# On the Analysis of Mixed-Index Time Fractional Differential Equation Systems

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Analytical Solutions

**Definition**

**1.**

**Remark**

**1.**

**Lemma**

**1.**

**Lemma**

**2.**

**Lemma**

**3.**

- (i)
- ${D}_{t}^{\alpha}{I}^{\alpha}y\left(t\right)=y\left(t\right)$
- (ii)
- ${I}^{\alpha}{D}_{t}^{\alpha}y\left(t\right)=y\left(t\right)-y\left(0\right)$
- (iii)
- ${D}_{t}^{\alpha}y\left(t\right)=\frac{1}{\Gamma (1-\alpha )}{\int}_{0}^{t}\frac{{y}^{\prime}\left(s\right)}{{(t-s)}^{\alpha}}ds={I}^{1-\alpha}{D}_{t}y\left(t\right).$

**Lemma**

**4.**

**Lemma**

**5.**

**Proof**

**Remark**

**2.**

**Theorem**

**1.**

#### 2.2. Asymptotic Stability of Multi-Index Systems

**Theorem**

**2.**

**Theorem**

**3.**

**Remark**

**3.**

**Remark**

**4.**

**Remark**

**5.**

**Theorem**

**4.**

## 3. Results

#### 3.1. The Solution of Mixed Index Linear Systems

**Theorem**

**5.**

**Remark**

**6.**

**Remark**

**7.**

**Remark**

**8.**

**Definition**

**2.**

**Theorem**

**6.**

**Proof**

#### 3.2. Study of Asymptotic Stability

**Theorem**

**7.**

**Theorem**

**8.**

**Remark**

**9.**

- (i)
- $\alpha =\beta :\phantom{\rule{1.em}{0ex}}\widehat{\theta}=\alpha \frac{\pi}{2}$, since in this case, ${\left(\frac{\theta}{d}\right)}^{2}={tan}^{2}\frac{\alpha \pi}{2}$.
- (ii)
- $\alpha +\beta =1:\phantom{\rule{1.em}{0ex}}\widehat{\theta}\in (\sqrt{sin\alpha \pi},\frac{\pi}{2}),\phantom{\rule{0.222222em}{0ex}}\alpha \in [\frac{1}{2},1]$. In the case $\alpha +\beta =1$, we see from (52) that:$${\left(\right)}^{\frac{\theta}{d}}2.$$Letting $\alpha =\frac{1}{2}+\u03f5$ with $\u03f5>0$ small, then $x={r}^{2\u03f5}$. This means that $\frac{{x}^{2}+1}{2x}$, as a function of r, is very shallow apart from when r is near the origin or very large. Hence, the asymptotic stability boundary will be almost constant over long periods of d when α and β are close together.
- (iii)
- $\alpha =2\beta :\phantom{\rule{1.em}{0ex}}\widehat{\theta}\in [\frac{sin\frac{\beta \pi}{2}\sqrt{2cos\frac{\beta \pi}{2}}}{cos\frac{3\beta \pi}{4}},\frac{\pi}{2}),\phantom{\rule{0.222222em}{0ex}}\beta \in (0,\frac{1}{2}].$

**Lemma**

**6.**

**Proof**

**Remark**

**10.**

- (i)
- $\beta =\frac{1}{2},\phantom{\rule{0.222222em}{0ex}}\alpha =1,\phantom{\rule{0.222222em}{0ex}}tan\tilde{\theta}=\sqrt{(1+d)(1+\sqrt{1+\frac{2}{d}})}$
- (ii)
- $\beta =\frac{1}{3},\phantom{\rule{0.222222em}{0ex}}\alpha =\frac{2}{3},\phantom{\rule{0.222222em}{0ex}}tan\tilde{\theta}=\sqrt{\frac{3}{8}}\sqrt{(1+\frac{d}{2})\sqrt{1+\frac{8}{3d}}+\frac{d}{2}-1}.$

**Theorem**

**9.**

**Remark**

**11.**

**Corollary**

**1.**

## 4. Simulations and Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Asymptotic stability region for single index scalar problem, for complex values of $\lambda $ (imaginary, vertical axis; real, horizontal axis).

**Figure 2.**Stability region, above the blue line, for choosing d and $\theta $, when the eigenvalues of A are $d\pm i\theta $, $\alpha =\frac{1}{2},\beta =1.$ The logarithmic scale is explored in the right-hand figure where the stability boundary dips below the angle $\frac{3\pi}{8}$.

**Figure 3.**Stability region, above the blue line, for choosing d and $\theta $, when the eigenvalues of A are $d\pm i\theta $, $\alpha =\frac{1}{3},\beta =\frac{2}{3}.$ The logarithmic scale is explored in the right-hand figure where the stability boundary dips below the angle $\frac{\pi}{4}$.

**Figure 4.**System dynamics with $(\alpha ,\beta )=(\frac{1}{2},1)$, top, and $(\alpha ,\beta )=(\frac{1}{3},\frac{2}{3})$, bottom. The left-hand column shows sustained dynamics with $d=1$ and $\theta $ chosen so that $(d,\theta )$ lies on the stability boundary. The right-hand column corresponds to the same d, but 0.3 has been added to the $\theta $ value.

**Figure 5.**Phase plots of ${y}_{1}$ versus ${y}_{2}$ for the decaying solutions in the right-hand column of Figure 4.

**Figure 6.**For A given by (70) with $d=-1,\phantom{\rule{0.166667em}{0ex}}\theta =\frac{1}{2}$ so that the eigenvalues are $-\frac{3}{2},\phantom{\rule{0.166667em}{0ex}}-\frac{1}{2}$, showing the effect of variation of $\alpha $ with fixed $\beta $ on the system dynamics.

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**MDPI and ACS Style**

Burrage, K.; Burrage, P.; Turner, I.; Zeng, F.
On the Analysis of Mixed-Index Time Fractional Differential Equation Systems. *Axioms* **2018**, *7*, 25.
https://doi.org/10.3390/axioms7020025

**AMA Style**

Burrage K, Burrage P, Turner I, Zeng F.
On the Analysis of Mixed-Index Time Fractional Differential Equation Systems. *Axioms*. 2018; 7(2):25.
https://doi.org/10.3390/axioms7020025

**Chicago/Turabian Style**

Burrage, Kevin, Pamela Burrage, Ian Turner, and Fanhai Zeng.
2018. "On the Analysis of Mixed-Index Time Fractional Differential Equation Systems" *Axioms* 7, no. 2: 25.
https://doi.org/10.3390/axioms7020025