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Mathematics
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Linear Elliptic PDEs
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Linear Elliptic PDEs

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "C1: Difference and Differential Equations".

Deadline for manuscript submissions: 30 November 2025 | Viewed by 1212

Special Issue Editors

Prof. Dr. Jochen Merker

E-Mail Website
Guest Editor
Centre for Mathematics and Natural Sciences, HTWK Leipzig University of Applied Sciences, 04251 Leipzig, Germany
Interests: applied analysis; partial differential equations; hamiltonian systems (with ports)
Special Issues, Collections and Topics in MDPI journals
Dr. Falko Baustian

E-Mail Website
Guest Editor
Institute of Mathematics, University of Rostock, 18057 Rostock, Germany
Interests: partial differential equations (PDEs); Fučík eigenvalue problems; basis properties of eigenfunctions; financial mathematics

Special Issue Information

Dear Colleagues,

Although the field of linear elliptical partial differential equations (PDEs) is one of the most-studied and best-understood topics in PDE theory and numerics; active research is still ongoing in this area. Regarding the analysis of linear elliptic operators, many open scientific problems can be found, e.g., in Section 5 of Vladimir Maz’ya, “Seventy Five (Thousand) Unsolved Problems in Analysis and Partial Differential Equations”, Integr. Equ. Oper. Theory (2018) 90:25, https://doi.org/10.1007/s00020-018-2460-8.

Among other things, new theoretical results about (systems of) linear elliptic PDEs and the regularity of solutions (for measure data or rough data) are suitable for this Special Issue. Furthermore, the numerical analysis of computational methods for the solution of linear elliptic PDEs (the finite element method, FEM; the finite difference method, FDM; and the finite volume method, FVM) and new trends in numerics, e.g., estimating Green’s function of a linear elliptic operator by machine learning or quantum algorithms for the solution of linear PDEs, fit into the scope of this Special Issue.

To summarize, this Special Issue gives authors the opportunity to present their latest findings related to linear elliptic PDEs.

Prof. Dr. Jochen Merker
Dr. Falko Baustian
Guest Editors

Manuscript Submission Information

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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

  • linear elliptic PDEs
  • regularity theory
  • eigenvalue problems
  • numerical analysis of PDE algorithms
  • machine learning for PDEs
  • quantum algorithms for PDEs

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Published Papers (1 paper)

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Research

25 pages, 11695 KiB  
Article
A Spacetime RBF-Based DNNs for Solving Unsaturated Flow Problems
by Chih-Yu Liu, Cheng-Yu Ku and Wei-Da Chen
Mathematics 2024, 12(18), 2940; https://doi.org/10.3390/math12182940 - 21 Sep 2024
Cited by 2 | Viewed by 705
Abstract
This study presents a novel approach for modeling unsaturated flow using deep neural networks (DNNs) integrated with spacetime radial basis functions (RBFs). Traditional methods for simulating unsaturated flow often face challenges in computational efficiency and accuracy, particularly when dealing with nonlinear soil properties [...] Read more.
This study presents a novel approach for modeling unsaturated flow using deep neural networks (DNNs) integrated with spacetime radial basis functions (RBFs). Traditional methods for simulating unsaturated flow often face challenges in computational efficiency and accuracy, particularly when dealing with nonlinear soil properties and complex boundary conditions. Our proposed model emphasizes the capabilities of DNNs in identifying complex patterns and the accuracy of spacetime RBFs in modeling spatiotemporal data. The training data comprise the initial data, boundary data, and radial distances used to construct the spacetime RBFs. The innovation of this approach is that it introduces spacetime RBFs, eliminating the need to discretize the governing equation of unsaturated flow and directly providing the solution of unsaturated flow across the entire time and space domain. Various error evaluation metrics are thoroughly assessed to validate the proposed method. This study examines a case where, despite incomplete initial and boundary data and noise contamination in the available boundary data, the solution of unsaturated flow can still be accurately determined. The model achieves RMSE, MAE, and MRE values of 10−4, 10−3, and 10−4, respectively, demonstrating that the proposed method is robust for solving unsaturated flow in soils, providing insights beyond those obtainable with traditional methods. Full article
(This article belongs to the Special Issue Linear Elliptic PDEs)
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