1. Neumann series of Bessel functions for inverse coefficient problems.
- Author
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Çetinkaya, Fatma Ayça, Khmelnytskaya, Kira V., and Kravchenko, Vladislav V.
- Subjects
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INVERSE problems , *VOCAL tract , *INVERSE functions , *BESSEL functions , *COMPLEX numbers - Abstract
Consider the Sturm–Liouville equation −y′′+q(x)y=ρ2y$$ -{y}^{\prime \prime }+q(x)y={\rho}^2y $$ with a real‐valued potential q∈L1(0,L),ρ∈ℂ,L>0$$ q\in {\mathcal{L}}_1\left(0,L\right),\rho \in \mathrm{\mathbb{C}},L>0 $$. Let u(ρ,x)$$ u\left(\rho, x\right) $$ be its solution satisfying certain initial conditions u(ρk,0)=ak,u′(ρk,0)=bk$$ u\left({\rho}_k,0\right)={a}_k,{u}^{\prime}\left({\rho}_k,0\right)={b}_k $$ for a number of ρk,k=1,2,...,K$$ {\rho}_k,k=1,2,\dots, K $$, where ρk,ak$$ {\rho}_k,{a}_k $$, and bk$$ {b}_k $$ are some complex numbers. Denote ℓk=u′(ρk,L)+Hu(ρk,L)$$ {\ell}_k={u}^{\prime}\left({\rho}_k,L\right)+ Hu\left({\rho}_k,L\right) $$, where H∈ℝ$$ H\in \mathrm{\mathbb{R}} $$. We propose a method for solving the inverse problem of the approximate recovery of the potential q(x)$$ q(x) $$ and number H$$ H $$ from the following data ρk,ak,bk,ℓkk=1K$$ {\left\{{\rho}_k,{a}_k,{b}_k,{\ell}_k\right\}}_{k=1}^K $$. In general, the problem is ill‐posed; however, it finds numerous practical applications. Such inverse problems as the recovery of the potential from a Weyl function or the inverse two‐spectra Sturm–Liouville problem are its special cases. Moreover, the inverse problem of determining the shape of a human vocal tract also reduces to the considered inverse problem. The proposed method is based on special Neumann series of Bessel functions representations for solutions of Sturm–Liouville equations. With their aid the problem is reduced to the classical inverse Sturm–Liouville problem of recovering q(x)$$ q(x) $$ from two spectra, which is solved again with the help of the same representations. The overall approach leads to an efficient numerical algorithm for solving the inverse problem. Its numerical efficiency is illustrated by several examples. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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