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Numerical Solution of Differential Equations and Their Applications
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Numerical Solution of Differential Equations and Their Applications

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

Deadline for manuscript submissions: 10 October 2025 | Viewed by 7325

Special Issue Editors

Dr. Zacharias Anastassi

E-Mail Website
Guest Editor
Institute of Artificial Intelligence, School of Computer Science and Informatics, De Montfort University, Leicester LE1 9BH, UK
Interests: numerical analysis of differential equations; oscillatory/periodic initial/boundary value problems; computational physics
Dr. Athinoula A. Kosti

E-Mail Website
Guest Editor
Institute of Artificial Intelligence, School of Computer Science and Informatics, De Montfort University, Leicester LE1 9BH, UK
Interests: numerical analysis; development and analysis of numerical algorithms; numerical solution of initial/boundary value problems
Dr. Mufutau Ajani Rufai

E-Mail Website
Guest Editor
Faculty of Engineering, Free University of Bozen-Bolzano, 39100 Bolzano, Italy
Interests: numerical analysis; differential equations and applications; applied fluid dynamics; computational mathematics; optimization

Special Issue Information

Dear Colleagues,

"Numerical Solution of Differential Equations and Their Applications" is an upcoming Special Issue of Mathematics that aims to explore recent developments in numerical methods for solving differential equations, and their applications in various fields of science and engineering.

The topics of interest for this Special Issue include, but are not limited to:

  1. Numerical methods for solving ordinary differential equations (ODEs), including Runge–Kutta methods, multistep methods, collocation methods, etc.;
  2. Numerical methods for solving partial differential equations (PDEs), including finite element methods, finite difference methods, spectral methods, etc.;
  3. High-order numerical methods for differential equations, including spectral methods, spectral collocation methods, finite difference methods, etc.;
  4. Error analysis and convergence of numerical methods for differential equations, etc.;
  5. Applications of numerical methods for differential equations, including fluid dynamics, solid mechanics, electromagnetics, and other fields of science and engineering, etc.;
  6. Adaptive numerical methods for differential equations, including adaptive mesh refinement and adaptive time-stepping, etc.

This Special Issue invites original research articles and review articles that present novel ideas and developments in the numerical solution of differential equations and/or their applications in science and engineering. The aim is to provide a platform for researchers to share their latest findings and to stimulate further research in this exciting field. The guest editors welcome submissions that address the above topics or related areas and provide new insights and solutions to the challenges faced by researchers and practitioners.

Dr. Zacharias Anastassi
Dr. Athinoula A. Kosti
Dr. Mufutau Ajani Rufai
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

  • numerical methods
  • differential equations
  • error analysis
  • convergence
  • applications

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Published Papers (6 papers)

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Research

24 pages, 1587 KiB  
Article
Chaos in Inverse Parallel Schemes for Solving Nonlinear Engineering Models
by Mudassir Shams and Bruno Carpentieri
Mathematics 2025, 13(1), 67; https://doi.org/10.3390/math13010067 - 27 Dec 2024
Viewed by 469
Abstract
Nonlinear equations are essential in research and engineering because they simulate complicated processes such as fluid dynamics, chemical reactions, and population growth. The development of advanced methods to address them becomes essential for scientific and applied research enhancements, as their resolution influences innovations [...] Read more.
Nonlinear equations are essential in research and engineering because they simulate complicated processes such as fluid dynamics, chemical reactions, and population growth. The development of advanced methods to address them becomes essential for scientific and applied research enhancements, as their resolution influences innovations by aiding in the proper prediction or optimization of the system. In this research, we develop a novel biparametric family of inverse parallel techniques designed to improve stability and accelerate convergence in parallel iterative algorithm. Bifurcation and chaos theory were used to find the best parameter regions that increase the parallel method’s effectiveness and stability. Our newly developed biparametric family of parallel techniques is more computationally efficient than current approaches, as evidenced by significant reductions in the number of iterations and basic operations each iterations step for solving nonlinear equations. Engineering applications examined with rough beginning data demonstrate high accuracy and superior convergence compared to existing classical parallel schemes. Analysis of global convergence further shows that the proposed methods outperform current methods in terms of error control, computational time, percentage convergence, number of basic operations per iteration, and computational order. These findings indicate broad usage potential in engineering and scientific computation. Full article
(This article belongs to the Special Issue Numerical Solution of Differential Equations and Their Applications)
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20 pages, 3306 KiB  
Article
Numerical Study of the Thermal Energy Storage Container Shape Impact on the NePCM Melting Process
by Obai Younis, Naef A. A. Qasem, Aissa Abderrahmane and Abdeldjalil Belazreg
Mathematics 2024, 12(24), 3954; https://doi.org/10.3390/math12243954 - 16 Dec 2024
Viewed by 512
Abstract
Recently, thermal energy storage has emerged as one of the alternative solutions to increase energy efficiency. The geometry of a thermal energy storage container holds a significant role in increasing the heat transmission rates in the container. In this article, we examined the [...] Read more.
Recently, thermal energy storage has emerged as one of the alternative solutions to increase energy efficiency. The geometry of a thermal energy storage container holds a significant role in increasing the heat transmission rates in the container. In this article, we examined the influence of the inner and outer tube shapes of a shell and tube LHTES on the thermal activity within the system. The gap between the inner and outer tube is loaded with nano-enhanced phase change material (NePCM); hot fluid is passed through the inner tube while the outer tube is insulated. COMSOL commercial software (version 6.2) was used to numerically simulate the NePCM melting process. Mainly, six different geometries were investigated with inner or outer tubes with trefoil, cinquefoil, and heptafoil shapes. The influences of nanoparticles volumetric fraction (ϕ = 0–8%) were also discussed. The findings are displayed and discussed in terms of the average Nusselt number, the average liquid fraction, the total energy generation, and the average temperature. The findings showed that the melting process is highly affected by the shape of the inner tube and ϕ, while the outer tube shape impact is less important. It was noticed that employing an inner tube with a trefoil improved the melting process by more than 25% while increasing the ϕ from 0 to 8% resulted in reducing the melting time by up to 20%. Full article
(This article belongs to the Special Issue Numerical Solution of Differential Equations and Their Applications)
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27 pages, 11433 KiB  
Article
Numerical Study of Shock Wave Interaction with V-Shaped Heavy/Light Interface
by Salman Saud Alsaeed and Satyvir Singh
Mathematics 2024, 12(19), 3131; https://doi.org/10.3390/math12193131 - 7 Oct 2024
Cited by 1 | Viewed by 868
Abstract
This paper investigates numerically the shock wave interaction with a V-shaped heavy/light interface. For numerical simulations, we choose six distinct vertex angles (θ=40∘,60∘,90∘,120∘,150∘, and [...] Read more.
This paper investigates numerically the shock wave interaction with a V-shaped heavy/light interface. For numerical simulations, we choose six distinct vertex angles (θ=40∘,60∘,90∘,120∘,150∘, and 170∘), five distinct shock wave strengths (Ms=1.12,1.22,1.30,1.60, and 2.0), and three different Atwood numbers (At=−0.32,−0.77, and −0.87). A two-dimensional space of compressible two-component Euler equations are solved using a third-order modal discontinuous Galerkin approach for the simulations. The present findings demonstrate that the vertex angle has a crucial influence on the shock wave interaction with the V-shaped heavy/light interface. The vertex angle significantly affects the flow field, interface deformation, wave patterns, spike generation, and vorticity production. As the vertex angle decreases, the vorticity production becomes more dominant. A thorough analysis of the vertex angle effect identifies the factors that propel the creation of vorticity during the interaction phase. Notably, smaller vertex angles lead to stronger vorticity generation due to a steeper density gradient, while larger angles result in weaker, more dispersed vorticity and a less complex interaction. Moreover, kinetic energy and enstrophy both dramatically rise with decreasing vortex angles. A detailed analysis is also carried out to analyze the vertex angle effects on the temporal variations of interface features. Finally, the impacts of different Mach and Atwood numbers on the V-shaped interface are briefly presented. Full article
(This article belongs to the Special Issue Numerical Solution of Differential Equations and Their Applications)
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23 pages, 9123 KiB  
Article
Modal Discontinuous Galerkin Simulations of Richtmyer–Meshkov Instability at Backward-Triangular Bubbles: Insights and Analysis
by Salman Saud Alsaeed and Satyvir Singh
Mathematics 2024, 12(13), 2005; https://doi.org/10.3390/math12132005 - 28 Jun 2024
Cited by 3 | Viewed by 864
Abstract
This paper investigates the dynamics of Richtmyer–Meshkov instability (RMI) in shocked backward-triangular bubbles through numerical simulations. Two distinct gases, He and SF6, are used within the backward-triangular bubble, surrounded by N2 gas. Simulations are conducted at two distinct strengths of [...] Read more.
This paper investigates the dynamics of Richtmyer–Meshkov instability (RMI) in shocked backward-triangular bubbles through numerical simulations. Two distinct gases, He and SF6, are used within the backward-triangular bubble, surrounded by N2 gas. Simulations are conducted at two distinct strengths of incident shock wave, including Ms=1.25 and 1.50. A third-order modal discontinuous Galerkin (DG) scheme is applied to simulate a physical conservation laws of two-component gas flows in compressible inviscid framework. Hierarchical Legendre modal polynomials are employed for spatial discretization in the DG platform. This scheme reduces the conservation laws into a semi-discrete set of ODEs in time, which is then solved using an explicit 3rd-order SSP Runge–Kutta scheme. The results reveal significant effects of bubble density and Mach numbers on the growth of RMI in the shocked backward-triangular bubble, a phenomenon not previously reported. These effects greatly influence flow patterns, leading to intricate wave formations, shock focusing, jet generation, and interface distortion. Additionally, a detailed analysis elucidates the mechanisms driving vorticity formation during the interaction process. The study also thoroughly examines these effects on the flow fields based on various integral quantities and interface characteristics. Full article
(This article belongs to the Special Issue Numerical Solution of Differential Equations and Their Applications)
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25 pages, 4246 KiB  
Article
Investigating the Dynamic Behavior of Integer and Noninteger Order System of Predation with Holling’s Response
by Kolade M. Owolabi, Sonal Jain and Edson Pindza
Mathematics 2024, 12(10), 1530; https://doi.org/10.3390/math12101530 - 14 May 2024
Cited by 1 | Viewed by 1145
Abstract
The paper’s primary objective is to examine the dynamic behavior of an integer and noninteger predator–prey system with a Holling type IV functional response in the Caputo sense. Our focus is on understanding how harvesting influences the stability, equilibria, bifurcations, and limit cycles [...] Read more.
The paper’s primary objective is to examine the dynamic behavior of an integer and noninteger predator–prey system with a Holling type IV functional response in the Caputo sense. Our focus is on understanding how harvesting influences the stability, equilibria, bifurcations, and limit cycles within this system. We employ qualitative and quantitative analysis methods rooted in bifurcation theory, dynamical theory, and numerical simulation. We also delve into studying the boundedness of solutions and investigating the stability and existence of equilibrium points within the system. Leveraging Sotomayor’s theorem, we establish the presence of both the saddle-node and transcritical bifurcations. The analysis of the Hopf bifurcation is carried out using the normal form theorem. The model under consideration is extended to the fractional reaction–diffusion model which captures non-local and long-range effects more accurately than integer-order derivatives. This makes fractional reaction–diffusion systems suitable for modeling phenomena with anomalous diffusion or memory effects, improving the fidelity of simulations in turn. An adaptable numerical technique for solving this class of differential equations is also suggested. Through simulation results, we observe that one of the Lyapunov exponents has a negative value, indicating the potential for the emergence of a stable-limit cycle via bifurcation as well as chaotic and complex spatiotemporal distributions. We supplement our analytical investigations with numerical simulations to provide a comprehensive understanding of the system’s behavior. It was discovered that both the prey and predator populations will continue to coexist and be permanent, regardless of the choice of fractional parameter. Full article
(This article belongs to the Special Issue Numerical Solution of Differential Equations and Their Applications)
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13 pages, 3770 KiB  
Article
A New Two-Step Hybrid Block Method for the FitzHugh–Nagumo Model Equation
by Mufutau Ajani Rufai, Athinoula A. Kosti, Zacharias A. Anastassi and Bruno Carpentieri
Mathematics 2024, 12(1), 51; https://doi.org/10.3390/math12010051 - 23 Dec 2023
Cited by 1 | Viewed by 1144
Abstract
This paper presents an efficient two-step hybrid block method (ETHBM) to obtain an approximate solution to the FitzHugh–Nagumo problem. The considered partial differential equation model problems are semi-discretized, reducing them to equivalent ordinary differential equations using the method of lines. In order to [...] Read more.
This paper presents an efficient two-step hybrid block method (ETHBM) to obtain an approximate solution to the FitzHugh–Nagumo problem. The considered partial differential equation model problems are semi-discretized, reducing them to equivalent ordinary differential equations using the method of lines. In order to evaluate the effectiveness of the proposed ETHBM, three numerical examples are presented and compared with the results obtained through existing methods. The results demonstrate that the proposed ETHBM produces more efficient results than some other numerical approaches in the literature. Full article
(This article belongs to the Special Issue Numerical Solution of Differential Equations and Their Applications)
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