# Inverse Problem for the Integral Dynamic Models with Discontinuous Kernels

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## Abstract

**:**

## 1. Introduction

#### Inverse Problem

## 2. Numerical Method

## 3. Arithmetic Complexity

- Calculation of the values of functions $f\left(t\right),{K}_{1}(t,s),{K}_{2}(t,s)$ and $x\left(t\right)$ at grid nodes and at midpoints:$$4N+{N}^{2};$$
- Calculation of the values ${\alpha}_{k},\phantom{\rule{0.277778em}{0ex}}k=1,2,\dots ,N$:$$12\xb7\stackrel{\mathrm{summation}\phantom{\rule{4.pt}{0ex}}{S}_{1},{S}_{2}}{\overbrace{3\xb7\sum _{k=1}^{N}2\xb7\frac{1+k-1}{2}(k-1)}}=36\sum _{k=1}^{N}k(k-1)$$$$=36\left(\frac{N(N+1)(2N+1)}{6}-\frac{1+N}{2}N\right)=12N({N}^{2}-1);$$
- Estimation of the number of arithmetic operations ${P}_{N}^{1}$ to iterate over possible values ${\alpha}_{k}$:$${P}_{N}^{1}\u2a7d\sum _{k=1}^{N}\left(1+2+\cdots +k\right)=\frac{1}{2}\sum _{k=1}^{N}k(k+1)=\frac{N(N+1)(N+2)}{6};$$
- Estimation of the number of comparison operations ${P}_{N}^{2}$ when iterating over possible values ${\alpha}_{k}$:$${P}_{N}^{2}\u2a7d2{P}_{N}^{1}=\frac{N(N+1)(N+2)}{3}.$$

## 4. General Case

## 5. Numerical Experiments

#### 5.1. Problem 1

#### 5.2. Problem 2

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Problem 1. Exact and approximate solutions $\alpha \left(t\right)$ and ${\alpha}_{N}\left(t\right)$ for $N=8,\phantom{\rule{0.166667em}{0ex}}N=16,\phantom{\rule{0.166667em}{0ex}}N=32$.

**Figure 2.**Problem 2. Exact and approximate solutions $\alpha \left(t\right)$ and ${\alpha}_{N}\left(t\right)$ for $N=8,\phantom{\rule{0.166667em}{0ex}}N=16,\phantom{\rule{0.166667em}{0ex}}N=32$.

N | 16 | 32 | 64 | 128 | 256 | 512 | 1024 | 2048 | 4096 |

${\epsilon}_{N}$ | 0.075 | 0.0094 | 0.00273 | 0.00079 | $5.43\times {10}^{-5}$ | $1.61\times {10}^{-5}$ | $4.21\times {10}^{-6}$ | $1.12\times {10}^{-6}$ | $3.11\times {10}^{-7}$ |

r | — | 2.99 | 1.78 | 1.79 | 3.86 | 1.75 | 1.93 | 1.91 | 1.85 |

N | 16 | 32 | 64 | 128 | 256 | 512 | 1024 | 2048 |

${\epsilon}_{N}$ | 0.005 | 0.00133 | 0.00049 | 0.00012 | $2.69\times {10}^{-5}$ | $8.32\times {10}^{-6}$ | $2.31\times {10}^{-6}$ | $6.49\times {10}^{-7}$ |

r | — | 1.91 | 1.43 | 2.07 | 2.13 | 1.69 | 1.85 | 1.83 |

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Tynda, A.N.; Sidorov, D.N. Inverse Problem for the Integral Dynamic Models with Discontinuous Kernels. *Mathematics* **2022**, *10*, 3945.
https://doi.org/10.3390/math10213945

**AMA Style**

Tynda AN, Sidorov DN. Inverse Problem for the Integral Dynamic Models with Discontinuous Kernels. *Mathematics*. 2022; 10(21):3945.
https://doi.org/10.3390/math10213945

**Chicago/Turabian Style**

Tynda, Aleksandr N., and Denis N. Sidorov. 2022. "Inverse Problem for the Integral Dynamic Models with Discontinuous Kernels" *Mathematics* 10, no. 21: 3945.
https://doi.org/10.3390/math10213945