Numerical Methods for Solving Differential Problems-II

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

Deadline for manuscript submissions: closed (31 January 2024) | Viewed by 4884

Special Issue Editor

1. Scientific Computing Group, Universidad de Salamanca, Plaza de la Merced, 37008 Salamanca, Spain
2. Escuela Politécnica Superior de Zamora, Universidad de Salamanca, Campus Viriato, 49029 Zamora, Spain
Interests: numerical solution of differential equations; numerical analysis
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Special Issue Information

Dear Colleagues,

It is well known that differential equations are present in numerous fields of science, engineering, economics, and many more. Many physical phenomena are modeled through differential equations, which have the appealing aspect of describing the world around us. There are many different types of such equations beyond the simple classifications between ordinary/partial or linear/nonlinear. Among other types, one can find singular, singularly perturbed, delay, integral or algebraic differential equations. If we add to the differential equation certain conditions that the solution must meet at one or more points, we will have initial value problems or boundary value problems. The combination of the different possibilities above gives rise to a wide catalog of differential problems, for which, in most cases, we cannot find an analytical solution.

The aim of this Special Issue is to update the numerical techniques for solving differential problems in a broad sense, with an emphasis on real-world applications. Contributions that involve a review of the state of the art for the numerical resolution of different types of differential problems will be welcome. As there are other Special Issues in this series devoted to fractional problems and numerical solution of partial differential equations, we have not included this kind of problems here.

Prof. Dr. Higinio Ramos
Guest Editor

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Keywords

  • numerical methods for differential equations
  • ordinary differential equations
  • initial value problems
  • boundary value problems
  • singular differential equations
  • singularly perturbed problems
  • delay differential equations
  • algebraic differential equations

Published Papers (4 papers)

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Research

16 pages, 544 KiB  
Article
Derivation of Three-Derivative Two-Step Runge–Kutta Methods
by Xueyu Qin, Jian Yu and Chao Yan
Mathematics 2024, 12(5), 711; https://doi.org/10.3390/math12050711 - 28 Feb 2024
Viewed by 400
Abstract
In this paper, we develop explicit three-derivative two-step Runge–Kutta (ThDTSRK) schemes, and propose a simpler general form of the order accuracy conditions (p7) by Albrecht’s approach, compared to the order conditions in terms of rooted trees. The parameters of [...] Read more.
In this paper, we develop explicit three-derivative two-step Runge–Kutta (ThDTSRK) schemes, and propose a simpler general form of the order accuracy conditions (p7) by Albrecht’s approach, compared to the order conditions in terms of rooted trees. The parameters of the general high-order ThDTSRK methods are determined by utilizing the order conditions. We establish a theory for the A-stability property of ThDTSRK methods and identify optimal stability coefficients. Moreover, ThDTSRK methods can achieve the intended order of convergence using fewer stages than other schemes, making them cost-effective for solving the ordinary differential equations. Full article
(This article belongs to the Special Issue Numerical Methods for Solving Differential Problems-II)
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8 pages, 486 KiB  
Article
Solving SIVPs of Lane–Emden–Fowler Type Using a Pair of Optimized Nyström Methods with a Variable Step Size
by Mufutau Ajani Rufai and Higinio Ramos
Mathematics 2023, 11(6), 1535; https://doi.org/10.3390/math11061535 - 22 Mar 2023
Cited by 2 | Viewed by 1132
Abstract
This research article introduces an efficient method for integrating Lane–Emden–Fowler equations of second-order singular initial value problems (SIVPs) using a pair of hybrid block methods with a variable step-size mode. The method pairs an optimized Nyström technique with a set of formulas applied [...] Read more.
This research article introduces an efficient method for integrating Lane–Emden–Fowler equations of second-order singular initial value problems (SIVPs) using a pair of hybrid block methods with a variable step-size mode. The method pairs an optimized Nyström technique with a set of formulas applied at the initial step to circumvent the singularity at the beginning of the interval. The variable step-size formulation is implemented using an embedded-type approach, resulting in an efficient technique that outperforms its counterpart methods that used fixed step-size implementation. The numerical simulations confirm the better performance of the variable step-size implementation. Full article
(This article belongs to the Special Issue Numerical Methods for Solving Differential Problems-II)
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13 pages, 5784 KiB  
Article
Dynamics and Global Bifurcations in Two Symmetrically Coupled Non-Invertible Maps
by Yamina Soula, Hadi Jahanshahi, Abdullah A. Al-Barakati and Irene Moroz
Mathematics 2023, 11(6), 1517; https://doi.org/10.3390/math11061517 - 21 Mar 2023
Viewed by 1053
Abstract
The theory of critical curves determines the main characteristics of a discrete dynamical system in two dimensions. One important property that has garnered recent attention is the problem of chaos synchronization, along with the location of its chaotic attractors, basin boundaries, and bifurcation [...] Read more.
The theory of critical curves determines the main characteristics of a discrete dynamical system in two dimensions. One important property that has garnered recent attention is the problem of chaos synchronization, along with the location of its chaotic attractors, basin boundaries, and bifurcation mechanisms. Varying the parameters of the maps reveals the instrumental role that these curves play, where the bifurcation leads to complex topological structures of the basins occurs by contact with the basin boundaries, resulting in the appearance or disappearance of some components of the basin. This study focuses on the properties of a discrete dynamical system consisting of two symmetrically coupled non-invertible maps, specifically those with an invariant one-dimensional submanifold (or one-dimensional maps). These maps exhibit a complex structure of basins with the coexistence of symmetric chaotic attractors, riddled basins, blow-out, on-off intermittency, and, most significantly, the appearance of chaotic synchronization with a correlation between all the characteristics. The numerical method of critical curves can be used to demonstrate a wide range of dynamic scenarios and explain the bifurcations that lead to their occurrence. These curves play a crucial role in a system of two symmetrically coupled maps, and their significance will be discussed. Full article
(This article belongs to the Special Issue Numerical Methods for Solving Differential Problems-II)
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6 pages, 242 KiB  
Article
Solution Spaces Associated to Continuous or Numerical Models for Which Integrable Functions Are Bounded
by Brian Villegas-Villalpando, Jorge E. Macías-Díaz and Qin Sheng
Mathematics 2022, 10(11), 1936; https://doi.org/10.3390/math10111936 - 06 Jun 2022
Viewed by 1173
Abstract
Boundedness is an essential feature of the solutions for various mathematical and numerical models in the natural sciences, especially those systems in which linear or nonlinear preservation or stability features are fundamental. In those cases, the boundedness of the solutions outside a set [...] Read more.
Boundedness is an essential feature of the solutions for various mathematical and numerical models in the natural sciences, especially those systems in which linear or nonlinear preservation or stability features are fundamental. In those cases, the boundedness of the solutions outside a set of zero measures is not enough to guarantee that the solutions are physically relevant. In this note, we will establish a criterion for the boundedness of integrable solutions of general continuous and numerical systems. More precisely, we establish a characterization of those measures over arbitrary spaces for which real-valued integrable functions are necessarily bounded at every point of the domain. The main result states that the collection of measures for which all integrable functions are everywhere bounded are exactly all of those measures for which the infimum of the measures for nonempty sets is a positive extended real number. Full article
(This article belongs to the Special Issue Numerical Methods for Solving Differential Problems-II)
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