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Advanced Numerical Methods for Differential Equations: Recent Developments, Analysis and...
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Advanced Numerical Methods for Differential Equations: Recent Developments, Analysis and Applications

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

Deadline for manuscript submissions: 31 December 2024 | Viewed by 2904

Special Issue Editor

Prof. Dr. Baccouch Mahboub

E-Mail Website
Guest Editor
Department of Mathematics, University of Nebraska at Omaha, Omaha, NE 68182, USA
Interests: numerical analysis; scientific computing; ordinary, partial, and stochastic differential equations

Special Issue Information

Dear Colleagues,

Numerical methods for differential equations are techniques used to approximate the solutions of differential equations that cannot be solved analytically. They are needed in practice since only a few differential equations can be mathematically solved. The goal of this Special Issue is to provide an overview of the recent progress in using numerical methods to solve differential equations. These include finite element methods and their extensions, including discontinuous Galerkin (DG) methods devoted to approximate the solutions for various real-world problems, such as fluid flow, solid mechanics, electromagnetics, and many others, as well as the analysis of these methods including error estimation, superconvergence, and adaptivity. Other topics include the design and analysis of new numerical schemes, as well as novel applications in any branch of engineering and science. While all contributions related to numerical methods are invited, the featured topics include: stability issues, efficient time integration, superconvergence phenomena, a priori and a posteriori error estimations, and mesh adaptivity. Contributions dealing with the applications of numerical methods for porous media flow, incompressible flow, solid mechanics, and elasticity are welcome.

Prof. Dr. Baccouch Mahboub
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

  • numerical methods
  • numerical analysis
  • scientific computing
  • computational mathematics
  • applied mathematics
  • differential equations
  • finite element methods
  • applications

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Published Papers (3 papers)

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Research

14 pages, 368 KiB  
Article
Fibonacci Wavelet Collocation Method for Solving Dengue Fever SIR Model
by Amit Kumar, Ayub Khan and Abdullah Abdullah
Mathematics 2024, 12(16), 2565; https://doi.org/10.3390/math12162565 - 20 Aug 2024
Cited by 1 | Viewed by 659
Abstract
The main focus in this manuscript is to find a numerical solution of a dengue fever disease model by using the Fibonacci wavelet method. The operational matrix of integration has been obtained using Fibonacci wavelets. The proposed method is called Fibonacci wavelet collocation [...] Read more.
The main focus in this manuscript is to find a numerical solution of a dengue fever disease model by using the Fibonacci wavelet method. The operational matrix of integration has been obtained using Fibonacci wavelets. The proposed method is called Fibonacci wavelet collocation method (FWCM). This biological model has been transformed into a system of nonlinear algebraic equations by using the Fibonacci wavelet collocation scheme. Afterward, this system has been solved by using the Newton–Raphson method. Finally, we provide evidence that our results are better than those obtained by various current approaches, both numerically and graphically, demonstrating the method’s accuracy and efficiency. Full article
(This article belongs to the Special Issue Advanced Numerical Methods for Differential Equations: Recent Developments, Analysis and Applications)
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10 pages, 1473 KiB  
Article
Multi-Soliton, Soliton–Cnoidal, and Lump Wave Solutions for the Supersymmetric Boussinesq Equation
by Peng-Fei Wei, Hao-Bo Zhang, Ye Liu, Si-Yu Lin, Rui-Yu Chen, Zi-Yi Xu, Wan-Li Wang and Bo Ren
Mathematics 2024, 12(13), 2002; https://doi.org/10.3390/math12132002 - 28 Jun 2024
Viewed by 813
Abstract
Based on the bosonization approach, the supersymmetric Boussinesq equation is converted into a coupled bosonic system. The symmetry group and the commutation relations of the corresponding bosonic system are determined through the Lie point symmetry theory. The group invariant solutions of the coupled [...] Read more.
Based on the bosonization approach, the supersymmetric Boussinesq equation is converted into a coupled bosonic system. The symmetry group and the commutation relations of the corresponding bosonic system are determined through the Lie point symmetry theory. The group invariant solutions of the coupled bosonic system are analyzed by the symmetry reduction technique. Special traveling wave solutions are generated by using the mapping and deformation method. Some novel solutions, such as multi-soliton, soliton–cnoidal interaction solutions, and lump waves, are given by utilizing the Hirota bilinear and the consistent tanh expansion methods. The methods in this paper can be effectively expanded to study rich localized waves for other supersymmetric systems. Full article
(This article belongs to the Special Issue Advanced Numerical Methods for Differential Equations: Recent Developments, Analysis and Applications)
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18 pages, 1082 KiB  
Article
Temporal High-Order Accurate Numerical Scheme for the Landau–Lifshitz–Gilbert Equation
by Jiayun He, Lei Yang and Jiajun Zhan
Mathematics 2024, 12(8), 1179; https://doi.org/10.3390/math12081179 - 15 Apr 2024
Viewed by 871
Abstract
In this paper, a family of temporal high-order accurate numerical schemes for the Landau–Lifshitz–Gilbert (LLG) equation is proposed. The proposed schemes are developed utilizing the Gauss–Legendre quadrature method, enabling them to achieve arbitrary high-order time discretization. Furthermore, the geometrical properties of the LLG [...] Read more.
In this paper, a family of temporal high-order accurate numerical schemes for the Landau–Lifshitz–Gilbert (LLG) equation is proposed. The proposed schemes are developed utilizing the Gauss–Legendre quadrature method, enabling them to achieve arbitrary high-order time discretization. Furthermore, the geometrical properties of the LLG equation, such as the preservation of constant magnetization magnitude and the Lyapunov structure, are investigated based on the proposed discrete schemes. It is demonstrated that the magnetization magnitude remains constant with an error of (2p+3) order in time when utilizing a (2p+2)th-order discrete scheme. Additionally, the preservation of the Lyapunov structure is achieved with a second-order error in the temporal step size. Numerical experiments and simulations effectively verify the performance of our proposed algorithm and validate our theoretical analysis. Full article
(This article belongs to the Special Issue Advanced Numerical Methods for Differential Equations: Recent Developments, Analysis and Applications)
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