# A Numerical Method for Weakly Singular Nonlinear Volterra Integral Equations of the Second Kind

## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

**contraction**if there exists a constant $0\le q<1$ such that

**Theorem**

**1.**

- (a)
- T has exactly one fixed point, which means equation $x=Tx$ has exactly one solution ${x}^{\ast}\in X$;
- (b)
- the sequence of successive approximations ${x}_{n+1}=T{x}_{n},n\in \mathbb{N},$ converges to the solution ${x}^{\ast}$, where ${x}_{0}$ can be any arbitrary point in X;
- (c)
- for every $n\in \mathbb{N}$, the following error estimate$$\begin{array}{ccc}\hfill \left|\right|{x}_{n}-{x}^{\ast}\left|\right|& \le & {\displaystyle \frac{{q}^{n}}{1-q}}\phantom{\rule{4pt}{0ex}}\left|\right|T{x}_{0}-{x}_{0}\left|\right|\hfill \end{array}$$

**Remark**

**1.**

## 2. Existence and Uniqueness of the Solution

**Remark**

**2.**

**Theorem**

**2.**

- (a)
- Equation (4) has a unique solution ${u}^{\ast}\in X$;
- (b)
- the sequence of successive approximations$${u}_{n+1}=F{u}_{n},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}n=0,1,\cdots $$converges to the solution ${u}^{\ast}$ for any ${u}_{0}\in X$;
- (c)
- for every $n\in \mathbb{N}$, the following error estimate$$\begin{array}{ccc}\hfill \left|\right|{u}_{n}-{u}^{\ast}{\left|\right|}_{\tau}& \le & {\displaystyle \frac{{q}^{n}}{1-q}}\phantom{\rule{4pt}{0ex}}\left|\right|F{u}_{0}-{u}_{0}{\left|\right|}_{\tau}\hfill \end{array}$$holds, where $q:={\displaystyle \frac{L\Gamma \left(\alpha \right)}{{\tau}^{\alpha}}}$ is the contraction constant.

**Proof**

**of**

**Theorem**

**2.**

**Theorem**

**3.**

**Theorem**

**4.**

**Proof**

**of**

**Theorem**

**4.**

## 3. Numerical Method

#### 3.1. Product Integration

**Remark**

**4.**

#### 3.2. Convergence and Error Analysis

**Theorem**

**5.**

**Proof**

**of**

**Theorem**

**5.**

## 4. Numerical Experiments

**Example**

**1.**

**Example**

**2.**

## 5. Conclusions

## Funding

## Conflicts of Interest

## References

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n | m | |
---|---|---|

12 | 24 | |

1 | 1.084348 × 10${}^{-1}$ | 1.882162 × 10${}^{-2}$ |

5 | 2.799553 × 10${}^{-4}$ | 5.567188 × 10${}^{-6}$ |

10 | 6.813960 × 10${}^{-7}$ | 4.690204 × 10${}^{-9}$ |

n | m | |
---|---|---|

12 | 24 | |

1 | 1.002977 × 10${}^{-1}$ | 3.014020 × 10${}^{-2}$ |

5 | 2.315358 × 10${}^{-4}$ | 4.412851 × 10${}^{-5}$ |

10 | 9.363611 × 10${}^{-7}$ | 5.525447 × 10${}^{-9}$ |

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Micula, S.
A Numerical Method for Weakly Singular Nonlinear Volterra Integral Equations of the Second Kind. *Symmetry* **2020**, *12*, 1862.
https://doi.org/10.3390/sym12111862

**AMA Style**

Micula S.
A Numerical Method for Weakly Singular Nonlinear Volterra Integral Equations of the Second Kind. *Symmetry*. 2020; 12(11):1862.
https://doi.org/10.3390/sym12111862

**Chicago/Turabian Style**

Micula, Sanda.
2020. "A Numerical Method for Weakly Singular Nonlinear Volterra Integral Equations of the Second Kind" *Symmetry* 12, no. 11: 1862.
https://doi.org/10.3390/sym12111862