# Analysis of Sequential Caputo Fractional Differential Equations versus Non-Sequential Caputo Fractional Differential Equations with Applications

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## Abstract

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## 1. Introduction

## 2. Preliminaries

**Definition 1.**

**Definition 2.**

**Definition 3.**

**Definition 4.**

**Definition 5.**

**Definition 6.**

**Definition 7.**

**Remark 1.**

**Definition 8.**

**Theorem 1.**

**Definition 9.**

**Definition 10.**

Laplace transform Table | |||

S.N | $f\left(t\right)={\mathcal{L}}^{-1}\left[F\left(s\right)\right]$ | $F\left(s\right)=\mathcal{L}\left(f\right(t\left)\right)$ | |

1. | ${E}_{q,1}\left(\pm \lambda {t}^{q}\right)$ | $\frac{{s}^{q-1}}{{s}^{q}\mp \lambda}$ | ${s}^{q}>\lambda ,q>-1$ |

2. | ${t}^{q-1}{E}_{q,q}\left(\pm \lambda {t}^{q}\right)$ | $\frac{1}{{s}^{q}\mp \lambda}$ | ${s}^{q}>\lambda ,q>-1$ |

3. | $\frac{{t}^{q}}{q}{E}_{q,q}\left(\pm \lambda {t}^{q}\right)$ | $\frac{{s}^{q-1}}{{({s}^{q}\mp \lambda )}^{2}}$ | ${s}^{q}>\lambda ,q>-1$ |

4. | $si{n}_{q,1}\left(\lambda {t}^{q}\right)$ | $\frac{\lambda {s}^{q-1}}{{s}^{2q}+{\lambda}^{2}}$ | $s>0$ |

5. | $co{s}_{q,1}\left(\lambda {t}^{q}\right)$ | $\frac{{s}^{2q-1}}{{s}^{2q}+{\lambda}^{2}}$ | $s>0$ |

6. | ${t}^{q-1}si{n}_{q,q}\left(\lambda {t}^{q}\right)$ | $\frac{\lambda}{{s}^{2q}+{\lambda}^{2}}$ | $s>0$ |

7. | ${t}^{q-1}co{s}_{q,q}\left(\lambda {t}^{q}\right)$ | $\frac{{s}^{q}}{{s}^{2q}+{\lambda}^{2}}$ | $s>0$ |

8. | ${E}_{q,1}\left(\lambda {t}^{q}\right)+\frac{\lambda {t}^{q}}{q}{E}_{q,q}\left(\lambda {t}^{q}\right)$ | $\frac{{s}^{2q-1}}{{({s}^{q}-\lambda )}^{2}}$ | |

9. | ${t}^{q-1}{\sum}_{k=0}^{\infty}\frac{(k+1){\lambda}^{k}{t}^{qk}}{\mathsf{\Gamma}(qk+q)}$ | $\frac{{s}^{q}}{{({s}^{q}-\lambda )}^{2}}$ | |

10. | ${t}^{2q-1}{\sum}_{k=0}^{\infty}\frac{(k+1){\lambda}^{k}{t}^{qk}}{\mathsf{\Gamma}(qk+2q)}$ | $\frac{1}{{({s}^{q}-\lambda )}^{2}}$ | |

11. | ${\sum}_{k=0}^{\infty}\frac{k(k+1)}{2}\frac{{\left(\lambda {t}^{q}\right)}^{k-1}}{\mathsf{\Gamma}\left(q\right(k-1)+1)}$ | $\frac{{s}^{3q-1}}{{({s}^{q}-\lambda )}^{3}}$ | |

12. | $Gco{s}_{q,1}\left\{(\lambda +i\mu ){t}^{q}\right\}$ | $\frac{{s}^{q-1}({s}^{q}-\lambda )}{{({s}^{q}-\lambda )}^{2}+{\mu}^{2}}$ | |

13. | $Gsi{n}_{q,1}\left\{(\lambda +i\mu ){t}^{q}\right\}$ | $\frac{\mu {s}^{q-1}}{{({s}^{q}-\lambda )}^{2}+{\mu}^{2}}$ | |

15. | ${t}^{q-1}Gco{s}_{q,q}\left\{(\lambda +i\mu ){t}^{q}\right\}$ | $\frac{{s}^{q}-\lambda}{{({s}^{q}-\lambda )}^{2}+{\mu}^{2}}$ | |

16. | ${t}^{q-1}Gsi{n}_{q,q}\left\{(\lambda +i\mu ){t}^{q}\right\}$ | $\frac{\mu}{{({s}^{q}-\lambda )}^{2}+{\mu}^{2}}$ |

## 3. Main Results

#### 3.1. Solution of Linear Sequential Caputo Fractional Differential Equations with Fractional Initial Conditions

- 1.
- Let $b\ne 0,\phantom{\rule{4pt}{0ex}}{b}^{2}-4c>0.$ In this case, the quadratic equation ${s}^{2q}+b{s}^{q}+c=0$ will have real and distinct roots, say ${\lambda}_{1}$ and ${\lambda}_{2}$. That is, we can factor ${s}^{2q}+b{s}^{q}+c=({s}^{q}-{\lambda}_{1})({s}^{q}-{\lambda}_{2}).$ In this case, using the partial fraction method, we can write (17) as$$\frac{{s}^{q-1}G\left(s\right)}{{s}^{2q}+b{s}^{q}+c}=\frac{{c}_{1}{s}^{q-1}}{({s}^{q}-{\lambda}_{1})}+\frac{{c}_{2}{s}^{q-1}}{({s}^{q}-{\lambda}_{2})}.$$Here, the constants ${c}_{i}$ for $i=1,2$ depend on $A,B,b,c,{\lambda}_{1}$, and ${\lambda}_{2}.$ Using the above relation in Equation (17) and taking the inverse Laplace transform, we can write the solutions of (14) and (15) as$$u\left(t\right)={c}_{1}{E}_{q,1}\left({\lambda}_{1}{t}^{q}\right)+{c}_{2}{E}_{q,1}\left({\lambda}_{2}{t}^{q}\right).$$Note that in the above case, if $b=0$ and $c<0,$ we will have two real and distinct roots. The solution can be obtained on the same lines as above.
- 2.
- $b\ne 0,\phantom{\rule{4pt}{0ex}}{b}^{2}-4c=0.$ In this case, the quadratic equation ${s}^{2q}+b{s}^{q}+c=0$ will have real and coincident roots, say $\lambda $. Then, we can factor as ${s}^{2q}+b{s}^{q}+c={({s}^{q}-\lambda )}^{2}.$ In this case, by algebraic manipulation, we can write (17) as$$\frac{{s}^{q-1}G\left(s\right)}{{s}^{2q}+b{s}^{q}+c}=\frac{A{s}^{q-1}}{({s}^{q}-\lambda )}+\frac{(\lambda A+B+bA){s}^{q-1}}{{({s}^{q}-\lambda )}^{2}}.$$
- 3.
- If $b\ne 0,then{b}^{2}-4c<0.$ In this case, the quadratic equation ${s}^{2q}+b{s}^{q}+c=0$ will have complex roots of the form $\lambda \pm i\mu .$ We can write ${s}^{2q}+b{s}^{q}+c={({s}^{q}-\lambda )}^{2}+{\mu}^{2}.$ Then by algebraic manipulation, we can write (17) as$$\frac{{s}^{q-1}G\left(s\right)}{{s}^{2q}+b{s}^{q}+c}=\frac{{g}_{1}{s}^{q-1}({s}^{q}-\lambda )}{{({s}^{q}-\lambda )}^{2}+{\mu}^{2}}+\frac{\mu {g}_{2}}{{({s}^{q}-\lambda )}^{2}+{\mu}^{2}}$$$$\frac{1}{{s}^{2q}+b{s}^{q}+c}=\frac{1}{\mu}\frac{\mu}{{({s}^{q}-\lambda )}^{2}+{\mu}^{2}},$$Now, using the table and the above relation, we can take the inverse Laplace transform of Relation (17) to obtain the solutions of (14) and (15) as$$u\left(t\right)={g}_{1}Gco{s}_{q,1}\left\{(\lambda +i\mu ){t}^{q}\right\}+{g}_{2}Gsi{n}_{q,1}\left\{(\lambda +i\mu ){t}^{q}\right\}.$$Further, if $\lambda =0$, then the solution of (14) and (15) will be$$u\left(t\right)={g}_{1}co{s}_{q,1}\left(\mu {t}^{q}\right)+{g}_{2}si{n}_{q,1}\left(\mu {t}^{q}\right).$$Furthermore, note that if $q=1,$ one can easily observe that the integer results can be obtained as a special case, except when the factors are complex numbers with the real part not equal to zero. Even in that situation, the results of the integer case can be obtained using the exponential rules.Note that we can also use the above technique to solve non-homogeneous sequential Caputo fractional differential equations when $f\left(t\right)\ne 0$. In this case, Mittag–Leffler functions ${E}_{q,q}\left(\lambda {t}^{q}\right)$ will be needed in the convolution integral.

**Remark 2.**

#### 3.2. Linear Sequential Caputo Fractional Boundary Value Problems with Fractional Boundary Conditions

**Remark 3.**

**Remark 4.**

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Vatsala, A.S.; Pageni, G.; Vijesh, V.A.
Analysis of Sequential Caputo Fractional Differential Equations versus Non-Sequential Caputo Fractional Differential Equations with Applications. *Foundations* **2022**, *2*, 1129-1142.
https://doi.org/10.3390/foundations2040074

**AMA Style**

Vatsala AS, Pageni G, Vijesh VA.
Analysis of Sequential Caputo Fractional Differential Equations versus Non-Sequential Caputo Fractional Differential Equations with Applications. *Foundations*. 2022; 2(4):1129-1142.
https://doi.org/10.3390/foundations2040074

**Chicago/Turabian Style**

Vatsala, Aghalaya S., Govinda Pageni, and V. Anthony Vijesh.
2022. "Analysis of Sequential Caputo Fractional Differential Equations versus Non-Sequential Caputo Fractional Differential Equations with Applications" *Foundations* 2, no. 4: 1129-1142.
https://doi.org/10.3390/foundations2040074