Submit to Special Issue Submit Abstract to Special Issue Review for Mathematics Propose a Special Issue

Journal Menu

Journal Browser

Numerical Methods and Applications for Differential Problems

Print Special Issue Flyer
Special Issue Editors
Special Issue Information
Keywords
Published Papers

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

Deadline for manuscript submissions: 30 April 2024 | Viewed by 7984

Share This Special Issue

Special Issue Editors

Dr. Justin B. Munyakazi

E-Mail Website
Guest Editor

Department of Mathematics and Applied Mathematics, University of the Western Cape, Cape Town, South Africa
Interests: numerical methods for differential problems; singularly perturbed problems and mathematical modeling

Prof. Dr. Gemechis File Duressa

E-Mail Website
Guest Editor

College of Natural Sciences, Jimma University, Jimma P.O. Box 378, Ethiopia
Interests: numerical methods for singularly perturbed problems; mathematical modeling

Prof. Dr. Musa Cakir

E-Mail Website
Guest Editor

Department of Mathematics, Faculty of Science, University of Van Yuzuncu Yil, Van, Turkey
Interests: numerical analysis and singularly perturbed problems

Special Issue Information

Dear Colleagues,

Differential equations arise in several research domains of applied sciences and engineering. These include but are not limited to fluid dynamics, heat conduction with mixed boundary conditions, population dynamics, and epidemiology. In most cases, it is difficult, if not impossible, to determine analytical solutions to the models listed above. Often, researchers have recourse to semi-analytical and numerical methods. However, obtaining a reliable semi-analytical solution involves a number of different elements and can become cumbersome. This difficulty may become more pronounced in the case of singularly perturbed problems. These includes a class of parameter-dependent problems and those presenting multiscale phenomena.

This Special Issue provides a platform where novel numerical methods that can be used to solve ordinary and partial differential equations as well as integro-differential equations will be presented. The stability and accuracy of those methods will be analyzed. Methods will be implemented on applied problems in various domains of science and engineering.

We solicit high-quality original research papers on numerical methods for differential problems, including the singularly perturbed ones. Survey articles are also welcome.

Dr. Justin B. Munyakazi
Prof. Dr. Gemechis File Duressa
Prof. Dr. Musa Cakir
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

singularly perturbed problems
ordinary differential equations
partial differential equations
integrodifferential equations
fitted numerical methods
boundary layers
interior layers
layer adapted meshes
uniform convergence
error bounds
convergence analysis
higher-order methods

Published Papers (6 papers)

Download All Papers

Order results

Result details

Show export options Show export options

Select all

Export citation of selected articles as:

Research

Jump to: Review, Other

12 pages, 402 KiB

Open AccessArticle

The Numerical Solution of the External Dirichlet Generalized Harmonic Problem for a Sphere by the Method of Probabilistic Solution

by Mamuli Zakradze, Zaza Tabagari, Nana Koblishvili, Tinatin Davitashvili, Jose Maria Sanchez and Francisco Criado-Aldeanueva

Mathematics 2023, 11(3), 539; https://doi.org/10.3390/math11030539 - 19 Jan 2023

Cited by 1 | Viewed by 1149

Abstract

In the present paper, an algorithm for the numerical solution of the external Dirichlet generalized harmonic problem for a sphere by the method of probabilistic solution (MPS) is given, where “generalized” indicates that a boundary function has a finite number of first kind discontinuity curves. The algorithm consists of the following main stages: (1) the transition from an infinite domain to a finite domain by an inversion; (2) the consideration of a new Dirichlet generalized harmonic problem on the basis of Kelvin’s theorem for the obtained finite domain; (3) the numerical solution of the new problem for the finite domain by the MPS, which in turn is based on a computer simulation of the Weiner process; (4) finding the probabilistic solution of the posed generalized problem at any fixed points of the infinite domain by the solution of the new problem. For illustration, numerical examples are considered and results are presented. Full article

(This article belongs to the Special Issue Numerical Methods and Applications for Differential Problems)

► Show Figures

Figure 1

36 pages, 3421 KiB

Open AccessArticle

On the Numerical Solution of 1D and 2D KdV Equations Using Variational Homotopy Perturbation and Finite Difference Methods

by Abey Sherif Kelil and Appanah Rao Appadu

Mathematics 2022, 10(23), 4443; https://doi.org/10.3390/math10234443 - 24 Nov 2022

Cited by 4 | Viewed by 1376

Abstract

The KdV equation has special significance as it describes various physical phenomena. In this paper, we use two methods, namely, a variational homotopy perturbation method and a classical finite-difference method, to solve 1D and 2D KdV equations with homogeneous and non-homogeneous source terms by considering five numerical experiments with initial and boundary conditions. The variational homotopy perturbation method is a semi-analytic technique for handling linear as well as non-linear problems. We derive classical finite difference methods to solve the five numerical experiments. We compare the performance of the two classes of methods for these numerical experiments by computing absolute and relative errors at some spatial nodes for short, medium and long time propagation. The logarithm of maximum error vs. time from the numerical methods is also obtained for the experiments undertaken. The stability and consistency of the finite difference scheme is obtained. To the best of our knowledge, a comparison between the variational homotopy perturbation method and the classical finite difference method to solve these five numerical experiments has not been undertaken before. The ideal extension of this work would be an application of the employed methods for fractional and stochastic KdV type equations and their variants. Full article

(This article belongs to the Special Issue Numerical Methods and Applications for Differential Problems)

► Show Figures

Figure 1

21 pages, 8718 KiB

Open AccessArticle

Novel Soliton Solutions of the Fractional Riemann Wave Equation via a Mathematical Method

by Shumaila Naz, Attia Rani, Muhammad Shakeel, Nehad Ali Shah and Jae Dong Chung

Mathematics 2022, 10(22), 4171; https://doi.org/10.3390/math10224171 - 08 Nov 2022

Viewed by 1195

Abstract

The Riemann wave equation is an intriguing nonlinear equation in the areas of tsunamis and tidal waves in oceans, electromagnetic waves in transmission lines, magnetic and ionic sound radiations in plasmas, static and uniform media, etc. In this innovative research, the analytical solutions of the fractional Riemann wave equation with a conformable derivative were retrieved as a special case, and broad-spectrum solutions with unknown parameters were established with the improved (G’/G)-expansion method. For the various values of these unknown parameters, the renowned periodic, singular, and anti-singular kink-shaped solitons were retrieved. Using the Maple software, we investigated the solutions by drawing the 3D, 2D, and contour plots created to analyze the dynamic behavior of the waves. The discovered solutions might be crucial in the disciplines of science and ocean engineering. Full article

(This article belongs to the Special Issue Numerical Methods and Applications for Differential Problems)

► Show Figures

Figure 1

Review

Jump to: Research, Other

16 pages, 845 KiB

Open AccessReview

A Systematic Review on the Solution Methodology of Singularly Perturbed Differential Difference Equations

by Gemechis File Duressa, Imiru Takele Daba and Chernet Tuge Deressa

Mathematics 2023, 11(5), 1108; https://doi.org/10.3390/math11051108 - 22 Feb 2023

Cited by 3 | Viewed by 1469

Abstract

This review paper contains computational methods or solution methodologies for singularly perturbed differential difference equations with negative and/or positive shifts in a spatial variable. This survey limits its coverage to singular perturbation equations arising in the modeling of neuronal activity and the methods developed by numerous researchers between 2012 and 2022. The review covered singularly perturbed ordinary delay differential equations with small or large negative shift(s), singularly perturbed ordinary differential–differential equations with mixed shift(s), singularly perturbed delay partial differential equations with small or large negative shift(s) and singularly perturbed partial differential–difference equations of the mixed type. The main aim of this review is to find out what numerical and asymptotic methods were developed in the last ten years to solve such problems. Further, it aims to stimulate researchers to develop new robust methods for solving families of the problems under consideration. Full article

(This article belongs to the Special Issue Numerical Methods and Applications for Differential Problems)

► Show Figures

Figure 1

19 pages, 364 KiB

Open AccessReview

A Fitted Operator Finite Difference Approximation for Singularly Perturbed Volterra–Fredholm Integro-Differential Equations

by Musa Cakir and Baransel Gunes

Mathematics 2022, 10(19), 3560; https://doi.org/10.3390/math10193560 - 29 Sep 2022

Cited by 7 | Viewed by 1389 | Correction

Abstract

This paper presents a ε-uniform and reliable numerical scheme to solve second-order singularly perturbed Volterra–Fredholm integro-differential equations. Some properties of the analytical solution are given, and the finite difference scheme is established on a non-uniform mesh by using interpolating quadrature rules and the linear basis functions. An error analysis is successfully carried out on the Boglaev–Bakhvalov-type mesh. Some numerical experiments are included to authenticate the theoretical findings. In this regard, the main advantage of the suggested method is to yield stable results on layer-adapted meshes. Full article

(This article belongs to the Special Issue Numerical Methods and Applications for Differential Problems)

► Show Figures