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Symmetry
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Symmetry in Differential - Difference Equations: Theory, Methods, Applications and...
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Symmetry in Differential - Difference Equations: Theory, Methods, Applications and Inverse Problem in Diffusion Equation

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (31 March 2023) | Viewed by 9553

Special Issue Editor

Prof. Dr. Fan Yang

E-Mail Website
Guest Editor
Department of Applied Mathematics, School of Science, Lanzhou University of Technology, Lanzhou 730050, China
Interests: partial differential equation and fractional reaction-diffusion equation; inverse problems of mathematical and physical equations
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Difference equations on the symmetry domain have been applied in many branches of engineering sciences—for example, heat conduction, reaction diffusion, pollutant detection, crack identification, geophysical prospecting, and electromagnetic theory.

Papers that employ the symmetry or asymmetry concept in experimental, theoretical, and computational investigations on combustion are welcomed. Papers related to the development and validation of reaction kinetics, reduction in reaction mechanisms, and modeling of combustion systems are welcome. We would particularly like to encourage papers on the inverse problem of identifying the unknown parameter on radially (spherically) symmetric difference equations and fractional diffusion equations.

Prof. Dr. Fan Yang
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Published Papers (7 papers)

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Research

23 pages, 662 KiB  
Article
Galerkin Method for a Backward Problem of Time-Space Fractional Symmetric Diffusion Equation
by Hongwu Zhang and Yong Lv
Symmetry 2023, 15(5), 1057; https://doi.org/10.3390/sym15051057 - 10 May 2023
Cited by 1 | Viewed by 1380
Abstract
We investigate a backward problem of the time-space fractional symmetric diffusion equation with a source term, wherein the negative Laplace operator −Δ contained in the main equation belongs to the category of uniformly symmetric elliptic operators. The problem is ill-posed because the [...] Read more.
We investigate a backward problem of the time-space fractional symmetric diffusion equation with a source term, wherein the negative Laplace operator −Δ contained in the main equation belongs to the category of uniformly symmetric elliptic operators. The problem is ill-posed because the solution does not depend continuously on the measured data. In this paper, the existence and uniqueness of the solution and the conditional stability for the inverse problem are given and proven. Based on the least squares technique, we construct a Galerkin regularization method to overcome the ill-posedness of the considered problem. Under a priori and a posteriori selection rules for the regularization parameter, the Hölder-type convergence results of optimal order for the proposed method are derived. Meanwhile, we verify the regularized effect of our method by carrying out some numerical experiments where the initial value function is a smooth function or a non-smooth one. Numerical results show that this method works well in dealing with the backward problem of the time-space fractional parabolic equation. Full article
(This article belongs to the Special Issue Symmetry in Differential - Difference Equations: Theory, Methods, Applications and Inverse Problem in Diffusion Equation)
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25 pages, 9570 KiB  
Article
Lumped Element Method Based Conductivity Reconstruction Algorithm for Localization Using Symmetric Discrete Operators on Coarse Meshes
by Zoltan Sari, Mihaly Klincsik, Peter Odry, Vladimir Tadic, Attila Toth and Zoltan Vizvari
Symmetry 2023, 15(5), 1008; https://doi.org/10.3390/sym15051008 - 30 Apr 2023
Cited by 2 | Viewed by 1387
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Symmetry in Differential - Difference Equations: Theory, Methods, Applications and Inverse Problem in Diffusion Equation)
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12 pages, 332 KiB  
Article
Qualitative Behavior of an Exponential Symmetric Difference Equation System
by Tarek F. Ibrahim, Somayah Refaei, Abdul Khaliq, Mohamed Abd El-Moneam, Bakri A. Younis, Waleed M. Osman and Bushra R. Al-Sinan
Symmetry 2022, 14(12), 2474; https://doi.org/10.3390/sym14122474 - 22 Nov 2022
Cited by 2 | Viewed by 1218
Abstract
We examine the unboundedness, persistence, boundedness, uniqueness, and existence of non-negative equilibrium of an exponential symmetric difference equations system: [...] Read more.
We examine the unboundedness, persistence, boundedness, uniqueness, and existence of non-negative equilibrium of an exponential symmetric difference equations system: Ωn+1=α1+β1Ωn+γ1Ωn−1e−(Ωn+ϖn), ϖn+1=α2+β2ϖn+γ2ϖn−1e−(Ωn+ϖn),n=0,1,⋯, whereby initial values Ω−1,ϖ−1,Ω0,ϖ0 and parameters α1,α2 are non-negative real numbers and β1,β2∈(0,1) and γ1,γ2≤0. We will discuss asymptotic global and local stability and the convergence rate of this system. Ultimately, to check our results, we set out some numerical explanations. Full article
(This article belongs to the Special Issue Symmetry in Differential - Difference Equations: Theory, Methods, Applications and Inverse Problem in Diffusion Equation)
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23 pages, 365 KiB  
Article
Jensen-Mercer Type Inequalities in the Setting of Fractional Calculus with Applications
by Bandar Bin-Mohsin, Muhammad Zakria Javed, Muhammad Uzair Awan, Marcela V. Mihai, Hüseyin Budak, Awais Gul Khan and Muhammad Aslam Noor
Symmetry 2022, 14(10), 2187; https://doi.org/10.3390/sym14102187 - 18 Oct 2022
Cited by 3 | Viewed by 1160
Abstract
The main objective of this paper is to establish some new variants of the Jensen–Mercer inequality via harmonically strongly convex function. We also propose some new fractional analogues of Hermite–Hadamard–Jensen–Mercer-like inequalities using AB fractional integrals. In order to obtain some of our main [...] Read more.
The main objective of this paper is to establish some new variants of the Jensen–Mercer inequality via harmonically strongly convex function. We also propose some new fractional analogues of Hermite–Hadamard–Jensen–Mercer-like inequalities using AB fractional integrals. In order to obtain some of our main results, we also derive new fractional integral identities. To demonstrate the significance of our main results, we present some interesting applications to special means and to error bounds as well. Full article
(This article belongs to the Special Issue Symmetry in Differential - Difference Equations: Theory, Methods, Applications and Inverse Problem in Diffusion Equation)
16 pages, 799 KiB  
Article
Fractional View Analysis of Fornberg–Whitham Equations by Using Elzaki Transform
by Faisal Haroon, Safyan Mukhtar and Rasool Shah
Symmetry 2022, 14(10), 2118; https://doi.org/10.3390/sym14102118 - 12 Oct 2022
Cited by 2 | Viewed by 1213
Abstract
We present analytical solutions of the Fornberg–Whitham equations with the aid of two well-known methods: Adomian decomposition transform and variational iteration transform involving fractional-order derivatives with the Atangana–Baleanu–Caputo derivative. The Elzaki transformation is used in the Atangana–Baleanu–Caputo derivative to find the solution to [...] Read more.
We present analytical solutions of the Fornberg–Whitham equations with the aid of two well-known methods: Adomian decomposition transform and variational iteration transform involving fractional-order derivatives with the Atangana–Baleanu–Caputo derivative. The Elzaki transformation is used in the Atangana–Baleanu–Caputo derivative to find the solution to the Fornberg–Whitham equations. Using certain exemplary situations, the proposed method’s viability is assessed. Comparative analysis for both integer and fractional-order results is established. For validation, the solutions of the suggested methods are compared with the actual results available in the literature. Two examples are considered to check the accuracy and effectiveness of the proposed techniques. Full article
(This article belongs to the Special Issue Symmetry in Differential - Difference Equations: Theory, Methods, Applications and Inverse Problem in Diffusion Equation)
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11 pages, 274 KiB  
Article
A Modified Regularization Method for a Spherically Symmetric Inverse Heat Conduction Problem
by Wei Cheng, Yi-Liang Liu and Fan Yang
Symmetry 2022, 14(10), 2102; https://doi.org/10.3390/sym14102102 - 10 Oct 2022
Cited by 2 | Viewed by 1112
Abstract
In this paper, we investigate a spherically symmetric inverse heat conduction problem, which determines the internal surface temperature distribution of the hollow sphere from measured data at the fixed location inside it. This problem is ill-posed, and a conditional stability result of its [...] Read more.
In this paper, we investigate a spherically symmetric inverse heat conduction problem, which determines the internal surface temperature distribution of the hollow sphere from measured data at the fixed location inside it. This problem is ill-posed, and a conditional stability result of its solution is given. A modified quasi-boundary value method is proposed to solve the ill-posed problem. Two Ho¨lder-type error estimates between the approximation solution and its exact solution are obtained under an a priori and an a posteriori regularization parameter selection rule, respectively. Full article
(This article belongs to the Special Issue Symmetry in Differential - Difference Equations: Theory, Methods, Applications and Inverse Problem in Diffusion Equation)
9 pages, 292 KiB  
Article
Global Stability of a Second-Order Exponential-Type Difference Equation
by Tarek Fawzi Ibrahim, Abdul Qadeer Khan, Fatima Mushyih Alshehri and Mohamed Abd El-Moneam
Symmetry 2022, 14(9), 1803; https://doi.org/10.3390/sym14091803 - 31 Aug 2022
Cited by 3 | Viewed by 1432
Abstract
In this work, we explore the boundedness and local and global asymptotic behavior of the solutions to a second-order difference formula of the exponential type [...] Read more.
In this work, we explore the boundedness and local and global asymptotic behavior of the solutions to a second-order difference formula of the exponential type ξn+1=a+bξn−1+cξn−1e−ρξn, where a,c,ρ∈(0,∞), b∈(0,1) and the initials ξ0,ξ−1 are non-negative real numbers. Some other special cases are given. We provide two concrete numerical examples to confirm the theoretical results. Full article
(This article belongs to the Special Issue Symmetry in Differential - Difference Equations: Theory, Methods, Applications and Inverse Problem in Diffusion Equation)
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