# On Some Iterative Numerical Methods for Mixed Volterra–Fredholm Integral Equations

## Abstract

**:**

## 1. Introduction

## 2. Solvability of the MVFIE in Equation (1)

**Definition**

**1.**

**$\mathit{q}\u2014$contraction**if there exists $0<q<1$ such that

**Theorem**

**1.**

- (a)
- equation $u=Fu$ has exactly one solution ${u}^{*}\in X$;
- (b)
- the sequence of successive approximations ${u}_{n+1}=F{u}_{n},n\in \mathbb{N},$ converges to the solution ${u}^{*}$, for any arbitrary choice of initial point ${u}_{0}\in X$; and
- (c)
- the error estimate$$\begin{array}{ccc}\hfill \left|\right|{u}_{n}-{u}^{*}\left|\right|& \le & {\displaystyle \frac{{q}^{n}}{1-q}}\phantom{\rule{4pt}{0ex}}\left|\right|{u}_{1}-{u}_{0}\left|\right|\hfill \end{array}$$

**Remark**

**1.**

**Theorem**

**2.**

- (i)
- there exists a constant $L>0$ such that$$\begin{array}{ccc}\hfill \left|K\right(x,t,s,u)-K(x,t,s,v\left)\right|& \le & L|u-v|,\hfill \end{array}$$
- (ii)
- $$\begin{array}{ccc}\hfill {M}_{K}{(b-a)}^{2}& \le & R,\hfill \end{array}$$
- (iii)
- $$q:=\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}L{(b-a)}^{2}<\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}1;$$

- (a)
- (b)
- the sequence of successive approximations$${u}_{n+1}=F{u}_{n},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}n=0,1,\dots $$
- (c)
- the error estimate$$\begin{array}{ccc}\hfill \left|\right|{u}_{n}-{u}^{*}\left|\right|& \le & {\displaystyle \frac{{q}^{n}}{1-q}}\phantom{\rule{4pt}{0ex}}\left|\right|{u}_{1}-{u}_{0}\left|\right|\hfill \end{array}$$

**Proof.**

**Remark**

**2.**

## 3. Numerical Approximation of the Solution

**Theorem**

**3.**

**Proof.**

#### Using the Trapezoidal Rule

**Theorem**

**4.**

## 4. Numerical Experiments

**Example**

**1.**

**Example**

**2.**

## 5. Conclusions

## Funding

## Conflicts of Interest

## References

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m | 12 | 24 | |
---|---|---|---|

n | |||

1 | $1.061540\times {10}^{-1}$ | $1.923056\times {10}^{-2}$ | |

5 | $9.798965\times {10}^{-4}$ | $2.386057\times {10}^{-4}$ | |

10 | $1.318075\times {10}^{-4}$ | $3.071171\times {10}^{-5}$ |

m | 12 | 24 | |
---|---|---|---|

n | |||

1 | $4.110239\times {10}^{-2}$ | $1.260717\times {10}^{-2}$ | |

5 | $2.253881\times {10}^{-4}$ | $5.973695\times {10}^{-5}$ | |

10 | $1.521347\times {10}^{-5}$ | $3.273994\times {10}^{-6}$ |

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**MDPI and ACS Style**

Micula, S.
On Some Iterative Numerical Methods for Mixed Volterra–Fredholm Integral Equations. *Symmetry* **2019**, *11*, 1200.
https://doi.org/10.3390/sym11101200

**AMA Style**

Micula S.
On Some Iterative Numerical Methods for Mixed Volterra–Fredholm Integral Equations. *Symmetry*. 2019; 11(10):1200.
https://doi.org/10.3390/sym11101200

**Chicago/Turabian Style**

Micula, Sanda.
2019. "On Some Iterative Numerical Methods for Mixed Volterra–Fredholm Integral Equations" *Symmetry* 11, no. 10: 1200.
https://doi.org/10.3390/sym11101200