Advanced Numerical Methods for Solving Differential Equations with Applications in Science and Engineering

A special issue of Computation (ISSN 2079-3197).

Deadline for manuscript submissions: 31 August 2024 | Viewed by 3777

Special Issue Editors


E-Mail Website
Guest Editor
Institute of Physics and Electrical Engineering, University of Miskolc, 3515 Miskolc, Hungary
Interests: heat transfer; diffusion-reaction equations; explicit and stable algorithms

E-Mail Website
Guest Editor
Youth Research Institute, Saint Petersburg Electrotechnical University “LETI”, 5, Professora Popova st., 197376 Saint Petersburg, Russia
Interests: numerical integration; symplectic integrators; symmetric methods; adaptive methods; chaotic systems; nonlinear dynamics
Special Issues, Collections and Topics in MDPI journals

E-Mail Website
Guest Editor
Saint Petersburg Electrotechnical University “LETI”, ul. Professora Popova 5, 197376 Saint Petersburg, Russia
Interests: nonlinear systems; discrete maps; memristors; neuron simulation; coupled oscillators; system identification

Special Issue Information

Dear Colleagues,

Numerical integration methods are an essential tool for implementing continuous systems described by differential equations in digital computers.

The developments in this field of mathematics have resulted in several large classes of so-called differential equation solvers, including single- and multistep algorithms with fixed and adaptive timesteps, as well as various space discretization methods, e.g., finite element analysis. In some areas of science and engineering, numerical simulation is the only way of obtaining reliable solutions. The choice of the numerical method and the simulation parameters for a certain problem is still a challenging task, which is commonly solved empirically. The time-efficient and reliable solution of stiff and/or nonlinear non-stationary equations with the required accuracy is a crucial goal for scholars in the field. One of the key branches in numerical methods is special solvers for nonlinear, e.g., chaotic systems, which are highly sensitive to all simulation parameters, including discretization operator type. A better understanding of the impact that numerical methods have on the properties of finite-difference models of continuous nonlinear systems could allow researchers to reveal the limits of computer simulation and predictability of nonlinear dynamics.

This Special Issue is dedicated to advanced numerical integration techniques, including single- and multistep methods, collocation methods, and composition and extrapolation ODE and PDE solvers. New finite-difference schemes for solving PDEs, adaptive stepsize control methods, parallelization on GPUs, and various applications of numerical simulation are also of interest.

Papers may report on original research in numerical integration, discuss methodological aspects, review the current state of the art, or offer perspectives on future prospects.

Specific methods and fields of applications include, but are not limited to:

  • Symplectic and symmetric integrators;
  • Hardware-targeted methods;
  • Multistep ODE solvers;
  • Semi-implicit integration and composition methods;
  • Nonlinear integration methods;
  • Step-control techniques and truncation error estimation;
  • Special techniques for solving nonlinear systems;
  • Numerical methods of solving time-dependent PDEs;
  • Large-scale and multiscale PDEs;
  • PDEs in spatially inhomogeneous media.

Papers on fractional, perturbed, or delay differential equations are welcomed only if they clearly show a direct application in science or engineering.

Manuscripts using only traditional methods are of interest only if they conduct an extensive comparison of the performance of different methods.   

Dr. Endre Kovács
Dr. Denis Butusov
Dr. Valerii Ostrovskii
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. Computation 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 1800 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 integration
  • differential equations
  • finite-difference methods
  • finite element method
  • stepsize control
  • partial differential equations
  • stability
  • local truncation error estimation
  • hardware-targeted methods
  • symmetric integration
  • symplectic integrator
  • stiff equations

Published Papers (3 papers)

Order results
Result details
Select all
Export citation of selected articles as:

Research

21 pages, 1582 KiB  
Article
Application of an Extended Cubic B-Spline to Find the Numerical Solution of the Generalized Nonlinear Time-Fractional Klein–Gordon Equation in Mathematical Physics
by Miguel Vivas-Cortez, M. J. Huntul, Maria Khalid, Madiha Shafiq, Muhammad Abbas and Muhammad Kashif Iqbal
Computation 2024, 12(4), 80; https://doi.org/10.3390/computation12040080 - 11 Apr 2024
Viewed by 805
Abstract
A B-spline function is a series of flexible elements that are managed by a set of control points to produce smooth curves. By using a variety of points, these functions make it possible to build and maintain complicated shapes. Any spline function of [...] Read more.
A B-spline function is a series of flexible elements that are managed by a set of control points to produce smooth curves. By using a variety of points, these functions make it possible to build and maintain complicated shapes. Any spline function of a certain degree can be expressed as a linear combination of the B-spline basis of that degree. The flexibility, symmetry and high-order accuracy of the B-spline functions make it possible to tackle the best solutions. In this study, extended cubic B-spline (ECBS) functions are utilized for the numerical solutions of the generalized nonlinear time-fractional Klein–Gordon Equation (TFKGE). Initially, the Caputo time-fractional derivative (CTFD) is approximated using standard finite difference techniques, and the space derivatives are discretized by utilizing ECBS functions. The stability and convergence analysis are discussed for the given numerical scheme. The presented technique is tested on a variety of problems, and the approximate results are compared with the existing computational schemes. Full article
Show Figures

Figure 1

18 pages, 3048 KiB  
Article
Extension of Cubic B-Spline for Solving the Time-Fractional Allen–Cahn Equation in the Context of Mathematical Physics
by Mubeen Fatima, Ravi P. Agarwal, Muhammad Abbas, Pshtiwan Othman Mohammed, Madiha Shafiq and Nejmeddine Chorfi
Computation 2024, 12(3), 51; https://doi.org/10.3390/computation12030051 - 5 Mar 2024
Cited by 1 | Viewed by 1262
Abstract
A B-spline is defined by the degree and quantity of knots, and it is observed to provide a higher level of flexibility in curve and surface layout. The extended cubic B-spline (ExCBS) functions with new approximation for second derivative and finite difference technique [...] Read more.
A B-spline is defined by the degree and quantity of knots, and it is observed to provide a higher level of flexibility in curve and surface layout. The extended cubic B-spline (ExCBS) functions with new approximation for second derivative and finite difference technique are incorporated in this study to solve the time-fractional Allen–Cahn equation (TFACE). Initially, Caputo’s formula is used to discretize the time-fractional derivative, while a new ExCBS is used for the spatial derivative’s discretization. Convergence analysis is carried out and the stability of the proposed method is also analyzed. The scheme’s applicability and feasibility are demonstrated through numerical analysis. Full article
Show Figures

Figure 1

19 pages, 17607 KiB  
Article
Systematic Investigation of the Explicit, Dynamically Consistent Methods for Fisher’s Equation
by Husniddin Khayrullaev, Issa Omle and Endre Kovács
Computation 2024, 12(3), 49; https://doi.org/10.3390/computation12030049 - 4 Mar 2024
Viewed by 1168
Abstract
We systematically investigate the performance of numerical methods to solve Fisher’s equation, which contains a linear diffusion term and a nonlinear logistic term. The usual explicit finite difference algorithms are only conditionally stable for this equation, and they can yield concentrations below zero [...] Read more.
We systematically investigate the performance of numerical methods to solve Fisher’s equation, which contains a linear diffusion term and a nonlinear logistic term. The usual explicit finite difference algorithms are only conditionally stable for this equation, and they can yield concentrations below zero or above one, even if they are stable. Here, we collect the stable and explicit algorithms, most of which we invented recently. All of them are unconditionally dynamically consistent for Fisher’s equation; thus, the concentration remains in the unit interval for arbitrary parameters. We perform tests in the cases of 1D and 2D systems to explore how the errors depend on the coefficient of the nonlinear term, the stiffness ratio, and the anisotropy of the system. We also measure running times and recommend which algorithms should be used in specific circumstances. Full article
Show Figures

Figure 1

Back to TopTop