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Symmetry
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Recent Advances in Numerical Methods for Partial Differential Equations
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Recent Advances in Numerical Methods for Partial Differential Equations

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

Deadline for manuscript submissions: closed (28 February 2022) | Viewed by 9032

Special Issue Editor

Prof. Yanren Hou

E-Mail Website
Guest Editor
Department of Computing Sciences, School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China
Interests: numerical methods for PDEs; numerical analysis

Special Issue Information

Dear Colleagues,

Numerical methods for partial differential equations always play an important role in applied mathematics and its engineering applications, especially in computational fluid dynamics and computational electromagnetics. The current Special Issue aims to publish original high-quality papers of recent advances in numerical methods for partial differential equations with engineering and other practical background. Research papers may focus on numerical simulations, new algorithms and  numerical analysis, based upon the finite element method, finite volume method, finite difference method, and spectral method, among others. The topics of research papers may include, but are not limited to, the following: Numerical Algorithms for Coupling of Multi-Physics; Parallel Algorithms in Space and/or Time; Interface Problems

Prof. Yanren Hou
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

  • numerical simulations
  • partial differential equations
  • numerical methods
  • numerical analysis
  • finite element method
  • computational fluid dynamics
  • Navier–Stokes equations
  • computational electromagnetics

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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

19 pages, 2270 KiB  
Article
Approximate Analytic–Numeric Fuzzy Solutions of Fuzzy Fractional Equations Using a Residual Power Series Approach
by Yousef Al-qudah, Mohammed Alaroud, Hamza Qoqazeh, Ali Jaradat, Sharifah E. Alhazmi and Shrideh Al-Omari
Symmetry 2022, 14(4), 804; https://doi.org/10.3390/sym14040804 - 12 Apr 2022
Cited by 10 | Viewed by 1426
Abstract
In this article, we consider a reliable analytical and numerical approach to create fuzzy approximated solutions for differential equations of fractional order with appropriate uncertain initial data by the means of a residual error function. The concept of strongly generalized differentiability is utilized [...] Read more.
In this article, we consider a reliable analytical and numerical approach to create fuzzy approximated solutions for differential equations of fractional order with appropriate uncertain initial data by the means of a residual error function. The concept of strongly generalized differentiability is utilized to introduce the fuzzy fractional derivatives. The proposed method provides a systematic scheme based on generalized Taylor expansion and minimization of the residual error function, so as to obtain the coefficients values of a fractional series based on the given initial data of triangular fuzzy numbers in the parametric form. The obtained approximated solutions are provided within an appropriate radius to the requisite domain in the form of rapidly convergent fractional series according to their parametric form. The method’s performance and applicability are verified by applying it on some numerical examples. The impact of r-levels and fractional order γ is presented quantitatively and graphically, showing the coincidence between the exact and the fuzzy approximated solutions. Moreover, for reliability and accuracy, our obtained results are numerically compared with the exact solutions and with results obtained using other methods described in the literature. This indicates that the proposed approach overcomes the difficulties that appear in other approaches to create fractional series solutions for varied uncertain natural problems arising within the fields of applied physics and engineering. Full article
(This article belongs to the Special Issue Recent Advances in Numerical Methods for Partial Differential Equations)
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17 pages, 347 KiB  
Article
On Conditions for L2-Dissipativity of an Explicit Finite-Difference Scheme for Linearized 2D and 3D Barotropic Gas Dynamics System of Equations with Regularizations
by Alexander Zlotnik
Symmetry 2021, 13(11), 2184; https://doi.org/10.3390/sym13112184 - 16 Nov 2021
Cited by 2 | Viewed by 1422
Abstract
We deal with 2D and 3D barotropic gas dynamics system of equations with two viscous regularizations: so-called quasi-gas dynamics (QGD) and quasi-hydrodynamics (QHD) ones. The system is linearized on a constant solution with any velocity, and an explicit two-level in time and symmetric [...] Read more.
We deal with 2D and 3D barotropic gas dynamics system of equations with two viscous regularizations: so-called quasi-gas dynamics (QGD) and quasi-hydrodynamics (QHD) ones. The system is linearized on a constant solution with any velocity, and an explicit two-level in time and symmetric three-point in each spatial direction finite-difference scheme on the uniform rectangular mesh is considered for the linearized system. We study L2-dissipativity of solutions to the Cauchy problem for this scheme by the spectral method and present a criterion in the form of a matrix inequality containing symbols of symmetric matrices of convective and regularizing terms. Analyzing these inequality and matrices, we also derive explicit sufficient conditions and necessary conditions in the Courant-type form which are rather close to each other. For the QHD regularization, such conditions are derived for the first time in 2D and 3D cases, whereas, for the QGD regularization, they improve those that have recently been obtained. Explicit formulas for a scheme parameter that guarantee taking the maximal time step are given for these conditions. An important moment is a new choice of an “average” spatial mesh step ensuring the independence of the conditions from the ratios of the spatial mesh steps and, for the QGD regularization, from the Mach number as well. Full article
(This article belongs to the Special Issue Recent Advances in Numerical Methods for Partial Differential Equations)
14 pages, 1593 KiB  
Article
The Analysis of Fractional-Order Kersten–Krasil Shchik Coupled KdV System, via a New Integral Transform
by Nehad Ali Shah, Asiful H. Seikh and Jae Dong Chung
Symmetry 2021, 13(9), 1592; https://doi.org/10.3390/sym13091592 - 30 Aug 2021
Cited by 4 | Viewed by 1887
Abstract
In this article, we use the homotopy perturbation transform method to find the fractional Kersten–Krasil’shchik coupled Korteweg–de Vries (KdV) non-linear system. This coupled non-linear system is typically used to describe electric circuits, traffic flow, shallow water waves, elastic media, electrodynamics, etc. The homotopy [...] Read more.
In this article, we use the homotopy perturbation transform method to find the fractional Kersten–Krasil’shchik coupled Korteweg–de Vries (KdV) non-linear system. This coupled non-linear system is typically used to describe electric circuits, traffic flow, shallow water waves, elastic media, electrodynamics, etc. The homotopy perturbation method is modified with the help of the ρ-Laplace transformation to investigate the solution of the given examples to show the accuracy of the current technique. The solution of the given technique and the actual results are shown and analyzed with figures. Full article
(This article belongs to the Special Issue Recent Advances in Numerical Methods for Partial Differential Equations)
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21 pages, 1313 KiB  
Article
Numerical Investigation of Fractional-Order Swift–Hohenberg Equations via a Novel Transform
by Kamsing Nonlaopon, Abdullah M. Alsharif, Ahmed M. Zidan, Adnan Khan, Yasser S. Hamed and Rasool Shah
Symmetry 2021, 13(7), 1263; https://doi.org/10.3390/sym13071263 - 14 Jul 2021
Cited by 72 | Viewed by 3232
Abstract
In this paper, the Elzaki transform decomposition method is implemented to solve the time-fractional Swift–Hohenberg equations. The presented model is related to the temperature and thermal convection of fluid dynamics, which can also be used to explain the formation process in liquid surfaces [...] Read more.
In this paper, the Elzaki transform decomposition method is implemented to solve the time-fractional Swift–Hohenberg equations. The presented model is related to the temperature and thermal convection of fluid dynamics, which can also be used to explain the formation process in liquid surfaces bounded along a horizontally well-conducting boundary. In the Caputo manner, the fractional derivative is described. The suggested method is easy to implement and needs a small number of calculations. The validity of the presented method is confirmed from the numerical examples. Illustrative figures are used to derive and verify the supporting analytical schemes for fractional-order of the proposed problems. It has been confirmed that the proposed method can be easily extended for the solution of other linear and non-linear fractional-order partial differential equations. Full article
(This article belongs to the Special Issue Recent Advances in Numerical Methods for Partial Differential Equations)
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