On the Qualitative Analysis of Solutions of Two Fractional Order Fractional Differential Equations

Abstract

In applied sciences, besides the importance of obtaining the analytical solutions of differential equations with constant coefficients, the qualitative analysis of the solutions of such equations is also very important. Due to this importance, in this study, a qualitative analysis of the solutions of a delayed and constant coefficient fractal differential equation with more than one fractional derivative was performed. In the equation under consideration, the derivatives are the Riemann–Liouville fractional derivatives. In the proof of the obtained results, Laplace transform formulas of the Riemann–Liouville fractional derivative and some inequalities are used. We also provide some examples to check the accuracy of our results.

1. Introduction

Fractional derivation and integration have many applications in different disciplines of sciences; for example, phenomena in physics, engineering, economics and mechanics (see the monographs [1,2,3,4,5,6] and the references therein). Therefore, it is seen that most of the mathematical models encountered in the applied sciences (for example, viscoelasticity, chemistry, electromagnetics, control, tumor growth models, coronas, pandemic diseases such as infections, etc.) emerge as fractional order differential equations or fractional order difference equations; see [7,8,9,10,11]. These studies are mainly based on the qualitative analysis of the solutions of fractional order differential equations with constant coefficients; see [12,13,14,15,16,17,18]. It will be seen from the studies found in these references that the equations considered are mathematical equations that include only one fractional derivative and model less complicated events. This area is still very open to new studies. Therefore, we tried to discuss an equation that has more general and more complex application areas than the equations discussed so far, the examination of which could not been seen in the literature until now, and to give some necessary and sufficient results on the oscillatory of delayed fractional differential equations in this study. Thus, we have also expanded the results obtained so far. For example, in [15], Meng et al. studied the linear fractional-order delay differential equation,

$${}^{C}D_{-}^{\alpha }x\left(t\right)-px(t-\tau )=0$$

(1)

and obtained the conclusion that

$$p^{{\displaystyle \frac{1}{\alpha }}}\tau >{\displaystyle \frac{\alpha }{e}}$$

is a sufficient and necessary condition of the oscillations for all solutions of Equation (1). In [16], Panigrahi and Chand obtained that

$${\left(\prod _{i=1}^n{a}_i\right)}^{{\displaystyle \frac{1}{n}}}\left(\sum _{i=1}^n{\tau }_i\right)>{\displaystyle \frac{1}{e}}\left({\lambda }^{\delta -1}\right)$$

is a sufficient condition for the oscillation of all solutions of the fractional delay differential equation (FDDE),

$$y^{\delta }\left(t\right)+\sum _{i=1}^n{a}_{i}y(t-{\tau }_i)=0.$$

In [18] Zhu and Xiang obtained that if

$$\tau p^{{\displaystyle \frac{1}{\alpha }}}>{\displaystyle \frac{1}{e}}$$

then every solution of

$${}^{C}D_t^{\alpha }x\left(t\right)+px(t-\tau )=0,t>0$$

oscillates.

In this manuscript, by Theorem 2, we extend the conditions mentioned above.

In this manuscript, we consider two fractional order delay differential equations with constant coefficients of the form

$$D_{+}^{\alpha }\omega \left(\iota \right)+D_{+}^{\beta }\omega \left(\iota \right)+p\omega (\iota -\tau )=0,$$

(2)

$$\omega \left(\iota \right)=\phi \left(\iota \right),\iota \in \left[-\tau ,0\right],$$

(3)

where $0<\beta <\alpha <1$, $\beta ={\displaystyle \frac{eveninteger}{oddinteger}}$, $\alpha =$${\displaystyle \frac{oddinteger}{oddinteger}}$, $p,\tau \in {\mathbb{R}}^{+}$ and $D_{+}^{\alpha }$ denotes Liouville fractional derivative of order $\alpha$. We use the Laplace transform formula of the Riemann–Liouville fractional derivative at the proof of the results.

2. Preliminaries

In this section, we give the following definitions of Liouville and Riemann–Liouville fractional integrals and fractional derivatives on the whole real axis $\mathbb{R}=\left(-\infty ,\infty \right)$ ([2] p. 87).

Definition 1. 

The Liouville fractional integral operators on $\mathbb{R}$ of order α of a function y is defined as

$$\begin{array}{ccc}\hfill \left(I_{+}^{\alpha }\omega \right)\left(\iota \right)& =& {\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{-\infty }^{\iota }{\left(\iota -s\right)}^{\alpha -1}\omega \left(s\right)ds,\hfill \\ \hfill \left(I_{-}^{\alpha }\omega \right)\left(\iota \right)& =& {\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{\iota }^{\infty }{\left(\iota -s\right)}^{\alpha -1}\omega \left(s\right)ds\hfill \end{array}$$

where $\Re \left(\alpha \right)>0$ and $\iota \in \mathbb{R}$.

Definition 2. 

The Liouville fractional derivative operators on $\mathbb{R}$ of order α of a function y is defined as

$$\begin{array}{ccc}\hfill \left(D_{+}^{\alpha }y\right)\left(\iota \right)& :& ={\left({\displaystyle \frac{d}{d\iota }}\right)}^n\left(I_{+}^{n-\alpha }y\right)\left(\iota \right)={\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\left({\displaystyle \frac{d}{d\iota }}\right)}^n{\int }_{-\infty }^{\iota }{\left(\iota -s\right)}^{n-\alpha -1}y\left(s\right)ds,\hfill \\ \hfill \left(D_{-}^{\alpha }y\right)\left(\iota \right)& :& ={\left(-{\displaystyle \frac{d}{d\iota }}\right)}^n\left(I_{-}^{n-\alpha }y\right)\left(\iota \right)={\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\left(-{\displaystyle \frac{d}{d\iota }}\right)}^n{\int }_{\iota }^{\infty }{\left(\iota -s\right)}^{n-\alpha -1}y\left(s\right)ds,\hfill \end{array}$$

where $n=\left[\Re \left(\alpha \right)\right]+1$, $\Re \left(\alpha \right)\geqq 0$ and $\iota \in \mathbb{R}$, respectively.

Let $\Re \left(\lambda \right)>0$; if $\Re \left(\alpha \right)\geqq 0$, then

$$D_{+}^{\alpha }{e}^{\lambda \iota }={\lambda }^{\alpha }{e}^{\lambda \iota }$$

see [8] (p. 88)

Let $\lambda \in \mathbb{R}$; if $\Re \left(\alpha \right)\geqq 0$, considering ${(\pm i)}^{\alpha }=e^{\pm i\alpha {\displaystyle \frac{\pi }{2}}}$, and by Definition 2, we have

$$\begin{array}{ccc}\hfill D_{+}^{\alpha }\left[sin\left(\lambda \iota \right)\right]& =& {\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\displaystyle \frac{d}{d\iota }}{\int }_{-\infty }^{\iota }{\left(\iota -s\right)}^{-\alpha }sin\left(\lambda s\right)ds\hfill \\ & =& {\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\displaystyle \frac{d}{d\iota }}{\int }_{-\infty }^{\iota }{\displaystyle \frac{e^{i\lambda s}-e^{-i\lambda s}}{2i{\left(\iota -s\right)}^{\alpha }}}ds={\lambda }^{\alpha }sin\left(\lambda \iota +{\displaystyle \frac{\pi \alpha }{2}}\right).\hfill \end{array}$$

Similarly, we have the following:

$$D_{+}^{\alpha }\left[cos\left(\lambda \iota \right)\right]={\lambda }^{\alpha }cos\left(\lambda \iota +{\displaystyle \frac{\pi \alpha }{2}}\right).$$

3. Main Results

In this section, we give some oscillation results for the fractional-order delay differential equation with constant coefficients of the form (2) and (3). Throughout this paper, we will assume that the following condition (4) has been satisfied. The solutions of Equations (2) and (3) are exponentially bounded of the form

$$\left|\omega \left(\iota \right)\right|\le A{e}^{a\iota },\mathrm{for}a\in {\mathbb{R}}^{-}\mathrm{and}\iota \ge 0.$$

(4)

Since $\omega \left(\iota \right)=e^{\lambda \iota }$ is such a function, by substituting this function into Equation (2) and using the Liouville fractional derivative $D_{+}^{\alpha }\omega \left(\iota \right)=e^{\lambda \iota }{\lambda }^{\alpha }$ for $\Re \left(\lambda \right)>0$, we obtain the following characteristic equation of the corresponding Equation (2) as

$${\lambda }^{\alpha }+{\lambda }^{\beta }+p{e}^{-\lambda \tau }=0.$$

(5)

Theorem 1. 

Suppose that condition (4) holds. Then, the following statements are equivalent:

(a) 

Every solution of Equation (2) oscillates.

(b) 

Equation (5) has no real roots.

Proof. 

$\left(a\right)\Rightarrow \left(b\right)$. If the characteristic Equation (5) has a real root ${\lambda }_0$, then obviously $\omega \left(\iota \right)=e^{{\lambda }_0\iota }$ is a nonoscillatory solution of Equation (2), which contradicts the statement $\left(a\right)$.

$\left(b\right)\Rightarrow \left(a\right)$. On the contrary, suppose that Equation (2) has an eventually positive solution $\omega \left(\iota \right)$ (when Equation (2) has eventually a negative solution $\omega \left(\iota \right)$, the proof will be similar). Then, $\omega \left(\iota \right)$$>0$ for $\iota >-\tau$. If we consider condition (4), the Laplace transform $\omega \left(s\right)$ of $\omega \left(\iota \right)$ exists, and ${\sigma }_0=inf\left\{s\in \mathbb{R}:\omega \left(s\right)\mathrm{exists}\right\}$ is the abscissa of convergence of $\omega \left(s\right)$.

Hence, Laplace transform $\omega \left(s\right)$ of $\omega \left(\iota \right)$ exists, and is given by

$$\omega \left(s\right)=\mathcal{L}\left(\omega \right)\left(s\right)={\int }_0^{\infty }{e}^{-s\iota }\omega \left(\iota \right)d\iota ,\Re \left(s\right)>a.$$

(6)

Let ${\sigma }_0$ be the abscissa of convergence of $\omega \left(s\right)$, i.e., ${\sigma }_0=inf\left\{s\in \mathbb{R}:\omega \left(s\right)\mathrm{exists}\right\}$. Then, by taking the Laplace transforms of both sides of Equation (2), we obtain

$$\mathcal{L}\left(D_{+}^{\alpha }\omega \right)\left(\iota \right)+\mathcal{L}\left(D_{+}^{\beta }\omega \right)\left(\iota \right)+p{e}^{-s\tau }\omega \left(s\right)+p{e}^{-s\tau }{\int }_{-\tau }^0{e}^{-s\iota }\omega \left(\iota \right)d\iota =0,\Re \left(s\right)>{\sigma }_0.$$

(7)

In [13], since the Laplace transform of the Riemann–Liouville fractional derivative is given by

$$\mathcal{L}\left\{D_{+}^{\alpha }\omega \left(\iota \right)\right\}\left(s\right)=s^{\alpha }\omega \left(s\right)-\left(D_{+}^{\alpha -1}\omega \right)\left(0\right)+{\displaystyle \frac{s}{\Gamma \left(1-\alpha \right)}}{\int }_0^{\infty }{\int }_{-\infty }^0{e}^{-s\iota }{\left(\iota -u\right)}^{-\alpha }y\left(u\right)dud\iota ,$$

from (7), we have

$$s^{\alpha }\omega \left(s\right)+s^{\beta }\omega \left(s\right)-\left(D_{+}^{\alpha -1}\omega \right)\left(0\right)-\left(D_{+}^{\beta -1}\omega \right)\left(0\right)+p{e}^{-s\tau }\omega \left(s\right)$$

$$+p{e}^{-s\tau }{\int }_{-\tau }^0{e}^{-s\iota }\omega \left(\iota \right)d\iota \le 0,\Re \left(s\right)>{\sigma }_0.$$

(8)

If we obtain

$$F\left(s\right)=s^{\alpha }+s^{\beta }+p{e}^{-s\tau }$$

and

$$Q\left(s\right)=\left(D_{+}^{\alpha -1}\omega \right)\left(0\right)+\left(D_{+}^{\beta -1}\omega \right)\left(0\right)-p{e}^{-s\tau }{\int }_{-\tau }^0{e}^{-s\iota }\omega \left(\iota \right)d\iota ,$$

from (8), we can write

$$F\left(s\right)\omega \left(s\right)\le Q\left(s\right),\Re \left(s\right)>{\sigma }_0,$$

(9)

where $F\left(s\right)$ and $Q\left(s\right)$ are entire functions, since $F\left(0\right)>0$ and $F\left(s\right)>0$ for all real $s>0$ from condition $\left(b\right)$. Hence, from (9), we can write

$$\omega \left(s\right)\le {\displaystyle \frac{Q\left(s\right)}{F\left(s\right)}},\Re \left(s\right)>{\sigma }_0.$$

(10)

Now, we claim that ${\sigma }_0=-\infty$. Assume the contrary, i.e., ${\sigma }_0>-\infty$. Then, ${\sigma }_0$ must be a root of $F\left(s\right)$, but this contradicts $F\left(s\right)>0$ for all real s. So, ${\sigma }_0=-\infty$, and from (10), we have

$$\omega \left(s\right)\le {\displaystyle \frac{Q\left(s\right)}{F\left(s\right)}},s\in \mathbb{R}.$$

(11)

As $s\to -\infty$ (11) leads to contradiction, because $\omega \left(s\right)$ and $F\left(s\right)$ are positive but $Q\left(s\right)$ is eventually negative, the proof of the theorem is complete. □

Theorem 2. 

Suppose that condition (4) holds. Then, the following statements are equivalent:

(a) 

Every solution of Equation (2) oscillates.

(b) 

$\tau p>{\left[{\displaystyle \frac{{\left(1-\alpha \right)}^{1-\alpha }}{{\left(1-\beta \right)}^{1-\beta }}}\right]}^{{\displaystyle \frac{1}{\alpha -\beta }}}{\displaystyle \frac{\alpha -\beta }{e}}$.

Proof. 

We showed that the characteristic equation

$$f\left(\lambda \right):={\lambda }^{\alpha }+{\lambda }^{\beta }+p{e}^{-\lambda \tau }$$

(12)

of Equation (2) has no real root by Theorem 1. We only need to prove that

$$\tau p>{\left[{\displaystyle \frac{{\left(1-\alpha \right)}^{1-\alpha }}{{\left(1-\beta \right)}^{1-\beta }}}\right]}^{{\displaystyle \frac{1}{\alpha -\beta }}}{\displaystyle \frac{\alpha -\beta }{e}}\iff f\left(\lambda \right)>0.$$

$\left(a\right)\Rightarrow \left(b\right)$. That is, $f\left(\lambda \right)>0\Rightarrow \tau p>{\left[{\displaystyle \frac{{\left(1-\alpha \right)}^{1-\alpha }}{{\left(1-\beta \right)}^{1-\beta }}}\right]}^{{\displaystyle \frac{1}{\alpha -\beta }}}{\displaystyle \frac{\alpha -\beta }{e}}$. Let us assume the contrary, that $\tau p\le {\left[{\displaystyle \frac{{\left(1-\alpha \right)}^{1-\alpha }}{{\left(1-\beta \right)}^{1-\beta }}}\right]}^{{\displaystyle \frac{1}{\alpha -\beta }}}{\displaystyle \frac{\alpha -\beta }{e}}$. Then, from (12), we have

$$0={\lambda }^{\alpha }+{\lambda }^{\beta }+p{e}^{-\lambda \tau }{\displaystyle \frac{\le }{\ge }}{\lambda }^{\alpha }+{\lambda }^{\beta }+{\left[{\displaystyle \frac{{\left(1-\alpha \right)}^{1-\alpha }}{{\left(1-\beta \right)}^{1-\beta }}}\right]}^{{\displaystyle \frac{1}{\alpha -\beta }}}{\displaystyle \frac{\alpha -\beta }{\tau }}{e}^{-\lambda \tau -1}$$

which contradicts $\lambda \in \mathbb{C}$.

$\left(b\right)\Rightarrow \left(a\right)$. That is, $\tau p>{\left[{\displaystyle \frac{{\left(1-\alpha \right)}^{1-\alpha }}{{\left(1-\beta \right)}^{1-\beta }}}\right]}^{{\displaystyle \frac{1}{\alpha -\beta }}}{\displaystyle \frac{\alpha -\beta }{e}}$$\Rightarrow f\left(\lambda \right)>0$. Let us assume the contrary, so that $f\left(\lambda \right)<0$. Since $p>0$, it is clear that $f\left(\lambda \right)<0$ is not satisfied for $\lambda \ge 0$. In that case, there exists a $\lambda ={\lambda }_0<0$ such that $f\left({\lambda }_0\right):={\lambda }_0^{\alpha }+{\lambda }_0^{\beta }+p{e}^{-{\lambda }_0\tau }<0$. Since $e^{-{\lambda }_0\tau }\ge \left(-{\lambda }_0\right)\tau e$, from (12), we have

$${\lambda }_0^{\alpha }+{\lambda }_0^{\beta }+p{e}^{-{\lambda }_0\tau }\le {\lambda }_0^{\alpha }+{\lambda }_0^{\beta }-{\lambda }_0\tau e\le 0.$$

From here, we obtain

$${\lambda }_0^{\alpha -1}+{\lambda }_0^{\beta -1}-\tau e\ge 0.$$

(13)

If we define in (13) that $g\left({\lambda }_0\right)={\lambda }_0^{\alpha -1}+{\lambda }_0^{\beta -1}$, $g\left({\lambda }_0\right)$ has a maximum value at ${\lambda }_0={\left(-{\displaystyle \frac{\beta -1}{\alpha -1}}\right)}^{{\displaystyle \frac{1}{\alpha -\beta }}}$, we have

$${\left[{\displaystyle \frac{{\left(\beta -1\right)}^{\beta -1}}{{\left(\alpha -1\right)}^{\alpha -1}}}\right]}^{{\displaystyle \frac{1}{\alpha -\beta }}}(\alpha -\beta )-p\tau e\ge 0$$

or

$$\tau p\le {\left[{\displaystyle \frac{{\left(1-\alpha \right)}^{1-\alpha }}{{\left(1-\beta \right)}^{1-\beta }}}\right]}^{{\displaystyle \frac{1}{\alpha -\beta }}}{\displaystyle \frac{\alpha -\beta }{e}}.$$

This contradicts condition $\tau p>{\left[{\displaystyle \frac{{\left(1-\alpha \right)}^{1-\alpha }}{{\left(1-\beta \right)}^{1-\beta }}}\right]}^{{\displaystyle \frac{1}{\alpha -\beta }}}{\displaystyle \frac{\alpha -\beta }{e}}$. The proof of the theorem is completed. □

Example 1. 

Consider the fractional-order delayed differential equation of the form

$$D_{+}^{{\displaystyle \frac{5}{7}}}\omega \left(\iota \right)+D_{+}^{{\displaystyle \frac{2}{3}}}\omega \left(\iota \right)+\left(2cos{\displaystyle \frac{\pi }{84}}\right)\omega (\iota -{\displaystyle \frac{13}{84}}\pi )=0,$$

(14)

where $\beta ={\displaystyle \frac{3}{2}}$, $\alpha ={\displaystyle \frac{5}{7}}$, $p=$$2cos{\displaystyle \frac{\pi }{84}}$, $\tau ={\displaystyle \frac{13}{84}}\pi$. Since $p\tau =\left(2cos{\displaystyle \frac{\pi }{84}}\right)\left({\displaystyle \frac{13}{84}}\pi \right)=0.97172>{\left[{\displaystyle \frac{{\left(\frac{2}{7}\right)}^{\frac{2}{7}}}{{\left(\frac{1}{3}\right)}^{\frac{1}{3}}}}\right]}^{21}{\displaystyle \frac{1}{21e}}=0.020841$, by Theorem 2, Equation (14) has oscillatory solutions. Truly, $\omega \left(\iota \right)=sin\iota$ is one of the oscillatory solutions of Equation (14) (Figure 1).

Example 2. 

Consider Equation (14). The characteristic equation of Equation (14) is ${\lambda }^{({\displaystyle \frac{5}{7}}+{\displaystyle \frac{13}{84}}\pi )}+{\lambda }^{({\displaystyle \frac{2}{3}}+{\displaystyle \frac{13}{84}}\pi )}+2cos\left({\displaystyle \frac{\pi }{84}}\right)=0$. One of the complex roots of this equation is $\lambda =-0.89180-0.45492i$. Therefore, all of the solutions of Equation (14) are oscillatory. So, Theorem 1 is verified (Figure 2).

4. Conclusions

Just as new and interesting results have been obtained so far on the behavior of solutions by adding a new term containing a higher derivative to a fixed or variable coefficient equation containing a first-order derivative, we have obtained results by adding a new term having a higher derivative to a fixed coefficient equation containing a single derivative. Similarly, researchers can also obtain very new and interesting results by taking at least one term of the equation we have discussed in this article as a variable coefficient. For example, one of the coefficients can be a positive-valued function, while it can be an oscillating function. For the qualitative analysis of solutions of two fractional order fractional differential equations, see [19,20].