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Mathematics
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Numerical Methods for Partial Differential Equation
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Numerical Methods for Partial Differential Equation

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: 31 December 2024 | Viewed by 2034

Special Issue Editor

Dr. Qingguang Guan

E-Mail Website
Guest Editor
School of Mathematics and Natural Sciences, University of Southern Mississippi, Hattiesburg, MS 39406, USA
Interests: computational neuroscience; high performance computing; deep learning; neural networks; weak Galerkin FEM and VEM; numerical analysis for nonlocal PDEs

Special Issue Information

Dear Colleagues,

Numerical methods for partial differential equations (PDEs) are a set of techniques used to solve PDE models computationally. Many problems in a wide range of sciences require such solutions. This Special Issue aims to collect original and novel contributions in the field of numerical methods for PDEs on polygonal/polyhedral meshes, including the Weak Galerkin method, the Hybrid Discontinuous method, the Virtual Element method, the Hybrid High-Order method, and other related methods. Topics will cover numerical analysis, mesh generation, and applications of these methods.

Dr. Qingguang Guan
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. Mathematics is an international peer-reviewed open access semimonthly 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 2600 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

  • polygonal/polyhedral meshes
  • weak Galerkin method
  • hybrid discontinuous method
  • virtual element method
  • hybrid high-order method
  • numerical analysis
  • partial differential equations (PDEs)
  • mesh generation

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Published Papers (2 papers)

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Research

27 pages, 1892 KiB  
Article
A Novel Centralized Allocation Data Envelopment Analysis Model for Carbon Emission Allocation Under a Heterogeneous Abatement Cost: Application Within the Chinese Industrial Sector
by Xiaohong Liu, Qingchun Meng, Ruiqi Sun and Xiangwei Zhang
Mathematics 2024, 12(21), 3409; https://doi.org/10.3390/math12213409 - 31 Oct 2024
Viewed by 525
Abstract
This paper presents a mathematical approach to analyzing carbon abatement costs and the allocation of carbon emission allowances in China’s industrial sectors. We utilize input–output data from 30 Chinese provinces between 2009 and 2018 to estimate carbon abatement costs by applying the slack-based [...] Read more.
This paper presents a mathematical approach to analyzing carbon abatement costs and the allocation of carbon emission allowances in China’s industrial sectors. We utilize input–output data from 30 Chinese provinces between 2009 and 2018 to estimate carbon abatement costs by applying the slack-based measure (SBM) efficiency model and its dual form. The SBM model captures inefficiencies and offers a rigorous framework for measuring abatement costs. Using these costs, we develop a centralized allocation data envelopment analysis (DEA) model, which maximizes sectoral benefits through optimal reallocation. This DEA model is formalized as a linear programming problem, with the aim of determining the efficient allocations of carbon allowances while maintaining the system’s economic productivity. Furthermore, we construct intertemporal, interregional, and spatiotemporal allocation DEA models to examine the dynamics of carbon emission allowance allocation over time, space, and combined spatiotemporal dimensions. These models offer insights into the efficiency of carbon markets under varying conditions. Our proposed new mathematical formulations reveal optimal allocation strategies that can balance emission reductions with industrial productivity. This study also provides novel mathematical frameworks for analyzing the carbon allowance distribution and contributions to both the theory and application of mathematical optimization in environmental policy design. Our findings reveal that China’s industrial carbon abatement costs exhibit significant interprovincial and regional differences. Developed provinces with higher levels of industrial development have higher carbon abatement costs, while provinces with less-developed industrial sectors have lower costs. Under the interregional allocation scenario of carbon emission allowances that consider abatement costs, developed provinces have smaller industrial carbon emission reductions, whereas less-developed provinces have larger reductions. In the intertemporal allocation scenario, provinces with larger industrial economies face greater emission reduction tasks. Under the combined interregional and intertemporal allocation scenario, industrial sectors in coastal developed provinces have lower carbon emission reductions, while those in inland less-developed provinces have higher reductions, mirroring the spatial allocation results of carbon emission allowances. Full article
(This article belongs to the Special Issue Numerical Methods for Partial Differential Equation)
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9 pages, 440 KiB  
Article
Existence of Kink and Antikink Wave Solutions of Singularly Perturbed Modified Gardner Equation
by Weifang Yan, Linlin Wang and Min Zhang
Mathematics 2024, 12(6), 928; https://doi.org/10.3390/math12060928 - 21 Mar 2024
Viewed by 972
Abstract
In this paper, the singularly perturbed modified Gardner equation is considered. Firstly, for the unperturbed equation, under certain parameter conditions, we obtain the exact expressions of kink wave solution and antikink wave solution by using the bifurcation method of dynamical systems. Then, the [...] Read more.
In this paper, the singularly perturbed modified Gardner equation is considered. Firstly, for the unperturbed equation, under certain parameter conditions, we obtain the exact expressions of kink wave solution and antikink wave solution by using the bifurcation method of dynamical systems. Then, the persistence of the kink and antikink wave solutions of the perturbed modified Gardner equation is studied by exploiting the geometric singular perturbation theory and the Melnikov function method. When the perturbation parameter is sufficiently small, we obtain the sufficient conditions to guarantee the existence of kink and antikink wave solutions. Full article
(This article belongs to the Special Issue Numerical Methods for Partial Differential Equation)
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