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Symmetry
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Symmetric Methods and Analysis for Time-Dependent Partial Differential Equations
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Symmetric Methods and Analysis for Time-Dependent Partial Differential Equations

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

Deadline for manuscript submissions: closed (30 June 2022) | Viewed by 22912

Special Issue Editor

Prof. Dr. Dongfang Li

grade E-Mail Website
Guest Editor
Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, China
Interests: numerical solutions to PDE and ODE; fractional partial differential equation
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Time-dependent differential equations are widely used to model many natural phenomena in different scientific fields. The evolution of the solutions can be well learned by certain high-performance numerical methods. Well-designed and high-performance numerical methods can considerably reduce the computational costs of long-term simulations of real-world problems. They can also perform better in the prediction of mathematical models. Therefore, it is very important to develop effective numerical schemes as well as their rigorous numerical analysis.

In light of the aforementioned regarding the significance of numerical schemes and analysis, the potential topics for this issue include but are not limited to the following:

The construction of effective numerical methods for solving time-dependent problems;
The convergence analysis of Symmetric schemes;
The stability analysis of Symmetric methods;
The dissipativity of Symmetric methods;
Iterative algorithms and their application;
Symmetric and structure-preserving numerical methods for time-dependent problems.
Please note that all submitted papers must be within the general scope of the Symmetry journal.

Submit your paper and select the Journal “Symmetry” and the Special Issue “Symmetric Methods and Analysis for Time-Dependent Partial Differential Equations” via: MDPI submission system. Our papers will be published on a rolling basis and we will be pleased to receive your submission once you have finished it.

Prof. Dr. Dongfang Li
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.

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Published Papers (12 papers)

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Research

12 pages, 293 KiB  
Article
Error Estimates of a Symmetric Spectral Method for a Linear Volterra Integral Equation
by Danna Wu, Weishan Zheng and Yanfeng Chen
Symmetry 2023, 15(1), 60; https://doi.org/10.3390/sym15010060 - 26 Dec 2022
Viewed by 1259
Abstract
A symmetric spectral method is applied to investigate the two-dimensional Volterra integral equation with weakly singular kernels and delays. In this work, the solution of the equation we considered is assumed to be sufficiently smooth so that the spectral method can be applied [...] Read more.
A symmetric spectral method is applied to investigate the two-dimensional Volterra integral equation with weakly singular kernels and delays. In this work, the solution of the equation we considered is assumed to be sufficiently smooth so that the spectral method can be applied naturally. Employing three couples of variable transformations, we apply the two-dimensional Gauss quadrature rule to approximate the weakly singular integral with delays and obtain the spectral discretization. Then we derive the convergence results of the proposed approximation scheme. We show that the errors of solution decay exponentially in both the infinity norm and weighted square norm. In the end, we carry out numerical experiments to verify the theoretical results. Full article
(This article belongs to the Special Issue Symmetric Methods and Analysis for Time-Dependent Partial Differential Equations)
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15 pages, 325 KiB  
Article
A Class of Adaptive Exponentially Fitted Rosenbrock Methods with Variable Coefficients for Symmetric Systems
by Tingting Qin, Yuchen Hua and Mengyao Zhang
Symmetry 2022, 14(8), 1708; https://doi.org/10.3390/sym14081708 - 17 Aug 2022
Cited by 1 | Viewed by 1331
Abstract
In several important scientific fields, the efficient numerical solution of symmetric systems of ordinary differential equations, which are usually characterized by oscillation and periodicity, has become an open problem of interest. In this paper, we construct a class of embedded exponentially fitted Rosenbrock [...] Read more.
In several important scientific fields, the efficient numerical solution of symmetric systems of ordinary differential equations, which are usually characterized by oscillation and periodicity, has become an open problem of interest. In this paper, we construct a class of embedded exponentially fitted Rosenbrock methods with variable coefficients and adaptive step size, which can achieve third order convergence. This kind of method is developed by performing the exponentially fitted technique for the two-stage Rosenbrock methods, and combining the embedded methods to estimate the frequency. By using Richardson extrapolation, we determine the step size control strategy to make the step size adaptive. Numerical experiments are given to verify the validity and efficiency of our methods. Full article
(This article belongs to the Special Issue Symmetric Methods and Analysis for Time-Dependent Partial Differential Equations)
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11 pages, 291 KiB  
Article
Convergence Analysis of a Symmetrical and Positivity-Preserving Finite Difference Scheme for 1D Poisson–Nernst–Planck System
by Weiwei Ling, Benchao Liu and Qian Guo
Symmetry 2022, 14(8), 1589; https://doi.org/10.3390/sym14081589 - 2 Aug 2022
Viewed by 1425
Abstract
The Poisson–Nernst–Planck (PNP) system is a nonlinear coupled system that describes the motion of ionic particles. As the exact solution of the system is not available, numerical investigations are essentially important, and there are quite a lot of numerical methods proposed in the [...] Read more.
The Poisson–Nernst–Planck (PNP) system is a nonlinear coupled system that describes the motion of ionic particles. As the exact solution of the system is not available, numerical investigations are essentially important, and there are quite a lot of numerical methods proposed in the existing literature. However, the theoretical analysis is usually neglected due to the complicated nature of the PNP system. In this paper, a theoretical investigation for a symmetrical finite difference method proposed in the previous literature was conducted. An L2 error estimate of O(τ+h2) was derived for the numerical scheme in 1D, where τ denotes the time step size and h denotes the spatial mesh size, respectively. Numerical results confirm the theoretical analysis. More importantly, a positivity-preserving condition for the scheme is provided with rigorously theoretical justification. Full article
(This article belongs to the Special Issue Symmetric Methods and Analysis for Time-Dependent Partial Differential Equations)
12 pages, 1790 KiB  
Article
Symmetric Spectral Collocation Method for a Kind of Nonlinear Volterra Integral Equation
by Nada Wu, Weishan Zheng and Wenjuan Gao
Symmetry 2022, 14(6), 1091; https://doi.org/10.3390/sym14061091 - 26 May 2022
Cited by 2 | Viewed by 1674
Abstract
In this paper, we develop an efficient spectral method for numerically solving the nonlinear Volterra integral equation with weak singularity and delays. Based on the symmetric collocation points, the spectral method is illustrated, and the convergence results are obtained. In the end, two [...] Read more.
In this paper, we develop an efficient spectral method for numerically solving the nonlinear Volterra integral equation with weak singularity and delays. Based on the symmetric collocation points, the spectral method is illustrated, and the convergence results are obtained. In the end, two numerical experiments are carried out to confirm the theoretical results. Full article
(This article belongs to the Special Issue Symmetric Methods and Analysis for Time-Dependent Partial Differential Equations)
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10 pages, 316 KiB  
Article
Parameter Estimation for a Type of Fractional Diffusion Equation Based on Compact Difference Scheme
by Wei Gu, Fang Wei and Min Li
Symmetry 2022, 14(3), 560; https://doi.org/10.3390/sym14030560 - 11 Mar 2022
Cited by 4 | Viewed by 1864
Abstract
Numerical solution and parameter estimation for a type of fractional diffusion equation are considered. Firstly, the symmetrical compact difference scheme is applied to solve the forward problem of the fractional diffusion equation. The stability and convergence of the symmetrical difference scheme are presented. [...] Read more.
Numerical solution and parameter estimation for a type of fractional diffusion equation are considered. Firstly, the symmetrical compact difference scheme is applied to solve the forward problem of the fractional diffusion equation. The stability and convergence of the symmetrical difference scheme are presented. Then, the Bayesian method is considered to estimate the unknown fractional order α of the fractional diffusion equation model. To validate the efficiency of the symmetrical numerical scheme and the estimation method, some simulation tests are considered. The simulation results demonstrate the accuracy of the compact difference scheme and show that the proposed estimation algorithm can provide effective statistical characteristics of the parameter. Full article
(This article belongs to the Special Issue Symmetric Methods and Analysis for Time-Dependent Partial Differential Equations)
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8 pages, 3200 KiB  
Article
A Quadruple Integral Involving Product of the Struve Hv(βt) and Parabolic Cylinder Du(αx) Functions
by Robert Reynolds and Allan Stauffer
Symmetry 2022, 14(1), 9; https://doi.org/10.3390/sym14010009 - 22 Dec 2021
Cited by 1 | Viewed by 2372
Abstract
The objective of the present paper is to obtain a quadruple infinite integral. This integral involves the product of the Struve and parabolic cylinder functions and expresses it in terms of the Hurwitz–Lerch Zeta function. Almost all Hurwitz-Lerch Zeta functions have an asymmetrical [...] Read more.
The objective of the present paper is to obtain a quadruple infinite integral. This integral involves the product of the Struve and parabolic cylinder functions and expresses it in terms of the Hurwitz–Lerch Zeta function. Almost all Hurwitz-Lerch Zeta functions have an asymmetrical zero distributionSpecial cases in terms fundamental constants and other special functions are produced. All the results in the work are new. Full article
(This article belongs to the Special Issue Symmetric Methods and Analysis for Time-Dependent Partial Differential Equations)
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9 pages, 269 KiB  
Article
A Note on a Fourier Sine Transform
by Robert Reynolds and Allan Stauffer
Symmetry 2021, 13(10), 1828; https://doi.org/10.3390/sym13101828 - 30 Sep 2021
Viewed by 1331
Abstract
This is a compilation of definite integrals of the product of the hyperbolic cosecant function and polynomial raised to a general power. In this work, we used our contour integral method to derive a Fourier sine transform in terms of the Lerch function. [...] Read more.
This is a compilation of definite integrals of the product of the hyperbolic cosecant function and polynomial raised to a general power. In this work, we used our contour integral method to derive a Fourier sine transform in terms of the Lerch function. Almost all Lerch functions have an asymmetrical zero-distribution. A summary table of the results are produced for easy reading. A vast majority of the results are new. Full article
(This article belongs to the Special Issue Symmetric Methods and Analysis for Time-Dependent Partial Differential Equations)
8 pages, 239 KiB  
Article
Double Integral of Logarithmic and Quotient Rational Functions Expressed in Terms of the Lerch Function
by Robert Reynolds and Allan Stauffer
Symmetry 2021, 13(9), 1708; https://doi.org/10.3390/sym13091708 - 15 Sep 2021
Viewed by 1571
Abstract
In this manuscript, the authors derive a double integral whose kernel involves the logarithmic function a polynomial raised to a power and a quotient function expressed it in terms of the Lerch function. All the results in this work are new. Full article
(This article belongs to the Special Issue Symmetric Methods and Analysis for Time-Dependent Partial Differential Equations)
11 pages, 4868 KiB  
Article
On Symmetrical Methods for Charged Particle Dynamics
by Renxuan Tang and Dongfang Li
Symmetry 2021, 13(9), 1626; https://doi.org/10.3390/sym13091626 - 3 Sep 2021
Cited by 3 | Viewed by 1405
Abstract
In this paper, we use the scalar auxiliary variable (SAV) approach to rewrite the charged particle dynamics as a new family of ODE systems. The systems own a conserved energy. It is shown that a family of symmetrical methods is energy-conserving for a [...] Read more.
In this paper, we use the scalar auxiliary variable (SAV) approach to rewrite the charged particle dynamics as a new family of ODE systems. The systems own a conserved energy. It is shown that a family of symmetrical methods is energy-conserving for a new ODE system but may not be for the original systems. Moreover, the methods have high-order accuracy. Numerical results are given to confirm the theoretical findings. Full article
(This article belongs to the Special Issue Symmetric Methods and Analysis for Time-Dependent Partial Differential Equations)
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7 pages, 211 KiB  
Article
Well-Posedness and Porosity for Symmetric Optimization Problems
by Alexander J. Zaslavski
Symmetry 2021, 13(7), 1253; https://doi.org/10.3390/sym13071253 - 13 Jul 2021
Viewed by 1465
Abstract
In the present work, we investigate a collection of symmetric minimization problems, which is identified with a complete metric space of lower semi-continuous and bounded from below functions. In our recent paper, we showed that for a generic objective function, the corresponding symmetric [...] Read more.
In the present work, we investigate a collection of symmetric minimization problems, which is identified with a complete metric space of lower semi-continuous and bounded from below functions. In our recent paper, we showed that for a generic objective function, the corresponding symmetric optimization problem possesses two solutions. In this paper, we strengthen this result using a porosity notion. We investigate the collection of all functions such that the corresponding optimization problem is well-posed and prove that its complement is a σ-porous set. Full article
(This article belongs to the Special Issue Symmetric Methods and Analysis for Time-Dependent Partial Differential Equations)
15 pages, 293 KiB  
Article
Preserving the Shape of Functions by Applying Multidimensional Schoenberg-Type Operators
by Camelia Liliana Moldovan and Radu Păltănea
Symmetry 2021, 13(6), 1016; https://doi.org/10.3390/sym13061016 - 5 Jun 2021
Viewed by 1685
Abstract
The paper presents a multidimensional generalization of the Schoenberg operators of higher order. The new operators are powerful tools that can be used for approximation processes in many fields of applied sciences. The construction of these operators uses a symmetry regarding the domain [...] Read more.
The paper presents a multidimensional generalization of the Schoenberg operators of higher order. The new operators are powerful tools that can be used for approximation processes in many fields of applied sciences. The construction of these operators uses a symmetry regarding the domain of definition. The degree of approximation by sequences of such operators is given in terms of the first and the second order moduli of continuity. Extending certain results obtained by Marsden in the one-dimensional case, the property of preservation of monotonicity and convexity is proved. Full article
(This article belongs to the Special Issue Symmetric Methods and Analysis for Time-Dependent Partial Differential Equations)
14 pages, 5947 KiB  
Article
Simulation and Analysis of Indoor Air Quality in Florida Using Time Series Regression (TSR) and Artificial Neural Networks (ANN) Models
by He Zhang, Ravi Srinivasan and Xu Yang
Symmetry 2021, 13(6), 952; https://doi.org/10.3390/sym13060952 - 27 May 2021
Cited by 15 | Viewed by 3422
Abstract
Exposures to air pollutants have been associated with various acute respiratory diseases and detrimental human health. Analysis and further interpretation of air pollutant patterns are correspondingly important as monitoring them. In the present study, the 24-h and four-month indoor and outdoor PM2.5 [...] Read more.
Exposures to air pollutants have been associated with various acute respiratory diseases and detrimental human health. Analysis and further interpretation of air pollutant patterns are correspondingly important as monitoring them. In the present study, the 24-h and four-month indoor and outdoor PM2.5, PM10, NO2, relative humidity, and temperature were measured simultaneously for a laboratory in Gainesville city, Florida. The indoor PM2.5, PM10, and NO2 concentrations were predicted using multiple linear regression (MLR), time series regression (TSR), and artificial neural networks (ANN) models. The modeling conducted in this study aims to perform a cross comparison study between these models in a symmetric environment. The value of root-mean-square error was improved by 18.33% in comparison with the MLR model. In addition, the value of the coefficient of determination was improved by 24.68%. The ANN model had the best performance and could predict the target air pollutants at 10-min intervals of the studied building with 90% accuracy levels. The TSR model showed slightly better performance compared to the MLR model. These results can be accordingly referred for studies analyzing indoor air quality in similar building types and climate zones. Full article
(This article belongs to the Special Issue Symmetric Methods and Analysis for Time-Dependent Partial Differential Equations)
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