# Picard’s Iterative Method for Caputo Fractional Differential Equations with Numerical Results

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminary Results

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Theorem**

**1.**

**Definition**

**4.**

**Definition**

**5.**

## 3. Main Result

**Theorem**

**2.**

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Proof**

**of Theorem 2.**

## 4. Numerical Results and Applications

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Graphs of ${}^{c}{D}^{q}u={u}^{2},u\left(0\right)=1$ from $t=0$ to $t=1$. (

**a**) Graph of ${}^{c}{D}^{.25}u={u}^{2},u\left(0\right)=1$. (

**b**) Graph of ${}^{c}{D}^{.5}u={u}^{2},u\left(0\right)=1$. (

**c**) Graph of ${}^{c}{D}^{.75}u={u}^{2},u\left(0\right)=1$.

**Figure 2.**Graphs of ${}^{c}{D}^{q}u={u}^{.5},u\left(0\right)=1$ from $t=0$ to $t=1$. (

**a**) Graph of ${}^{c}{D}^{.25}u={u}^{.5},u\left(0\right)=1$. (

**b**) Graph of ${}^{c}{D}^{.5}u={u}^{.5},u\left(0\right)=1$. (

**c**) Graph of ${}^{c}{D}^{.75}u={u}^{.5},u\left(0\right)=1$.

**Figure 3.**Graphs of ${}^{c}{D}^{q}u=-{u}^{2},u\left(0\right)=1$ from $t=0$ to $t=1$. (

**a**) Graph of ${}^{c}{D}^{.25}u=-{u}^{2},u\left(0\right)=1$. (

**b**) Graph of ${}^{c}{D}^{.5}u=-{u}^{2},u\left(0\right)=1$. (

**c**) Graph of ${}^{c}{D}^{.75}u=-{u}^{2},u\left(0\right)=1$.

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**MDPI and ACS Style**

Lyons, R.; Vatsala, A.S.; Chiquet, R.A.
Picard’s Iterative Method for Caputo Fractional Differential Equations with Numerical Results. *Mathematics* **2017**, *5*, 65.
https://doi.org/10.3390/math5040065

**AMA Style**

Lyons R, Vatsala AS, Chiquet RA.
Picard’s Iterative Method for Caputo Fractional Differential Equations with Numerical Results. *Mathematics*. 2017; 5(4):65.
https://doi.org/10.3390/math5040065

**Chicago/Turabian Style**

Lyons, Rainey, Aghalaya S. Vatsala, and Ross A. Chiquet.
2017. "Picard’s Iterative Method for Caputo Fractional Differential Equations with Numerical Results" *Mathematics* 5, no. 4: 65.
https://doi.org/10.3390/math5040065