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Mathematics
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Differential Equations with Boundary Value Problems: Theory and Applications
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Differential Equations with Boundary Value Problems: Theory and Applications

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

Deadline for manuscript submissions: closed (31 October 2024) | Viewed by 4261

Special Issue Editor

Dr. Yu Tian

E-Mail Website
Guest Editor
School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China
Interests: differential equations; dynamical systems; variational methods and applications; nonlinear analysis

Special Issue Information

Dear Colleagues,

Boundary-value problems (BVPs) have an important place in engineering, environmental phenomena, physical and engineering sciences. The purpose of this Special Issue is to gather contributions pertaining to topics including (but not limited to) the existence of solutions, analytical solutions, and minimizers for functionals of boundary value problems. The differential equations include ordinary differential equations, fractional differential equations, difference equations, etc.

Dr. Yu Tian
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. 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

  • ordinary differential equation
  • fractional differential equation
  • difference equation
  • boundary value problem
  • existence
  • multiplicity solutions
  • critical point theory
  • variational methods

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Published Papers (4 papers)

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Research

34 pages, 403 KiB  
Article
Unraveling the Complexity of Inverting the Sturm–Liouville Boundary Value Problem to Its Canonical Form
by Natanael Karjanto and Peter Sadhani
Mathematics 2024, 12(9), 1329; https://doi.org/10.3390/math12091329 - 26 Apr 2024
Viewed by 676
Abstract
The Sturm–Liouville boundary value problem (SLBVP) stands as a fundamental cornerstone in the realm of mathematical analysis and physical modeling. Also known as the Sturm–Liouville problem (SLP), this paper explores the intricacies of this classical problem, particularly the relationship between its canonical and [...] Read more.
The Sturm–Liouville boundary value problem (SLBVP) stands as a fundamental cornerstone in the realm of mathematical analysis and physical modeling. Also known as the Sturm–Liouville problem (SLP), this paper explores the intricacies of this classical problem, particularly the relationship between its canonical and Liouville normal (Schrödinger) forms. While the conversion from the canonical to Schrödinger form using Liouville’s transformation is well known in the literature, the inverse transformation seems to be neglected. Our study attempts to fill this gap by investigating the inverse of Liouville’s transformation, that is, given any SLP in the Schrödinger form with some invariant function, we seek the SLP in its canonical form. By closely examining the second Paine–de Hoog–Anderson (PdHA) problem, we argue that retrieving the SLP in its canonical form can be notoriously difficult and can even be impossible to achieve in its exact form. Finding the inverse relationship between the two independent variables seems to be the main obstacle. We confirm this claim by considering four different scenarios, depending on the potential and density functions that appear in the corresponding invariant function. In the second PdHA problem, this invariant function takes a reciprocal quadratic binomial form. In some cases, the inverse Liouville transformation produces an exact expression for the SLP in its canonical form. In other situations, however, while an exact canonical form is not possible to obtain, we successfully derived the SLP in its canonical form asymptotically. Full article
(This article belongs to the Special Issue Differential Equations with Boundary Value Problems: Theory and Applications)
12 pages, 271 KiB  
Article
The Role of Data on the Regularity of Solutions to Some Evolution Equations
by Maria Michaela Porzio
Mathematics 2024, 12(5), 761; https://doi.org/10.3390/math12050761 - 4 Mar 2024
Viewed by 745
Abstract
In this paper, we study the influence of the initial data and the forcing terms on the regularity of solutions to a class of evolution equations including linear and semilinear parabolic equations as the model cases, together with the nonlinear p-Laplacian equation. We [...] Read more.
In this paper, we study the influence of the initial data and the forcing terms on the regularity of solutions to a class of evolution equations including linear and semilinear parabolic equations as the model cases, together with the nonlinear p-Laplacian equation. We focus our study on the regularity (in terms of belonging to appropriate Lebesgue spaces) of the gradient of the solutions. We prove that there are cases where the regularity of the solutions as soon as t>0 is not influenced at all by the initial data. We also derive estimates for the gradient of these solutions that are independent of the initial data and reveal, once again, that for this class of evolution problems, the real “actors of the regularity” are the forcing terms. Full article
(This article belongs to the Special Issue Differential Equations with Boundary Value Problems: Theory and Applications)
10 pages, 478 KiB  
Article
Solitary Wave Solutions of a Hyperelastic Dispersive Equation
by Yuheng Jiang, Yu Tian and Yao Qi
Mathematics 2024, 12(4), 564; https://doi.org/10.3390/math12040564 - 13 Feb 2024
Viewed by 910
Abstract
This paper explores solitary wave solutions arising in the deformations of a hyperelastic compressible plate. Explicit traveling wave solution expressions with various parameters for the hyperelastic compressible plate are obtained and visualized. To analyze the perturbed equation, we employ geometric singular perturbation theory, [...] Read more.
This paper explores solitary wave solutions arising in the deformations of a hyperelastic compressible plate. Explicit traveling wave solution expressions with various parameters for the hyperelastic compressible plate are obtained and visualized. To analyze the perturbed equation, we employ geometric singular perturbation theory, Melnikov methods, and invariant manifold theory. The solitary wave solutions of the hyperelastic compressible plate do not persist under small perturbations for wave speed c>−βk2. Further exploration of nonlinear models that accurately depict the persistence of solitary wave solution on the significant physical processes under the K-S perturbation is recommended. Full article
(This article belongs to the Special Issue Differential Equations with Boundary Value Problems: Theory and Applications)
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18 pages, 274 KiB  
Article
Solutions of Umbral Dirac-Type Equations
by Hongfen Yuan and Valery Karachik
Mathematics 2024, 12(2), 344; https://doi.org/10.3390/math12020344 - 20 Jan 2024
Viewed by 1060
Abstract
The aim of this work is to study the method of the normalized systems of functions. The normalized systems of functions with respect to the Dirac operator in the umbral Clifford analysis are constructed. Furthermore, the solutions of umbral Dirac-type equations are investigated [...] Read more.
The aim of this work is to study the method of the normalized systems of functions. The normalized systems of functions with respect to the Dirac operator in the umbral Clifford analysis are constructed. Furthermore, the solutions of umbral Dirac-type equations are investigated by the normalized systems. Full article
(This article belongs to the Special Issue Differential Equations with Boundary Value Problems: Theory and Applications)
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