# Recovery of Inhomogeneity from Output Boundary Data

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

**Problem 1.**

## 2. Preliminaries

**Theorem 1**

## 3. Solution of Problem 1

#### 3.1. Computation of Coefficients ${g}_{n}\left(L\right)$ and ${s}_{n}\left(L\right)$

#### 3.2. One Spectrum and Multiplier Constants Inverse Problem

**Problem 2.**

**Theorem 2.**

**Proof.**

## 4. Algorithm

- (1)
- Choose $N\in \mathbb{N}$, such that $2(N+1)\le m$, and compute the sets of coefficients ${\left\{{g}_{n}\left(L\right)\right\}}_{n=0}^{N}$ and ${\left\{{s}_{n}\left(L\right)\right\}}_{n=0}^{N}$ from (18).
- (2)
- Compute a number ${N}_{N}\ge N$ of zeros ${\left\{{\alpha}_{j}\right\}}_{j=0}^{{N}_{N}}$ of the function ${\phi}_{N}(\rho ,L)$, which approximate the Neumann–Dirichlet singular numbers, and a number ${N}_{D}\ge N+1$ of zeros ${\left\{{\gamma}_{j}\right\}}_{j=1}^{{N}_{D}}$ of the function ${S}_{N}(\rho ,L)$, which approximate the Dirichlet–Dirichlet singular numbers of $q\left(x\right)$.
- (3)
- Compute the coefficients ${\left\{{t}_{n}\left(0\right)\right\}}_{n=0}^{N}$ from (23).
- (4)
- Compute the constants ${\left\{{\beta}_{j}\right\}}_{j=0}^{{N}_{N}}$ from (22).
- (5)
- Choose ${N}_{c}\in \mathbb{N}$, such that $2({N}_{c}+1)\le {N}_{N}+1$, and a number of points ${x}_{m}\in (0,L)$. For each ${x}_{m}$, solve (24) to find ${g}_{0}\left({x}_{m}\right)$.
- (6)
- Compute $q\left(x\right)$ from (14).

## 5. Numerical Examples

**Example 1.**

**Example 2.**

**Example 3.**

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Potential from Example 1 recovered from the Weyl function given at 15 points and with N = 4.

**Figure 2.**Potential ${q}_{2}\left(x\right)$ recovered from the data of type (3) with 25 points ${\rho}_{k}$ distributed uniformly on the segment in the complex plane $\left[-1.5+0.1i,1.5+0.1i\right]$ and $N=9$.

**Figure 3.**Potential ${q}_{2}\left(x\right)$ recovered from the data (8) with 25 points ${\rho}_{k}$ distributed uniformly on the segment in the complex plane $\left[-1.5+0.1i,1.5+0.1i\right]$, $a\left(k\right)=sin{\rho}_{k}$ and $b\left(k\right)=cos{\rho}_{k}$, corresponding to ${l}_{k}$, and $N=9$.

**Figure 4.**Potential ${q}_{3}\left(x\right)$ recovered from the data of type (3) given at 60 points ${\rho}_{k}$ distributed uniformly on the segment $\left[0.1+0.1i,15+0.1i\right]$ and $N=24$.

j | ${\mathit{\alpha}}_{\mathit{j}}$ | ${\tilde{\mathit{\alpha}}}_{\mathit{j}}$ | $\left|{\mathit{\alpha}}_{\mathit{j}}-{\tilde{\mathit{\alpha}}}_{\mathit{j}}\right|$ |
---|---|---|---|

0 | $1.11803398$ | $1.11803296$ | $1.02\xb7{10}^{-6}$ |

10 | $10.54751155$ | $10.54751173$ | $1.8\xb7{10}^{-7}$ |

20 | $20.52437575$ | $20.52437601$ | $2.6\xb7{10}^{-7}$ |

39 | $39.51265620$ | $39.51265636$ | $1.6\xb7{10}^{-7}$ |

j | ${\mathit{\gamma}}_{\mathit{j}}$ | ${\tilde{\mathit{\gamma}}}_{\mathit{j}}$ | $\left|{\mathit{\gamma}}_{\mathit{j}}-{\tilde{\mathit{\gamma}}}_{\mathit{j}}\right|$ |
---|---|---|---|

1 | $1.41421356$ | $1.41421215$ | $1.41\xb7{10}^{-6}$ |

11 | $11.04536101$ | $11.04536156$ | $5.43\xb7{10}^{-7}$ |

21 | $21.02379604$ | $21.02379651$ | $4.74\xb7{10}^{-7}$ |

39 | $39.01281840$ | $39.01281869$ | $2.90\xb7{10}^{-7}$ |

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**MDPI and ACS Style**

Kravchenko, V.V.; Khmelnytskaya, K.V.; Çetinkaya, F.A. Recovery of Inhomogeneity from Output Boundary Data. *Mathematics* **2022**, *10*, 4349.
https://doi.org/10.3390/math10224349

**AMA Style**

Kravchenko VV, Khmelnytskaya KV, Çetinkaya FA. Recovery of Inhomogeneity from Output Boundary Data. *Mathematics*. 2022; 10(22):4349.
https://doi.org/10.3390/math10224349

**Chicago/Turabian Style**

Kravchenko, Vladislav V., Kira V. Khmelnytskaya, and Fatma Ayça Çetinkaya. 2022. "Recovery of Inhomogeneity from Output Boundary Data" *Mathematics* 10, no. 22: 4349.
https://doi.org/10.3390/math10224349