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Axioms
Special Issues
Numerical Methods for Fractional and Integer PDEs
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Numerical Methods for Fractional and Integer PDEs

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: closed (31 December 2022) | Viewed by 6104

Special Issue Editors

Prof. Dr. Yang Liu

E-Mail Website
Guest Editor
School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China
Interests: finite element method; finite difference method; LDG methods; numerical methods for fractional PDEs
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Hong Li

E-Mail Website
Co-Guest Editor
School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China
Interests: space–time finite element method; spectral method
Prof. Dr. Linghua Kong

E-Mail Website
Co-Guest Editor
School of Mathematics and Statistics, Jiangxi Normal University, Nanchang 330022, China
Interests: numerical methods for PDEs; structure-preserving algorithms
Dr. Zhichao Fang

E-Mail Website
Co-Guest Editor
School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China
Interests: finite volume element method; finite element method; numerical computing for fractional PDEs

Special Issue Information

Dear Colleagues,

Fractional and integer PDEs can describe a large number of practical problems in the field of science and engineering. For solving this kind of PDEs, numerical methods including the finite element method, finite difference method, local discontinuous Galerkin method, finite volume method, spectral method, meshless method, and others have played a very important role. Numerical analysis such as stability and convergence, fast algorithms, and new discrete methods have attracted the attention of many scholars. In addition, numerical experiments have been carried out to confirm the feasibility, effectiveness, and computational efficiency of the algorithms.

 The principal areas of interest of the Special Issue include but are not limited to the following:

  1. Numerical methods for fractional and integer PDEs;
  2. Stability and convergence based on numerical methods for PDEs;
  3. Construction of a new approximation formula for fractional calculus;
  4. Fast numerical algorithms for solving PDEs.

Prof. Dr. Yang Liu
Prof. Dr. Hong Li
Prof. Dr. Linghua Kong
Dr. Zhichao Fang
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. Axioms 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 method
  • numerical analysis
  • fractional partial differential equation
  • integer partial differential equation
  • fractional calculus
  • fast algorithm
  • structure-preserving algorithm
  • convergence

Published Papers (4 papers)

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Research

12 pages, 944 KiB  
Article
A Space-Time Legendre-Petrov-Galerkin Method for Third-Order Differential Equations
by Siqin Tang and Hong Li
Axioms 2023, 12(3), 281; https://doi.org/10.3390/axioms12030281 - 08 Mar 2023
Viewed by 979
Abstract
In this article, a space-time spectral method is considered to approximate third-order differential equations with non-periodic boundary conditions. The Legendre-Petrov-Galerkin discretization is employed in both space and time. In the theoretical analysis, rigorous proof of error estimates in the weighted space-time norms is [...] Read more.
In this article, a space-time spectral method is considered to approximate third-order differential equations with non-periodic boundary conditions. The Legendre-Petrov-Galerkin discretization is employed in both space and time. In the theoretical analysis, rigorous proof of error estimates in the weighted space-time norms is obtained for the fully discrete scheme. We also formulate the matrix form of the fully discrete scheme by taking appropriate test and trial functions in both space and time. Finally, extensive numerical experiments are conducted for linear and nonlinear problems, and spectral accuracy is derived for both space and time. Moreover, the numerical results are compared with those computed by other numerical methods to confirm the efficiency of the proposed method. Full article
(This article belongs to the Special Issue Numerical Methods for Fractional and Integer PDEs)
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16 pages, 514 KiB  
Article
A High-Order Approximate Solution for the Nonlinear 3D Volterra Integral Equations with Uniform Accuracy
by Zi-Qiang Wang, Ming-Dan Long and Jun-Ying Cao
Axioms 2022, 11(9), 476; https://doi.org/10.3390/axioms11090476 - 16 Sep 2022
Viewed by 1134
Abstract
In this paper, we present a high-order approximate solution with uniform accuracy for nonlinear 3D Volterra integral equations. This numerical scheme is constructed based on the three-dimensional block cubic Lagrangian interpolation method. At the same time, we give the local truncation error analysis [...] Read more.
In this paper, we present a high-order approximate solution with uniform accuracy for nonlinear 3D Volterra integral equations. This numerical scheme is constructed based on the three-dimensional block cubic Lagrangian interpolation method. At the same time, we give the local truncation error analysis of the numerical scheme based on Taylor’s theorem. Through theoretical analysis, we reach the conclusion that the optimal convergence order of this high-order numerical scheme is 4. Finally, we verify the effectiveness and applicability of the method through four numerical examples. Full article
(This article belongs to the Special Issue Numerical Methods for Fractional and Integer PDEs)
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14 pages, 485 KiB  
Article
Stability of a Nonlinear ML-Nonsingular Kernel Fractional Langevin System with Distributed Lags and Integral Control
by Kaihong Zhao
Axioms 2022, 11(7), 350; https://doi.org/10.3390/axioms11070350 - 21 Jul 2022
Cited by 20 | Viewed by 1303
Abstract
The fractional Langevin equation has more advantages than its classical equation in representing the random motion of Brownian particles in complex viscoelastic fluid. The Mittag–Leffler (ML) fractional equation without singularity is more accurate and effective than Riemann–Caputo (RC) and Riemann–Liouville (RL) fractional equation [...] Read more.
The fractional Langevin equation has more advantages than its classical equation in representing the random motion of Brownian particles in complex viscoelastic fluid. The Mittag–Leffler (ML) fractional equation without singularity is more accurate and effective than Riemann–Caputo (RC) and Riemann–Liouville (RL) fractional equation in portraying Brownian motion. This paper focuses on a nonlinear ML-fractional Langevin system with distributed lag and integral control. Employing the fixed-point theorem of generalised metric space established by Diaz and Margolis, we built the Hyers–Ulam–Rassias (HUR) stability along with Hyers–Ulam (HU) stability of this ML-fractional Langevin system. Applying our main results and MATLAB software, we have carried out theoretical analysis and numerical simulation on an example. By comparing with the numerical simulation of the corresponding classical Langevin system, it can be seen that the ML-fractional Langevin system can better reflect the stationarity of random particles in the statistical sense. Full article
(This article belongs to the Special Issue Numerical Methods for Fractional and Integer PDEs)
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22 pages, 1270 KiB  
Article
TT-M Finite Element Algorithm for the Coupled Schrödinger–Boussinesq Equations
by Jiale Tian, Ziyu Sun, Yang Liu and Hong Li
Axioms 2022, 11(7), 314; https://doi.org/10.3390/axioms11070314 - 28 Jun 2022
Cited by 3 | Viewed by 1382
Abstract
In this article, the coupled Schrödinger–Boussinesq equations are solved numerically using the finite element method combined with the time two-mesh (TT-M) fast algorithm. The spatial direction is discretized by the standard Galerkin finite element method, the temporal direction is approximated by the TT-M [...] Read more.
In this article, the coupled Schrödinger–Boussinesq equations are solved numerically using the finite element method combined with the time two-mesh (TT-M) fast algorithm. The spatial direction is discretized by the standard Galerkin finite element method, the temporal direction is approximated by the TT-M Crank–Nicolson scheme, and then the numerical scheme of TT-M finite element (FE) system is formulated. The method includes three main steps: for the first step, the nonlinear system is solved on the coarse time mesh; for the second step, by an interpolation formula, the numerical solutions at the fine time mesh point are computed based on the numerical solutions on the coarse mesh system; for the last step, the linearized temporal fine mesh system is constructed based on Taylor’s formula for two variables, and then the TT-M FE solutions can be obtained. Furthermore, theory analyses on the TT-M system including the stability and error estimations are conducted. Finally, a large number of numerical examples are provided to verify the accuracy of the algorithm, the correctness of theoretical results, and the computational efficiency with a comparison to the numerical results calculated by using the standard FE method. Full article
(This article belongs to the Special Issue Numerical Methods for Fractional and Integer PDEs)
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