# Comparison of Selected Numerical Methods for Solving Integro-Differential Equations with the Cauchy Kernel

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Proposed Solution Methods

#### 2.1. Method Using Taylor Series

#### 2.2. Approach with Metaheuristic Optimization Algorithms

## 3. Metaheuristic Algorithms for Optimization Problems

#### 3.1. Whale Optimization Algorithm

- In each of iteration (t is number of iteration), the best individual in the population ${\mathbf{x}}_{best}^{t}$ is determined. This individual’s position is closest to the prey, and other individuals move towards it according to the following formulas$$\mathbf{d}=|\mathbf{c}\ast {\mathbf{x}}_{best}^{t}-{\mathbf{x}}^{t}|,$$$${\mathbf{x}}^{t+1}={\mathbf{x}}_{best}^{t}-\mathbf{a}\ast \mathbf{d},$$$$\mathbf{a}=2\mathbf{w}\ast \mathbf{r}-\mathbf{w},$$$$\mathbf{c}=2\mathbf{r},$$
- The next important stage is the mechanism of spiral-shaped movement of whales. Mathematically, we describe this process with the following formula$${\mathbf{x}}^{t+1}={\mathbf{d}}^{\prime}{e}^{br}cos\left(2\pi r\right)+{\mathbf{x}}_{best}^{t},$$
- Whales move using both a shrinking encircling mechanism and a spiral-shaped movement. In the algorithm, this behavior is simulated by the formula$${\mathbf{x}}^{t+1}=\left\{\begin{array}{cc}{\mathbf{x}}_{best}^{t}-\mathbf{a}\ast \mathbf{d}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{Equation}\phantom{\rule{3.33333pt}{0ex}}\left(4\right),\hfill & \mathrm{for}p\le 0.5,\hfill \\ {\mathbf{d}}^{\prime}{e}^{br}cos\left(2\pi r\right)+{\mathbf{x}}_{best}^{t}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{Equation}\phantom{\rule{3.33333pt}{0ex}}\left(7\right),\hfill & \mathrm{for}p>0.5.\hfill \end{array}\right.$$In the above formula $p\in [0,1]$, therefore, there is a $50\%$ chance of making each move. It is possible to control behavior of the algorithm by the level of probability p. Standard approach assumes a level of $0.5$.
- During the exploration phase, whales behave similarly to in Equation (4), with the difference that for the vector $\mathbf{a}$, we assume $\left|\mathbf{a}\right|>1$, which simulates the exploration phase and the whales move along with a random individual in the population, not the best individual. This stage of the algorithm is described mathematically by the formulas$$\mathbf{d}=|\mathbf{c}\ast {\mathbf{x}}_{rand}^{t}-{\mathbf{x}}^{t}|,$$$${\mathbf{x}}^{t+1}={\mathbf{x}}_{rand}^{t}-\mathbf{a}\ast \mathbf{d},$$
- The vector $\mathbf{w}$, whose values change from iteration to iteration in the range from 2 to 0, is responsible for the transition from the exploration phase to the exploitation phase. For values of the vector $\mathbf{w}$ in the range $(1,2]$, there is an exploration phase, while for the range $[0,1]$, there is an exploitation phase.

Algorithm 1: Pseudocode of WOA |

#### 3.2. Artificial Bee Colony

- working bees—these are bees whose job is to look for a food source. Important information for these bees consists of the following: the distance between the hive and the food source, the direction the bee should follow to reach the food source and the amount of nectar in the source.
- bees unclassified—these are bees that search for new food sources. We can divide them into two groups: scouts and onlookers. Scouts, after leaving a food source, look for another one in randomly way, while viewers look for visited sources based on the information provided.

- abandons the source, becomes an onlooker and watches the bees conveying information,
- transmits information through dance and recruits other bees,
- continues to explore on its own, without hiring other bees.

- the locations of food sources correspond to potential solutions of the optimized problem.
- the quantity of nectar in the source corresponds to the quality of the solution.
- the number of working bees is equal to the number of onlookers, which is denoted by
**SN**. - Food sources are modified according to the formula$${\mathbf{v}}_{i}^{t}={\mathbf{x}}_{i}^{t}+\mathbf{r}\ast ({\mathbf{x}}_{i}^{t}-{\mathbf{x}}_{k}^{t}),$$Compare positions ${\mathbf{v}}_{i}^{t}$ with ${\mathbf{x}}_{i}^{t}$. If $\mathcal{F}\left({\mathbf{v}}_{i}^{t}\right)\le \mathcal{F}\left({\mathbf{x}}_{i}^{t}\right)$. The position of ${\mathbf{x}}_{i}^{t}$ in the population t is then replaced by ${\mathbf{v}}_{i}^{t}$. Otherwise, the position ${\mathbf{x}}_{i}^{t}$ remains in the population.
- Each item in population is assigned a probability according to the formula$${p}_{i}=\frac{fit\left({\mathbf{x}}^{i}\right)}{{\displaystyle \sum _{j=1}^{SN}}fit\left({\mathbf{x}}^{j}\right)},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}i=1,2,\dots ,SN,$$$$fit\left({\mathbf{x}}_{i}\right)=\left\{\begin{array}{cc}\frac{1}{1+\mathcal{F}\left({\mathbf{x}}_{i}\right)},\hfill & \mathrm{if}\phantom{\rule{4pt}{0ex}}\mathcal{F}\left({\mathbf{x}}_{i}\right)\ge 0,\hfill \\ 1+\left|\mathcal{F}\right({\mathbf{x}}_{i}\left)\right|,\hfill & \mathrm{if}\phantom{\rule{4pt}{0ex}}\mathcal{F}\left({\mathbf{x}}_{i}\right)<0.\hfill \end{array}\right.$$
- Each onlooker bee selects one source according to the probability ${p}_{i}$ and starts searching near it according to the Formula (11). Then the bee compares two locations—the new and previous one.
- If, after performing the previous step of the algorithm, any of the food sources have not changed their position, then they are omitted and replaced with a new random source$${\mathbf{x}}_{i}={\mathbf{x}}_{min}+\mathbf{r}\ast ({\mathbf{x}}_{max}-{\mathbf{x}}_{min}),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}i=1,2,\dots ,SN,$$

Algorithm 2: Pseudocode of ABC |

## 4. Numerical Examples

#### 4.1. Example 1

#### 4.2. Example 2

#### 4.3. Example 3

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**Absolute error $\Delta \left(x\right)$ plot for a polynomial of degree 3 and the WOA (

**a**) and ABC (

**b**) algorithm (example 1).

**Figure 5.**Absolute error $\Delta \left(x\right)$ plot for a polynomial of degree 4 and the WOA (

**a**) and ABC (

**b**) algorithm (example 1).

**Figure 6.**Absolute error $\Delta \left(x\right)$ plot for a polynomial of degree 5 and the WOA (

**a**) and ABC (

**b**) algorithm (example 1).

**Figure 7.**The exact solution (solid green line) and approximate solution (dashed red line) for $n=4$ (

**a**) and the absolute errors ${\Delta}^{\prime}$ of this approximation (

**b**) (example 2).

**Figure 8.**The exact solution (solid green line) and approximate solution (dashed red line) for $n=6$ (

**a**) and the absolute errors ${\Delta}^{\prime}$ of this approximation (

**b**) (example 2).

**Figure 9.**The exact solution (solid green line) and approximate solution (dashed red line) in the case of WOA (

**a**) and the absolute errors ${\Delta}^{\prime}$ of this approximation (

**b**) (example 2).

**Figure 10.**The exact solution (solid green line) and approximate solution (dashed red line) in the case of ABC (

**a**) and the absolute errors ${\Delta}^{\prime}$ of this approximation (

**b**) (example 2).

**Figure 11.**The exact solution (solid green line) and approximate solution (dashed red line) for $n=6$ (

**a**) and the absolute errors ${\Delta}^{\prime}$ of this approximation (

**b**) (example 3).

**Figure 12.**The exact solution (solid green line) and approximate solution (dashed red line) for $n=16$ (

**a**) and the absolute errors ${\Delta}^{\prime}$ of this approximation (

**b**) (example 3).

**Figure 13.**The exact solution (solid green line) and approximate solution (dashed red line) for a polynomial of degree 3 (

**a**) and the absolute errors ${\Delta}^{\prime}$ of this approximation in the case of WOA (

**b**) (example 3).

**Figure 14.**The exact solution (solid green line) and approximate solution (dashed red line) for a polynomial of degree 3 (

**a**) and the absolute errors ${\Delta}^{\prime}$ of this approximation in the case of ABC (

**b**) (example 3).

**Figure 15.**The exact solution (solid green line) and approximate solution (dashed red line) for a polynomial of degree 4 (

**a**) and the absolute errors ${\Delta}^{\prime}$ of this approximation in the case of WOA (

**b**) (example 3).

**Figure 16.**The exact solution (solid green line) and approximate solution (dashed red line) for a polynomial of degree 4 (

**a**) and the absolute errors ${\Delta}^{\prime}$ of this approximation in the case of ABC (

**b**) (example 3).

**Figure 17.**The exact solution (solid green line) and approximate solution (dashed red line) for a polynomial of degree 5 (

**a**) and the absolute errors ${\Delta}^{\prime}$ of this approximation in the case of WOA (

**b**) (example 3).

**Figure 18.**The exact solution (solid green line) and approximate solution (dashed red line) for a polynomial of degree 5 (

**a**) and the absolute errors ${\Delta}^{\prime}$ of this approximation in the case of ABC (

**b**) (example 3).

**Table 1.**Sought coefficient values obtained using WOA and ABC methods. ${a}_{i}\phantom{\rule{4pt}{0ex}}(i=0,1,2,3,4)$—obtained coefficient value, $\mathcal{F}$—objective function value.

${\mathit{a}}_{0}$ | ${\mathit{a}}_{1}$ | ${\mathit{a}}_{2}$ | ${\mathit{a}}_{3}$ | ${\mathit{a}}_{4}$ | $\mathcal{F}$ | |
---|---|---|---|---|---|---|

WOA | $2.6771$ | $-3.0077$ | $-3.9007$ | $1.0142$ | $-1.7346$ | $0.1631$ |

ABC | $2.0051$ | $-2.9996$ | $-3.9976$ | $0.9996$ | $-1.9981$ | $0.0034$ |

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**MDPI and ACS Style**

Brociek, R.; Pleszczyński, M.
Comparison of Selected Numerical Methods for Solving Integro-Differential Equations with the Cauchy Kernel. *Symmetry* **2024**, *16*, 233.
https://doi.org/10.3390/sym16020233

**AMA Style**

Brociek R, Pleszczyński M.
Comparison of Selected Numerical Methods for Solving Integro-Differential Equations with the Cauchy Kernel. *Symmetry*. 2024; 16(2):233.
https://doi.org/10.3390/sym16020233

**Chicago/Turabian Style**

Brociek, Rafał, and Mariusz Pleszczyński.
2024. "Comparison of Selected Numerical Methods for Solving Integro-Differential Equations with the Cauchy Kernel" *Symmetry* 16, no. 2: 233.
https://doi.org/10.3390/sym16020233