Reconstruction of Potential in Discrete Sturm–Liouville Problem

Koyunbakan, Hikmet

doi:10.1007/s12346-021-00548-9

Cited by 7 publications

(5 citation statements)

References 23 publications

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“…It is interesting to note that the challenges with CIPs are already difficult for ordinary differential equations. However, there are published papers in the literature in this field that provide specific solutions to the mentioned problems [37,38].…”

Section: Related Workmentioning

confidence: 99%

Lumped Element Method Based Conductivity Reconstruction Algorithm for Localization Using Symmetric Discrete Operators on Coarse Meshes

et al. 2023

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The inverse conductivity problem in electrical impedance tomography involves the solving of a nonlinear and under-determined system of equations. This paper presents a new approach, which leads to a quadratic and overdetermined system of equations. The aim of the paper is to establish new research directions in handling of the inverse conductivity problem. The basis of the proposed method is that the material, which can be considered as an isotropic continuum, is modeled as a linear network with concentrated parameters. The weights of the obtained graph represent the properties of the discretized continuum. Further, the application of the developed procedure allows for the dielectric constant to be used in the multi-frequency approach, as a result of which the optimized system of equations always remains overdetermined. Through case studies, the efficacy of the reconstruction method by changing the mesh resolution applied for discretizing is presented and evaluated. The presented results show, that, due to the application of discrete, symmetric mathematical structures, the new approach even at coarse mesh resolution is capable of localizing the inhomogeneities of the material.

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Section: Related Workmentioning

confidence: 99%

Lumped Element Method Based Conductivity Reconstruction Algorithm for Localization Using Symmetric Discrete Operators on Coarse Meshes

et al. 2023

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“…In [12,13,15], the authors studied some Sturm-Liouville problems with impulsive conditions. The inverse problem for the Sturm-Liouville equation in the discrete state is worked out in [17]. A Sturm-Liouville problem with some nonlocal boundary conditions is studied in [16].…”

Section: Introductionmentioning

confidence: 99%

Теорема Существования Фрактальной Задачи Штурма - Лиувилля

Allahverdiev,

Tuna

2024

Владикавказский Математический Журнал

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In this article, using a new calculus defined on fractal subsets of the set of real numbers, a Sturm--Lioville type problem is discussed, namely the fractal Sturm--Liouville problem. The existence and uniqueness theorem has been proved for such equations. In this context, the historical development of the subject is discussed in the introduction. In Section 2, the basic concepts of $F^{\alpha}$-calculus defined on fractal subsets of real numbers are given, i. e., $F^{\alpha}$-continuity, $F^{\alpha}$-derivative and fractal integral definitions are given and some theorems to be used in the article are given. In Section 3, the existence and uniqueness of the solutions for the fractal Sturm--Liouville problem are obtained by using the successive approximations method. Thus, the well-known existence and uniqueness problem for Sturm--Liouville equations in ordinary calculus is handled on the fractal calculus axis, and the existing results are generalized.

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“…Other important studies that deal with the properties of difference equations are [10] and [11]. In addition, there are many studies that examine discrete versions of Sturm-Liouville, Bessel, Dirac on time scales (see [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29]). To give basic results, we should recall substantial concepts of time scale theory.…”

Section: Introductionmentioning

confidence: 99%

On Some Spectral Properties of Discrete Sturm-Liouville Problem

YAR

Yılmaz

Gulsen

2023

Fundamental Journal of Mathematics and Applications

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Time scale theory helps us to combine differential equations with difference equations. Especially in models such as biology, medicine and economics, since the independent variable is handled discrete, it requires us to analyze in discrete clusters. In these cases, the difference equations defined in Z are considered. Boundary value problems (BVP’s) are used to solve and model problems in many physical area. In this study, we examined spectral features of discrete Sturm-Liouville problem. We have given some examples to make subject understandable.

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Reconstruction of Potential in Discrete Sturm–Liouville Problem

Cited by 7 publications

References 23 publications

Lumped Element Method Based Conductivity Reconstruction Algorithm for Localization Using Symmetric Discrete Operators on Coarse Meshes

Lumped Element Method Based Conductivity Reconstruction Algorithm for Localization Using Symmetric Discrete Operators on Coarse Meshes

Теорема Существования Фрактальной Задачи Штурма - Лиувилля

On Some Spectral Properties of Discrete Sturm-Liouville Problem

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