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Symmetry
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Symmetry in Mathematical Theory and Simulation Methods for Backward Problems
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Symmetry in Mathematical Theory and Simulation Methods for Backward Problems

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

Deadline for manuscript submissions: 28 February 2025 | Viewed by 4291

Special Issue Editor

Dr. Chih-Wen Chang

E-Mail Website
Guest Editor
Department of Mechanical Engineering, National United University, Miaoli 36063, Taiwan
Interests: numerical analysis, fluid mechanics, thermodynamics, dynamic system, plasticity, friction dynamics, inverse/backward problems, ordinary differential equations, partial differential equations, Lie algebra, Lie-group numerical methods
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Backward problems have been investigated in science, mathematics, and engineering, and reveal an unknown property of an object from their experimentation or observation. Backward problems conform to the Symmetry journal's ideology as they are the opposite of the associated forward issue, which concerns the cause­–effect relationship.

Backward problems have a wide range of applications, including mechanics, heat conduction, acoustics, semiconductors, medical imaging, nondestructive testing, physics, systems biology, finance, robotics, computer vision, radar, thermoelastics, and groundwater.

This Special Issue of Symmetry concentrates on the present mathematical theory and simulation regarding backward problems and how they relate to their applications in engineering and science.

Dr. Chih-Wen Chang
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.

Keywords

  • backward problems
  • inverse problems
  • numerical analysis
  • mathematical modeling
  • fractional problems
  • ordinary/partial differential equations
  • meshless methods
  • applications
  • symmetry operators

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Further information on MDPI's Special Issue polices can be found here.

Published Papers (4 papers)

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Research

22 pages, 312 KiB  
Article
Four Classes of Symmetric Sums over Cyclically Binomial Products
by Marta Na Chen and Wenchang Chu
Symmetry 2025, 17(2), 209; https://doi.org/10.3390/sym17020209 - 29 Jan 2025
Viewed by 194
Abstract
Four classes of multiple symmetric sums over cyclic products of binomial coefficients are examined. By incorporating the generating function approach and recursive construction method, they are expressed analytically as coefficients of rational functions. Several recurrence relations and generating functions are explicitly determined when [...] Read more.
Four classes of multiple symmetric sums over cyclic products of binomial coefficients are examined. By incorporating the generating function approach and recursive construction method, they are expressed analytically as coefficients of rational functions. Several recurrence relations and generating functions are explicitly determined when the dimension of the multiple sums does not exceed five. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Theory and Simulation Methods for Backward Problems)
15 pages, 871 KiB  
Article
High-Accuracy Solutions to the Time-Fractional KdV–Burgers Equation Using Rational Non-Polynomial Splines
by Miguel Vivas-Cortez, Majeed A. Yousif, Bewar A. Mahmood, Pshtiwan Othman Mohammed, Nejmeddine Chorfi and Alina Alb Lupas
Symmetry 2025, 17(1), 16; https://doi.org/10.3390/sym17010016 - 25 Dec 2024
Viewed by 432
Abstract
A non-polynomial spline is a technique that utilizes information from symmetric functions to solve mathematical or physical models numerically. This paper introduces a novel non-polynomial spline construct incorporating a rational function term to develop an efficient numerical scheme for solving time-fractional differential equations. [...] Read more.
A non-polynomial spline is a technique that utilizes information from symmetric functions to solve mathematical or physical models numerically. This paper introduces a novel non-polynomial spline construct incorporating a rational function term to develop an efficient numerical scheme for solving time-fractional differential equations. The proposed method is specifically applied to the time-fractional KdV–Burgers (TFKdV) equation. and time-fractional differential equations are crucial in physics as they provide a more accurate description of various complex processes, such as anomalous diffusion and wave propagation, by capturing memory effects and non-local interactions. Using Taylor expansion and truncation error analysis, the convergence order of the numerical scheme is derived. Stability is analyzed through the Fourier stability criterion, confirming its conditional stability. The accuracy and efficiency of the rational non-polynomial spline (RNPS) method are validated by comparing numerical results from a test example with analytical and previous solutions, using norm errors. Results are presented in 2D and 3D graphical formats, accompanied by tables highlighting performance metrics. Furthermore, the influences of time and the fractional derivative are examined through graphical analysis. Overall, the RNPS method has demonstrated to be a reliable and effective approach for solving time-fractional differential equations. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Theory and Simulation Methods for Backward Problems)
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19 pages, 425 KiB  
Article
Inverse Boundary Conditions Interface Problems for the Heat Equation with Cylindrical Symmetry
by Miglena N. Koleva and Lubin G. Vulkov
Symmetry 2024, 16(8), 1065; https://doi.org/10.3390/sym16081065 - 18 Aug 2024
Cited by 1 | Viewed by 1141
Abstract
In this paper, we study inverse interface problems with unknown boundary conditions, using point observations for parabolic equations with cylindrical symmetry. In the one-dimensional, two-layer interface problem, the left interval 0<r<l1, i.e., the zero degeneracy, causes serious [...] Read more.
In this paper, we study inverse interface problems with unknown boundary conditions, using point observations for parabolic equations with cylindrical symmetry. In the one-dimensional, two-layer interface problem, the left interval 0<r<l1, i.e., the zero degeneracy, causes serious solution difficulty. For this, we investigate the well-posedness of the direct (forward) problem. Next, we formulate and solve five inverse boundary condition problems for the interface heat equation with cylindrical symmetry from internal measurements. The finite volume difference method is developed to construct second-order schemes for direct and inverse problems. The correctness of the proposed numerical solution decomposition algorithms for the inverse problems is discussed. Several numerical examples are presented to illustrate the efficiency of the approach. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Theory and Simulation Methods for Backward Problems)
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28 pages, 418 KiB  
Article
Noncommutative Integration of Generalized Diffusion PDE
by Sergey Victor Ludkowski
Symmetry 2022, 14(10), 2049; https://doi.org/10.3390/sym14102049 - 1 Oct 2022
Viewed by 1263
Abstract
The article is devoted to the noncommutative integration of a diffusion partial differential equation (PDE). Its generalizations are also studied. This is motivated by the fact that many existing approaches for solutions of PDEs are based on evolutionary operators obtained as solutions of [...] Read more.
The article is devoted to the noncommutative integration of a diffusion partial differential equation (PDE). Its generalizations are also studied. This is motivated by the fact that many existing approaches for solutions of PDEs are based on evolutionary operators obtained as solutions of the corresponding stochastic PDEs. However, this is restricted to PDEs of an order not higher than 2 over the real or complex field. This article is aimed at extending such approaches to PDEs of an order higher than 2. For this purpose, measures and random functions having values in modules over complexified Cayley–Dickson algebras are investigated. Noncommutative integrals of hypercomplex random functions are investigated. By using them, the noncommutative integration of the generalized diffusion PDE is scrutinized. Possibilities are indicated for a subsequent solution of higher-order PDEs using their decompositions and noncommutative integration. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Theory and Simulation Methods for Backward Problems)
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