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Mathematics
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Advances in Partial Differential Equations: Methods and Applications
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Advances in Partial Differential Equations: Methods and Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E: Applied Mathematics".

Deadline for manuscript submissions: closed (10 April 2025) | Viewed by 2289

Special Issue Editors

Prof. Dr. Xiaojie Hou

E-Mail Website
Guest Editor
Department of Mathematics and Statistics, University of North Carolina Wilmington, Wilmington, NC 28403, USA
Interests: finite and infinite dimensional dynamical systems; traveling waves in reaction diffusion systems
Prof. Dr. Yi Li

E-Mail Website
Guest Editor
Department of Mathematics and Computer Science, John Jay College of Criminal Justice, City University of New York, New York, NY 10019, USA
Interests: nonlinear elliptic and parabolic differential equations; with their applications in physics, biology, and medicine
Prof. Dr. Shuangjie Peng

E-Mail Website
Guest Editor
School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China
Interests: nonlinear partial differential equations; nonlinear functional analysis; singular perturbations

Special Issue Information

Dear Colleagues, 

During the last few decades, partial differential equations have achieved many fascinating results in theory as well as in real world applications. In this Special Issue, we aim to provide a platform for mathematicians to exchange and demonstrate the newest ideas, theories and applications in this research field.  

The topics of papers of this Special Issue include the following: the existence and uniqueness and bifurcations of solutions of partial differential equations and systems,  the stability of solutions of reaction diffusion equations and systems; singular perturbation as well as geometric singular perturbation methods in differential equations and their applications; periodic solutions as well as turing instability for steady states and periodic solutions; traveling waves, trains, pulses in reaction diffusion systems and hyperbolic systems as well as their bifurcations and stabilities; applications in biology, ecology, cell biology and medical fields, including modeling, analysis of models and numerical analysis of model systems. 

Prof. Dr. Xiaojie Hou
Prof. Dr. Yi Li
Prof. Dr. Shuangjie Peng
Guest Editors

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

  • existence
  • uniqueness
  • stability
  • bifurcation
  • singular perturbation
  • pattern

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Published Papers (3 papers)

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Research

18 pages, 573 KiB  
Article
Finite Element Method for Solving the Screened Poisson Equation with a Delta Function
by Liang Tang and Yuhao Tang
Mathematics 2025, 13(8), 1360; https://doi.org/10.3390/math13081360 - 21 Apr 2025
Abstract
This paper presents a Finite Element Method (FEM) framework for solving the screened Poisson equation with a Dirac delta function as the forcing term. The singularity introduced by the delta function poses challenges for standard numerical methods, particularly in higher dimensions. To address [...] Read more.
This paper presents a Finite Element Method (FEM) framework for solving the screened Poisson equation with a Dirac delta function as the forcing term. The singularity introduced by the delta function poses challenges for standard numerical methods, particularly in higher dimensions. To address this, we employ integrated Legendre basis functions, which yield sparse and structured system matrices characterized by a Banded-Block-Banded-Arrowhead (B3-Arrowhead) form. In one dimension, the resulting linear system can be solved directly. In two and three dimensions, the equation can be efficiently solved using a generalized Alternating Direction Implicit (ADI) method combined with reverse Cholesky factorization. Numerical results in 1D, 2D, and 3D confirm that the method accurately captures the localized impulse response and reproduces the expected Green’s function behavior. The proposed approach offers a robust and scalable solution framework for partial differential equations with singular source terms and has potential applications in physics, engineering, and computational science. Full article
(This article belongs to the Special Issue Advances in Partial Differential Equations: Methods and Applications)
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26 pages, 1038 KiB  
Article
Deep Learning Artificial Neural Network for Pricing Multi-Asset European Options
by Zhiqiang Zhou, Hongying Wu, Yuezhang Li, Caijuan Kang and You Wu
Mathematics 2025, 13(4), 617; https://doi.org/10.3390/math13040617 - 13 Feb 2025
Viewed by 570
Abstract
This paper studies a p-layers deep learning artificial neural network (DLANN) for European multi-asset options. Firstly, a p-layers DLANN is constructed with undetermined weights and bias. Secondly, according to the terminal values of the partial differential equation (PDE) and the points [...] Read more.
This paper studies a p-layers deep learning artificial neural network (DLANN) for European multi-asset options. Firstly, a p-layers DLANN is constructed with undetermined weights and bias. Secondly, according to the terminal values of the partial differential equation (PDE) and the points that satisfy the PDE of multi-asset options, some discrete data are fed into the p-layers DLANN. Thirdly, using the least square error as the objective function, the weights and bias of the DLANN are trained well. In order to optimize the objective function, the partial derivatives for the weights and bias of DLANN are carefully derived. Moreover, to improve the computational efficiency, a time-segment DLANN is proposed. Numerical examples are presented to confirm the accuracy, efficiency, and stability of the proposed p-layers DLANN. Computational examples show that the DLANN’s relative error is less than 0.5% for different numbers of assets d=1,2,3,4. In the future, the p-layers DLANN can be extended into American options, Asian options, Lookback options, and so on. Full article
(This article belongs to the Special Issue Advances in Partial Differential Equations: Methods and Applications)
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10 pages, 255 KiB  
Article
Exact Null Controllability of a One-Dimensional Wave Equation with a Mixed Boundary
by Lizhi Cui and Jing Lu
Mathematics 2023, 11(18), 3855; https://doi.org/10.3390/math11183855 - 9 Sep 2023
Viewed by 972
Abstract
In this paper, exact null controllability of one-dimensional wave equations in non-cylindrical domains was discussed. It is different from past papers, as we consider boundary conditions for more complex cases. The wave equations have a mixed Dirichlet–Neumann boundary condition. The control is put [...] Read more.
In this paper, exact null controllability of one-dimensional wave equations in non-cylindrical domains was discussed. It is different from past papers, as we consider boundary conditions for more complex cases. The wave equations have a mixed Dirichlet–Neumann boundary condition. The control is put on the fixed endpoint with a Neumann boundary condition. By using the Hilbert Uniqueness Method, exact null controllability can be obtained. Full article
(This article belongs to the Special Issue Advances in Partial Differential Equations: Methods and Applications)
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