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Axioms
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Higher Order Differential Equations
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Higher Order Differential Equations

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: closed (30 November 2022) | Viewed by 8610

Special Issue Editor

Dr. Carlos Escudero

E-Mail Website
Guest Editor
Departamento de Matemáticas Fundamentales, Universidad Nacional de Educación a Distancia, 28040 Madrid, Spain
Interests: stochastic analysis; partial differential equations; mathematical statistical mechanics

Special Issue Information

Dear Colleagues,

This Special Issue is devoted to the theory and applications of higher-order differential equations. The last decades have shown an impressive advancement in the mathematical analysis of these equations. Issues like well-posedness, uniqueness and multiplicity of solutions, finite-time blow-up, global existence, and the positivity and regularity of solutions have been addressed in many different models. At the same time, mathematical modelling has shown the presence of newer equations of this type in an increasing number of fields, among which one finds geometry, physics, engineering, etc. This fact has in turn demanded more progress on the mathematical analytical side. Overall, the current situation is multifaceted and has benefited from interdisciplinarity. Abstract developments and mathematical analyses of particular models has deepened our theoretical knowledge of these equations. Applied research has proven the utility of such advances in areas both within and outside mathematics, and also contributed to expand the theoretical framework to include topics like control theory and numerical analysis. Physical scientists and engineers, among others, have found new higher-order equations that combine their interest in the specific field of application with an appealing mathematical structure. This Special Issue aims to facilitate the transfer of knowledge between these different communities working on or with higher-order differential equations and therefore welcomes high-quality contributions on any of the aforementioned topics.

Dr. Carlos Escudero
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. Axioms 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 2400 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

  • higher-order ordinary differential equations
  • higher-order partial differential equations
  • higher-order linear differential equations
  • higher-order nonlinear differential equations
  • higher-order initial value problems
  • higher-order boundary value problems
  • higher-order differential operators
  • applications of higher-order differential equations

Published Papers (6 papers)

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Research

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17 pages, 418 KiB  
Article
Spectral Treatment of High-Order Emden–Fowler Equations Based on Modified Chebyshev Polynomials
by Waleed Mohamed Abd-Elhameed, Mohamed Salem Al-Harbi, Amr Kamel Amin and Hany M. Ahmed
Axioms 2023, 12(2), 99; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms12020099 - 17 Jan 2023
Cited by 10 | Viewed by 1018
Abstract
This paper is devoted to proposing numerical algorithms based on the use of the tau and collocation procedures, two widely used spectral approaches for the numerical treatment of the initial high-order linear and non-linear equations of the singular type, especially those of the [...] Read more.
This paper is devoted to proposing numerical algorithms based on the use of the tau and collocation procedures, two widely used spectral approaches for the numerical treatment of the initial high-order linear and non-linear equations of the singular type, especially those of the high-order Emden–Fowler type. The class of modified Chebyshev polynomials of the third-kind is constructed. This class of polynomials generalizes the class of the third-kind Chebyshev polynomials. A new formula that expresses the first-order derivative of the modified Chebyshev polynomials in terms of their original modified polynomials is established. The establishment of this essential formula is based on reducing a certain terminating hypergeometric function of the type 5F4(1). The development of our suggested numerical algorithms begins with the extraction of a new operational derivative matrix from this derivative formula. Expansion’s convergence study is performed in detail. Some illustrative examples of linear and non-linear Emden–Flower-type equations of different orders are displayed. Our proposed algorithms are compared with some other methods in the literature. This confirms the accuracy and high efficiency of our presented algorithms. Full article
(This article belongs to the Special Issue Higher Order Differential Equations)
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22 pages, 381 KiB  
Article
Multidimensional Markovian BSDEs with Jumps and Continuous Generators
by Mhamed Eddahbi, Anwar Almualim, Nabil Khelfallah and Imène Madoui
Axioms 2023, 12(1), 26; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms12010026 - 26 Dec 2022
Viewed by 1028
Abstract
We deal with a multidimensional Markovian backward stochastic differential equation driven by a Poisson random measure and independent Brownian motion (BSDEJ for short). As a first result, we prove, under the Lipschitz condition, that the BSDEJ’s adapted solution can be represented in terms [...] Read more.
We deal with a multidimensional Markovian backward stochastic differential equation driven by a Poisson random measure and independent Brownian motion (BSDEJ for short). As a first result, we prove, under the Lipschitz condition, that the BSDEJ’s adapted solution can be represented in terms of a given Markov process and some deterministic functions. Then, by means of this representation, we show existence results for such equations assuming that their generators are totally or partially continuous with respect to their variables and satisfy the usual linear growth conditions. The ideas of the proofs are to approximate the generator by a suitable sequence of Lipschitz functions via convolutions with mollifiers and make use of the L2–domination condition, on the law of the underlying Markov process, for which several examples are given. Full article
(This article belongs to the Special Issue Higher Order Differential Equations)
15 pages, 1606 KiB  
Article
Solving Biharmonic Equations with Tri-Cubic C1 Splines on Unstructured Hex Meshes
by Jeremy Youngquist and Jörg Peters
Axioms 2022, 11(11), 633; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms11110633 - 10 Nov 2022
Cited by 1 | Viewed by 1490
Abstract
Unstructured hex meshes are partitions of three spaces into boxes that can include irregular edges, where n≠4 boxes meet along an edge, and irregular points, where the box arrangement is not consistent with a tensor-product grid. A new class of tri-cubic [...] Read more.
Unstructured hex meshes are partitions of three spaces into boxes that can include irregular edges, where n≠4 boxes meet along an edge, and irregular points, where the box arrangement is not consistent with a tensor-product grid. A new class of tri-cubic C1 splines is evaluated as a tool for solving elliptic higher-order partial differential equations over unstructured hex meshes. Convergence rates for four levels of refinement are computed for an implementation of the isogeometric Galerkin approach applied to Poisson’s equation and the biharmonic equation. The ratios of error are contrasted and superior to an implementation of Catmull-Clark solids. For the trivariate Poisson problem on irregularly partitioned domains, the reduction by 24 in the L2 norm is consistent with the optimal convergence on a regular grid, whereas the convergence rate for Catmull-Clark solids is measured as O(h3). The tri-cubic splines in the isogeometric framework correctly solve the trivariate biharmonic equation, but the convergence rate in the irregular case is lower than O(h4). An optimal reduction of 24 is observed when the functions on the C1 geometry are relaxed to be C0. Full article
(This article belongs to the Special Issue Higher Order Differential Equations)
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13 pages, 263 KiB  
Article
Regularity and Decay of Global Solutions for the Generalized Benney-Lin Equation Posed on Bounded Intervals and on a Half-Line
by Nikolai A. Larkin
Axioms 2022, 11(11), 596; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms11110596 - 28 Oct 2022
Viewed by 857
Abstract
Initial-boundary value problems for the generalized Benney-Lin equation posed on bounded intervals and on the right half-line were considered. The existence and uniqueness of global regular solutions on arbitrary intervals as well as their exponential decay for small solutions and for a special [...] Read more.
Initial-boundary value problems for the generalized Benney-Lin equation posed on bounded intervals and on the right half-line were considered. The existence and uniqueness of global regular solutions on arbitrary intervals as well as their exponential decay for small solutions and for a special choice of a bounded interval have been established. Full article
(This article belongs to the Special Issue Higher Order Differential Equations)
10 pages, 291 KiB  
Article
Finite Time Blowup in a Fourth-Order Dispersive Wave Equation with Nonlinear Damping and a Non-Local Source
by Stella Vernier-Piro
Axioms 2022, 11(5), 212; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms11050212 - 01 May 2022
Viewed by 1502
Abstract
In this work, we consider a class of initial boundary value problems for fourth-order dispersive wave equations with superlinear damping and non-local source terms as well as time-dependent coefficients in Ω×(t>0), where Ω is a bounded [...] Read more.
In this work, we consider a class of initial boundary value problems for fourth-order dispersive wave equations with superlinear damping and non-local source terms as well as time-dependent coefficients in Ω×(t>0), where Ω is a bounded domain in RN and N≥2. We prove that there exists a safe time interval of existence in the solution [0,T], with T being a lower bound of the blowup time t*. Moreover, we find an explicit lower bound of t*, assuming the coefficients are positive constants. Full article
(This article belongs to the Special Issue Higher Order Differential Equations)

Review

Jump to: Research

12 pages, 286 KiB  
Review
On the Method of Differential Invariants for Solving Higher Order Ordinary Differential Equations
by Winter Sinkala and Molahlehi Charles Kakuli
Axioms 2022, 11(10), 555; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms11100555 - 14 Oct 2022
Cited by 1 | Viewed by 1147
Abstract
There are many routines developed for solving ordinary differential Equations (ODEs) of different types. In the case of an nth-order ODE that admits an r-parameter Lie group (3≤r≤n), there is a powerful method of [...] Read more.
There are many routines developed for solving ordinary differential Equations (ODEs) of different types. In the case of an nth-order ODE that admits an r-parameter Lie group (3≤r≤n), there is a powerful method of Lie symmetry analysis by which the ODE is reduced to an (n−r)th-order ODE plus r quadratures provided that the Lie algebra formed by the infinitesimal generators of the group is solvable. It would seem this method is not widely appreciated and/or used as it is not mentioned in many related articles centred around integration of higher order ODEs. In the interest of mainstreaming the method, we describe the method in detail and provide four illustrative examples. We use the case of a third-order ODE that admits a three-dimensional solvable Lie algebra to present the gist of the integration algorithm. Full article
(This article belongs to the Special Issue Higher Order Differential Equations)
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