\), \(z\in \mathbb {R}\), \(m\in \mathbb {N}\), where \(\imath=\sqrt {-1}\), we can write
$$ E_{q}(z)=\frac{E_{\frac{q}{2}}(\sqrt{z})+E_{\frac{q}{2}}(-\sqrt{z})}{2} $$
(19)
for any \(z>0\). Next, from [27, Theorem 2.1] we know
$$E_{\alpha}(z)=\frac{-z\sin\pi\alpha}{\pi\alpha} \int_{0}^{\infty}\frac{e^{-r^{\frac{1}{\alpha}}}\,dr}{r^{2}-2rz\cos\pi\alpha+z^{2}} $$
for any \(\alpha>0\), \(z<0\). So, using \(\cos\frac{\pi q}{2}\geq-1/2\) for \(q\in(1,4/3)\), we get
$$\begin{aligned} E_{\frac{q}{2}}(-\sqrt{z})&=\frac{2\sqrt{z}\sin\frac{\pi q}{2}}{\pi q} \int_{0}^{\infty}\frac{e^{-r^{\frac{2}{q}}}\,dr}{r^{2}+2r\sqrt{z}\cos\frac {\pi q}{2}+z} \\ &\leq\frac{2\sqrt{z}\sin\frac{\pi q}{2}}{\pi q} \int_{0}^{\infty}\frac{dr}{r^{2}-r\sqrt{z}+z} =\frac{2\sqrt{z}\sin\frac{\pi q}{2}}{\pi q} \frac{4\sqrt{3}\pi }{9\sqrt{z}} =\frac{8\sqrt{3}\sin\frac{\pi q}{2}}{9q}. \end{aligned}$$
Finally, applying this estimation and Lemma 2.1 to (19) results in
$$E_{q}\bigl(\kappa t^{q}\bigr)\leq\frac{1}{2} \biggl( \frac{2e^{\kappa^{\frac{1}{q}}t}}{q} +\frac{8\sqrt{3}\sin\frac{\pi q}{2}}{9q} \biggr) =\frac{e^{\kappa^{\frac{1}{q}}t}}{q}+ \frac{4\sqrt{3}\sin\frac{\pi q}{2}}{9q}. $$
 □
Since by (17),
$$\int_{0}^{t} \biggl\vert 1-\frac{1}{\varGamma(q)}s^{q-1} \biggr\vert \,ds\le \biggl\vert t-\frac{t^{q}}{\varGamma(q+1)} \biggr\vert -2 \lambda(q) $$
for all \(t\in \mathbb {R}_{+}\), Lemma 3.1 implies
$$ \begin{aligned}[b] \theta_{q}(t)&\leq \biggl( \frac{e^{L^{\frac{1}{q}}t}}{q} +\frac{4\sqrt{3}\sin\frac{\pi q}{2}}{9q} \biggr) \biggl( \biggl\vert t - \frac{t^{q}}{\varGamma(q+1)} \biggr\vert -2\lambda(q) \biggr) \\ &\leq \biggl(e^{t\overline{L}}+\frac{4\sqrt{3}}{9} \biggr) \biggl( \biggl\vert t- \frac{t^{q}}{\varGamma(q+1)} \biggr\vert -2\lambda(q) \biggr) \end{aligned} $$
(20)
for \(q\in(1,4/3)\), where \(\overline{L}=\max\{L,L^{3/4}\}\). So we obtain a result on the uniform convergence.

Theorem 3.2

Under assumption (H), the solution \(x(t)\) of (15) uniformly converges on any finite interval \([0,T]\), \(T>0\), of \(\mathbb {R}_{+}\) to the solution \(y(t)\) of (16) if \(q\to1_{+}\) and \(x_{0}=y_{0}\), \(x_{1}=y_{1}\).

Proof

The statement can be proved analogously to Theorem 2.2. □

Next, we consider equation (8) for an arbitrary \(\epsilon>0\) and \(q\in(1,4/3)\). Clearly, \(\theta_{q}(t)\) is increasing on \(\mathbb {R}_{+}\) from 0 to ∞, implying that (8) has the only solution \(\bar{t}(\epsilon,q)\in \mathbb {R}_{+}\) for which \(\lim_{\epsilon\to0_{+}}\bar{t}(\epsilon,q)=0\) and \(\lim_{q\to 1_{+}}\bar{t}(\epsilon,q)=\infty\) hold. Moreover, the function \(t\mapsto t-\frac{t^{q}}{\varGamma(q+1)}\) is nonnegative on \([0,r_{0}]\) and nonpositive on \([r_{0},\infty)\) for \(r_{0}\) given by (9). Note that \(r_{0}(q)\) is increasing on \((1,\infty )\) from \(\lim_{q\to1_{+}}r_{0}(q)=e^{1-\gamma}\doteq1.526205\) to ∞.

Next, we consider the function \(\phi_{t}(q):=\frac{t^{q}}{\varGamma(q+1)}\) on \((1,4/3)\) for \(t>0\). From (10), we obtain
$$\bigl\vert \phi_{t}'(q) \bigr\vert \leq- \frac{t\ln t}{\varGamma(q+1)}+\frac{\varGamma '(q+1)}{\varGamma^{2}(q+1)} \leq-t\ln t+\frac{\varGamma'(\frac{7}{3})}{\varGamma^{2}(\frac {7}{3})}\leq1.038041 $$
for \(t\in(0,1]\), and
$$\bigl\vert \phi'_{t}(q) \bigr\vert \leq T^{q}\ln T+0.51902T^{q} $$
for \(t\in(1,T]\), \(T>1\). As a consequence, we have
$$\bigl\vert \phi_{t}'(q) \bigr\vert \leq1.038041+T^{q}\ln T+0.51902T^{q} $$
for all \(t\in(0,T]\), \(T>1\), \(q\in(1,4/3)\). This implies
$$ \biggl\vert t-\frac{t^{q}}{\varGamma(q+1)} \biggr\vert = \bigl\vert \phi_{t}(1)-\phi_{t}(q) \bigr\vert \leq \bigl(1.038041+T^{q}\ln T+0.51902T^{q}\bigr) (q-1) $$
(21)
for \(t\in(0,T]\), \(T>1\), \(q\in(1,4/3)\). Using (18), (20), and (21), we arrive at
$$\theta_{q}(t)\leq \biggl(e^{T\overline{L}}+\frac{4\sqrt{3}}{9} \biggr) \bigl(3+T^{q}\ln T+T^{q}\bigr) (q-1) $$
for \(t\in(0,T]\), \(T>1\), \(q\in(1,4/3)\). Now, we consider the equation
$$ \mu_{L,q}(T):= \biggl(e^{T\overline{L}}+\frac{4\sqrt{3}}{9} \biggr) \bigl(3+T^{q}\ln T+T^{q}\bigr)=\frac{1}{\sqrt{q-1}}. $$
(22)
The function \(\mu_{L,q}(T)\) is increasing from \(4(e^{\overline {L}}+4\sqrt{3}/9)\) to ∞ on \([1,\infty)\). So, for any
$$ q< 1+\frac{1}{16 (e^{\overline{L}}+\frac{4\sqrt{3}}{9} )^{2}},\quad q\in(1,4/3), $$
(23)
(22) has a unique solution \(T_{L}(q)>1\). Note that \(\lim_{q\to 1_{+}}T_{L}(q)=\infty\). Summarizing, we have the following result.

Theorem 3.3

Under assumption (H) and for any q fulfilling (23), the solutions \(x(t)\) and \(y(t)\) of (15) and (16) with \(x_{0}=y_{0}\), \(x_{1}=y_{1}\), respectively, satisfy
$$ \bigl\Vert x(t)-y(t) \bigr\Vert \le M\sqrt{q-1} $$
(24)
for any \(t\in[0,T_{L}(q)]\), where \(T_{L}(q)>1\) is the unique solution of (22).

Next, we present a simple example illustrating the convergence results when the order q is close to 1.

Example 3.4

Let us consider the following initial-value problems:
$$\begin{aligned}& \begin{gathered} D_{0}^{q}x(t) = px(t),\quad t\in \mathbb {R}_{+}, \\ x(0) = x_{0}, \end{gathered} \end{aligned}$$
(25)
$$\begin{aligned}& \begin{gathered} y'(t) = py(t), \quad t\in \mathbb {R}_{+}, \\ y(0) = y_{0}, \end{gathered} \end{aligned}$$
(26)
$$\begin{aligned}& \begin{gathered} D_{0}^{q}u(t) = pu(t), \quad t\in \mathbb {R}_{+}, \\ u(0) = u_{0}, \\ u'(0) = u_{1}, \end{gathered} \end{aligned}$$
(27)
$$\begin{aligned}& \begin{gathered} v'(t) = pv(t)+v_{1}, \quad t\in \mathbb {R}_{+}, \\ v(0) = v_{0}, \end{gathered} \end{aligned}$$
(28)
where \(q\in(0,1)\) in (25) and \(q\in(1,2)\) in (27). The ODEs have the solutions \(y(t)=y_{0}e^{pt}\), \(v(t)=e^{pt}(v_{0}+v_{1}/p)-v_{1}/p\). From [2, Theorem 4.3], the other solutions are \(x(t)=x_{0}E_{q}(pt^{q})\) and \(u(t)=u_{0}E_{q}(pt^{q})+u_{1}tE_{q,2}(pt^{q})\). To see the convergence, we set all the initial conditions and the parameter equal to 1, i.e., \(x_{0}=y_{0}=u_{0}=v_{0}=u_{1}=v_{1}=p=1\). Figure 1 depicts the convergences \(x\to y\) and \(u\to v\) as \(q\to1_{-}\) and \(q\to1_{+}\), respectively.
Figure 1
Figure 1

Convergence of solutions of Caputo fractional DEs (25) (dashed blue), (27) (dot-dashed red) to solutions of ODEs (26), (28), respectively. The closer \(q\in\{ 0.6,0.7,0.8,0.9,1.1,1.2,1.3,1.4\}\) is to 1, the more saturated the colors are

The physical significance of Fig. 1 relies on demonstration of transition of q through 1. Since (25) is a one-dimensional system depending just on \(x_{0}\), its limit (26) is also one-dimensional. But passing to (27), we get a two-dimensional system depending on \(u_{0}\) and \(u_{1}\). Then its limit (28) as \(q\to1_{+}\) is also two-dimensional. This makes the difference. Note that (28) is equivalent to a second order ODE
$$\begin{gathered} v''(t) = pv'(t), \quad t\in \mathbb {R}_{+}, \\ v(0) = v_{0}, \\ v'(0) = \tilde{v}_{1}:=pv_{0}+v_{1}. \end{gathered} $$
The above arguments are more visible for \(p<0\). Then by [27, Formula (7)] we see that solutions of (25), (26), and (27) asymptotically tend to zero, while the one of (28) tends to \(-\frac{v_{1}}{p}\). So all these equations are dissipative. But the limit of (27) as \(q\to2_{-}\) is
$$ \begin{gathered} z''(t) = pz(t), \quad t\in \mathbb {R}_{+}, \\ z(0) = z_{0}, \\ z'(0) = z_{1}, \end{gathered} $$
(29)
which has all solutions oscillating for \(p<0\). Consequently, the dissipation of (25)–(28) is changing to oscillation on finite intervals as \(q\to2_{-}\). This is presented in Figs. 2 and 3.
Figure 2
Figure 2

The solutions of Caputo fractional DEs (25) for \(q\in\{0.2,0.4,0.6,0.8\}\) and (26) for \(p=-5\) and \(x_{0}=y_{0}=1\)

Figure 3
Figure 3

The solutions of Caputo fractional DEs (27) for \(q\in\{1.2,1.4,1.6,1.8\}\) and (29) for \(p=-5\) and \(u_{0}=z_{0}=u_{1}=z_{1}=1\)

These figures also support the fact that comparison estimates can be done in general only on finite intervals.

4 Conclusion

Solutions of q-order fractional differential equations of Caputo type for q near 1 are compared to solutions of the corresponding 1-order ordinary differential equations, by establishing the effective convergence results. As a result we get that the limit cases \(q\to1_{+}\) and \(q\to1_{-}\) are different. Theoretical results are demonstrated on a simple illustrative example. Our method can be directly extended to any order q near a natural number.

Declarations

Acknowledgements

The authors are grateful to the referees for their careful reading of the manuscript and their valuable comments.

Availability of data and materials

Not applicable.

Funding

This work is supported by the National Natural Science Foundation of China (11661016), Training Object of High Level and Innovative Talents of Guizhou Province ((2016)4006), Science and Technology Program of Guizhou Province ([2017]5788-10), Major Research Project of Innovative Group in Guizhou Education Department ([2018]012), the Slovak Research and Development Agency under the contract No. APVV-14-0378, and the Slovak Grant Agency VEGA No. 2/0153/16 and No. 1/0078/17.

Authors’ contributions

All authors contributed equally to this paper. All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Bratislava, Slovakia
(2)
Mathematical Institute, Slovak Academy of Sciences, Bratislava, Slovakia
(3)
Department of Mathematics, Guizhou University, Guiyang, China
(4)
School of Mathematical Sciences, Qufu Normal University, Qufu, China

References

  1. Fečkan, M., Wang, J.R., Pospíšil, M.: Fractional-Order Equations and Inclusions. de Gruyter, Berlin (2017) View ArticleGoogle Scholar
  2. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006) View ArticleGoogle Scholar
  3. Li, C., Zeng, F.: Numerical Methods for Fractional Calculus. CRC Press, Boca Raton (2015) View ArticleGoogle Scholar
  4. Zhou, Y.: Fractional Evolution Equations and Inclusions: Analysis and Control. Academic Press, San Diego (2016) MATHGoogle Scholar
  5. Wang, Y., Liu, L., Wu, Y.: Positive solutions for a nonlocal fractional differential equation. Nonlinear Anal. 74, 3599–3605 (2011) MathSciNetView ArticleGoogle Scholar
  6. Zhang, X., Liu, L., Wu, Y.: Existence results for multiple positive solutions of nonlinear higher order perturbed fractional differential equations with derivatives. Appl. Math. Comput. 219, 1420–1433 (2012) MathSciNetMATHGoogle Scholar
  7. Wang, Y., Liu, L., Zhang, X., Wu, Y.: Positive solutions of a fractional semipositone differential system arising from the study of HIV infection models. Appl. Math. Comput. 258, 312–324 (2015) MathSciNetMATHGoogle Scholar
  8. Li, M., Wang, J.: Exploring delayed Mittag-Leffler type matrix functions to study finite time stability of fractional delay differential equations. Appl. Math. Comput. 324, 254–265 (2018) MathSciNetGoogle Scholar
  9. Wang, J., Ibrahim, A.G., O’Regan, D.: Topological structure of the solution set for fractional non-instantaneous impulsive evolution inclusions. J. Fixed Point Theory Appl. 20(2), Article ID 59 (2018) MathSciNetView ArticleGoogle Scholar
  10. Wang, J., Ibrahim, A.G., O’Regan, D.: Hilfer type fractional differential switched inclusions with noninstantaneous impulsive and nonlocal conditions. Nonlinear Anal., Model. Control 23(6), 921–941 (2018) MathSciNetView ArticleGoogle Scholar
  11. Li, M., Debbouche, A., Wang, J.: Relative controllability in fractional differential equations with pure delay. Math. Methods Appl. Sci. 41, 8906–8914 (2018) MathSciNetView ArticleGoogle Scholar
  12. Zhang, X., Mao, C., Liu, L., Wu, Y.: Exact iterative solution for an abstract fractional dynamic system model for bioprocess. Qual. Theory Dyn. Syst. 16, 205–222 (2017) MathSciNetView ArticleGoogle Scholar
  13. Zhang, X., Liu, L., Wu, Y., Wiwatanapataphee, B.: Nontrivial solutions for a fractional advection dispersion equation in anomalous diffusion. Appl. Math. Lett. 66, 1–8 (2017) MathSciNetView ArticleGoogle Scholar
  14. Jiang, J., Liu, L., Wu, Y.: Multiple positive solutions of singular fractional differential system involving Stieltjes integral conditions. Electron. J. Qual. Theory Differ. Equ. 43, 1–18 (2012) MathSciNetMATHGoogle Scholar
  15. Klimek, M.: Sequential fractional differential equations with Hadamard derivative. Commun. Nonlinear Sci. Numer. Simul. 16, 4689–4697 (2011) MathSciNetView ArticleGoogle Scholar
  16. Ma, Q., Wang, R., Wang, J., Ma, Y.: Qualitative analysis for solutions of a certain more generalized two-dimensional fractional differential system with Hadamard derivative. Appl. Math. Comput. 257, 436–445 (2014) MathSciNetMATHGoogle Scholar
  17. Singh, J., Kumar, D., Baleanu, D.: New aspects of fractional Biswas–Milovic model with Mittag-Leffler law. Math. Model. Nat. Phenom. 14, 303 (2019) MathSciNetView ArticleGoogle Scholar
  18. Singh, J., Kumar, D., Baleanu, D., Rathored, S.: An efficient numerical algorithm for the fractional Drinfeld–Sokolov–Wilson equation. Appl. Math. Comput. 335, 12–24 (2018) MathSciNetGoogle Scholar
  19. Kumar, D., Singh, J., Baleanu, D., Rathore, S.: Analysis of a fractional model of Ambartsumian equation. Eur. Phys. J. Plus 133, 259 (2018) View ArticleGoogle Scholar
  20. Kumar, D., Singh, J., Baleanu, D., Sushila: Analysis of regularized long-wave equation associated with a new fractional operator with Mittag-Leffler type kernel. Physica A 492, 155–167 (2018) MathSciNetView ArticleGoogle Scholar
  21. Kumar, D., Tchier, F., Singh, J., Baleanu, D.: An efficient computational technique for fractal vehicular traffic flow. Entropy 20, 259 (2018) View ArticleGoogle Scholar
  22. Heymans, N., Podlubny, I.: Physical interpretation of initial conditions for fractional differential equations with Riemann–Liouville fractional derivatives. Rheol. Acta 45, 765–771 (2006) View ArticleGoogle Scholar
  23. Ye, H., Gao, J., Ding, Y.: A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328, 1075–1081 (2007) MathSciNetView ArticleGoogle Scholar
  24. Gorenflo, R., Kilbas, A.A., Mainardi, F., Rogosin, S.V.: Mittag-Leffler Functions, Related Topics and Applications. Springer, Berlin (2014) View ArticleGoogle Scholar
  25. Cong, N.D., Doan, T.S., Siegmund, S., Tuan, H.T.: On stable manifolds for planar fractional differential equations. Appl. Math. Comput. 226, 157–168 (2014) MathSciNetMATHGoogle Scholar
  26. Dzherbashyan, M.M.: Integral Transforms and Representations of Functions in the Complex Plane. Nauka, Moscow (1966) (in Russian) Google Scholar
  27. Gorenflo, R., Loutchko, J., Luchko, Y.: Computation of the Mittag-Leffler function \(E_{\alpha,\beta }(z)\) and its derivative. Fract. Calc. Appl. Anal. 5(4), 491–518 (2002) Correction: Fract. Calc. Appl. Anal. 6(1), 111–112 (2003) MathSciNetMATHGoogle Scholar

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