# 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 ${\left(\right)}_{{g}_{n}}^{\left(L\right)}$ and ${\left(\right)}_{{s}_{n}}^{\left(L\right)}$ from (18).
- (2)
- Compute a number ${N}_{N}\ge N$ of zeros ${\left(\right)}_{{\alpha}_{j}}^{}$ of the function ${\phi}_{N}(\rho ,L)$, which approximate the Neumann–Dirichlet singular numbers, and a number ${N}_{D}\ge N+1$ of zeros ${\left(\right)}_{{\gamma}_{j}}^{}$ of the function ${S}_{N}(\rho ,L)$, which approximate the Dirichlet–Dirichlet singular numbers of $q\left(x\right)$.
- (3)
- Compute the coefficients ${\left(\right)}_{{t}_{n}}^{\left(0\right)}$ from (23).
- (4)
- Compute the constants ${\left(\right)}_{{\beta}_{j}}^{}$ 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

## References

- Bondarenko, N.P. Inverse Sturm—Liouville problem with analytical functions in the boundary condition. Open Math.
**2020**, 18, 512–528. [Google Scholar] [CrossRef] - Bondarenko, N.P. Solvability and stability of the inverse Sturm–Liouville problem with analytical functions in the boundary condition. Math. Methods Appl. Sci.
**2020**, 43, 7009–7021. [Google Scholar] [CrossRef] - Gesztesy, F.; del Rio, R.; Simon, B. Inverse spectral analysis with partial information on the potential, III. Updating boundary conditions. Int. Math. Res. Notices
**1997**, 15, 751–758. [Google Scholar] - Horváth, M. Inverse spectral problems and closed exponential systems. Ann. Math.
**2005**, 162, 885–918. [Google Scholar] [CrossRef] [Green Version] - Yurko, V.A. Introduction to the Theory of Inverse Spectral Problems; Fizmatlit: Moscow, Russia, 2007. [Google Scholar]
- Gao, Q.; Cheng, X.; Huang, Z. On a boundary value method for computing Sturm–Liouville potentials from two spectra. Int. J. Comput. Math.
**2014**, 91, 490–513. [Google Scholar] [CrossRef] - Guliyev, N.J. On two-spectra inverse problems. Proc. Am. Math. Soc.
**2020**, 148, 4491–4502. [Google Scholar] [CrossRef] - Kammanee, A.; Böckmann, C. Boundary value method for inverse Sturm–Liouville problems. Appl. Math. Comput.
**2009**, 214, 342–352. [Google Scholar] [CrossRef] - Savchuk, A.M.; Shkalikov, A.A. Inverse problem for Sturm–Liouville operators with distribution potentials: Reconstruction from two spectra. Russ. J. Math. Phys.
**2005**, 12, 507–514. [Google Scholar] - Kravchenko, V.V.; Navarro, L.J.; Torba, S.M. Representation of solutions to the one-dimensional Schrödinger equation in terms of Neumann series of Bessel functions. Appl. Math. Comput.
**2017**, 314, 173–192. [Google Scholar] [CrossRef] [Green Version] - Levitan, B.M. Inverse Sturm–Liouville Problems; VSP: Zeist, The Netherlands, 1987. [Google Scholar]
- Marchenko, V.A. Sturm–Liouville Operators and Applications: Revised Edition; AMS Chelsea Publishing: New York, NY, USA, 2011. [Google Scholar]
- Shishkina, E.L.; Sitnik, S.M. Transmutations, Singular and Fractional Differential Equations with Applications to Mathematical Physics; Elsevier: Amsterdam, The Netherlands, 2020. [Google Scholar]
- Avdonin, S.; Kravchenko, V.V. Method for solving inverse spectral problems on quantum star graphs. J. Inverse-Ill-Pose P. 2022; in press. [Google Scholar]
- Kravchenko, V.V. Spectrum completion and inverse Sturm-Liouville problems. Math. Method Appl. Sci.
**2022**, in press. [Google Scholar] [CrossRef] - Abramovitz, M.; Stegun, I.A. Handbook of Mathematical Functions; Unites States Department of Commerce: New York, NY, USA, 1972. [Google Scholar]
- Rundell, W.; Sacks, P.E. Reconstruction techniques for classical inverse Sturm–Liouville problems. Math. Comput.
**1992**, 58, 161–183. [Google Scholar] [CrossRef] - Chadan, K.; Colton, D.; Päivxaxrinta, L.; Rundell, W. An Introduction to Inverse Scattering and Inverse Spectral Problems; SIAM: Philadelphia, PA, USA, 1997. [Google Scholar]
- Brown, B.M.; Samko, V.S.; Knowles, I.W.; Marletta, M. Inverse spectral problem for the Sturm—Liouville equation. Inverse Probl.
**2003**, 19, 235–252. [Google Scholar] [CrossRef] - Kravchenko, V.V.; Torba, S.M. Analytic approximation of transmutation operators and applications to highly accurate solution of spectral problems. J. Comput. Appl. Math.
**2015**, 275, 1–26. [Google Scholar] [CrossRef]

**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 $\left(\right)$ 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 $\left(\right)$, $a\left(k\right)=sin{\rho}_{k}$ 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 $\left(\right)$ and $N=24$.

j | ${\mathit{\alpha}}_{\mathit{j}}$ | ${\tilde{\mathit{\alpha}}}_{\mathit{j}}$ | $\left(\right)open="|"\; close="|">{\mathit{\alpha}}_{\mathit{j}}-{\tilde{\mathit{\alpha}}}_{\mathit{j}}$ |
---|---|---|---|

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 | ${\mathit{\gamma}}_{\mathit{j}}$ | ${\tilde{\mathit{\gamma}}}_{\mathit{j}}$ | $\left(\right)open="|"\; close="|">{\mathit{\gamma}}_{\mathit{j}}-{\tilde{\mathit{\gamma}}}_{\mathit{j}}$ |
---|---|---|---|

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

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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