# On Construction of Solutions of Linear Differential Systems with Argument Deviations of Mixed Type

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## Abstract

**:**

## 1. Introduction

## 2. Auxiliary Problems with Two-Point Conditions

## 3. Iteration Process

**Lemma**

**1.**

**Proof.**

## 4. Applicability Conditions

**Theorem**

**1.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## 5. Proof of Theorem 1

**Lemma**

**2.**

**Lemma**

**3.**

**Proof.**

## 6. Some Estimates

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Proof.**

**Lemma**

**6.**

**Proof.**

## 7. Practical Realisation

- Solve the zeroth approximate determining system$$\begin{array}{c}{\Delta}_{0}(\xi ,\eta )=0,\\ h({u}_{0}(\xb7,\xi ,\eta ))=d.\end{array}$$This approximate determining system has the simplest form and its root $({\xi}^{(0)},{\eta}^{(0)})$ serves as a rough approximation of the unknown values of $(\xi ,\eta )$. The function$${U}_{0}(t):={u}_{0}({\xi}^{(0)},{\eta}^{(0)}),\phantom{\rule{2.em}{0ex}}t\in [a,b],$$
- Analytically construct the function ${u}_{1}(\xb7,\xi ,\eta )$ according to the recurrence Formula (14), keeping $\xi $ and $\eta $ as parameters. Numerically solve the corresponding first approximate determining system$$\begin{array}{c}{\Delta}_{1}(\xi ,\eta )=0,\\ h({u}_{1}(\xb7,\xi ,\eta ))=d\end{array}$$$${U}_{1}(t):={u}_{1}({\xi}^{(1)},{\eta}^{(1)}),\phantom{\rule{2.em}{0ex}}t\in [a,b].$$
- Choose a certain ${m}_{0}\ge 1$ and continue by analogy to step 2 for $m=1,2,\cdots ,{m}_{0}$ by analytically constructing the functions ${u}_{m}(\xb7,\xi ,\eta )$, $m=1,2,\cdots ,{m}_{0}$. Computer algebra systems are very helpful for this purpose. Numerically solve every mth approximate determining system in a neighbourhood of the root of the $(m-1)$th one, $({\xi}^{(m-1)},{\eta}^{(m-1)})$, $m=1,2,\cdots ,{m}_{0}$. Collect the values $({\xi}^{(m)},{\eta}^{(m)})$, $m=1,2,\cdots ,{m}_{0}$, into a table, construct the approximations$${U}_{m}(t):={u}_{m}({\xi}^{(m)},{\eta}^{(m)}),\phantom{\rule{2.em}{0ex}}t\in [a,b],$$$$\begin{array}{cccc}\hfill {U}_{m}(a)& ={\xi}^{(m)},\hfill & \hfill {U}_{m}(b)& ={\eta}^{(m)}.\hfill \end{array}$$Multiple roots of system (58) usually indicate the existence of multiple solutions of the problem. In such cases, in order to select a particular one, we specify a suitable neighbourhood when solving the approximate determining equations numerically.
- Analyse the results of step 3 and decide whether the computaton should be continued.

- Clear signs of convergence and a good degree of coincidence (${U}_{{m}_{0}}$ satisfies the set accuracy requirements; it remains only to check the solvability rigorously as mentioned above).
- There are signs of convergence but the accuracy requirements are not met (continue the computation with $m={m}_{0}+1$).
- There are signs of divergence, usually accompanied by failure to solve some of the equations (either there is no solution or the convergence boundary is trespassed; the scheme is inapplicable).
- Failure to carry out symbolic computations at a certain point (software or hardware limitations; some simplifications should be used).

## 8. A Numerical Example

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Table 1.**Values of the parameters computed from the approximate determining equations with $0\le m\le 10$ (11 nodes)

m | ${\mathit{\xi}}_{1}$ | ${\mathit{\xi}}_{2}$ | ${\mathit{\xi}}_{3}$ | ${\mathit{\eta}}_{1}$ | ${\mathit{\eta}}_{2}$ | ${\mathit{\eta}}_{3}$ |
---|---|---|---|---|---|---|

0 | 0.630827 | −0.682613 | 1 | −0.416667 | 3.40206 | 0.143882 |

1 | 0.516729 | −0.0189486 | 1 | −0.416667 | 4.08704 | −0.412343 |

2 | 0.486264 | 0.00196245 | 1 | −0.416667 | 4.04751 | −0.675279 |

3 | 0.425443 | −0.0020287 | 1 | −0.416667 | 4.05544 | −0.624705 |

4 | 0.439424 | −0.00126683 | 1 | −0.416667 | 4.05392 | −0.634359 |

5 | 0.43643 | −0.00141226 | 1 | −0.416667 | 4.05421 | −0.632516 |

6 | 0.437032 | −0.0013845 | 1 | −0.416667 | 4.05416 | −0.632868 |

7 | 0.436914 | −0.0013898 | 1 | −0.416667 | 4.05417 | −0.632801 |

8 | 0.436937 | −0.00138879 | 1 | −0.416667 | 4.05417 | −0.632813 |

9 | 0.436932 | −0.00138898 | 1 | −0.416667 | 4.05417 | −0.632811 |

10 | 0.436933 | −0.00138895 | 1 | −0.416667 | 4.05417 | −0.632811 |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

∞ | 0.436933 | −0.00138889 | 1 | −0.416667 | 4.05417 | −0.63281 |

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Rontó, A.; Rontóová, N.
On Construction of Solutions of Linear Differential Systems with Argument Deviations of Mixed Type. *Symmetry* **2020**, *12*, 1740.
https://doi.org/10.3390/sym12101740

**AMA Style**

Rontó A, Rontóová N.
On Construction of Solutions of Linear Differential Systems with Argument Deviations of Mixed Type. *Symmetry*. 2020; 12(10):1740.
https://doi.org/10.3390/sym12101740

**Chicago/Turabian Style**

Rontó, András, and Natálie Rontóová.
2020. "On Construction of Solutions of Linear Differential Systems with Argument Deviations of Mixed Type" *Symmetry* 12, no. 10: 1740.
https://doi.org/10.3390/sym12101740