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Applications of Partial Differential Equations in Mathematical Physics, 2nd Edition
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Applications of Partial Differential Equations in Mathematical Physics, 2nd Edition

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: 31 January 2025 | Viewed by 5396

Special Issue Editor

Dr. Panayiotis Vafeas

E-Mail Website
Guest Editor
Department of Chemical Engineering, University of Patras, University Campus, 26504 Patras, Greece
Interests: applied mathematics and mathematical physics; partial differential equations and applications in physical science and engineering; mathematical modelling and boundary value problems
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Partial differential equations in mathematical physics provide a suitable platform for the development of original research in the fields of applied mathematics and physical sciences for the solution of boundary value problems with the introduction of partial differential equations and related methodologies. The purpose of this Special Issue is to gather contributions from experts on analytical and semi-analytical techniques with application domains, including, but not limited to, fluid dynamics, creeping hydrodynamics, and magnetic fluids; direct and inverse scattering problems in wave phenomena; electromagnetism and low-frequency scattering; electric and magnetic activity of the brain; scattering of elastic waves from isotropic and anisotropic materials; mathematical modelling of cancer tumour growth; and interaction with cold atmospheric pressure plasma jet systems and actuators. Contributions with a main emphasis on numerical methods for the application of partial differential equations in mathematical physics are also welcome, provided they exploit analytical means at certain stages of the procedures employed for the derivations of the solutions.

Dr. Panayiotis Vafeas
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

  • mathematical physics
  • partial differential equations
  • boundary value problems
  • applications in science and engineering

Related Special Issue

Published Papers (6 papers)

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Research

14 pages, 300 KiB  
Article
Existence and Uniqueness of Weak Solutions to Frictionless-Antiplane Contact Problems
by Besma Fadlia, Mohamed Dalah and Delfim F. M. Torres
Mathematics 2024, 12(3), 434; https://0-doi-org.brum.beds.ac.uk/10.3390/math12030434 - 29 Jan 2024
Viewed by 710
Abstract
We investigate a quasi-static-antiplane contact problem, examining a thermo-electro-visco-elastic material with a friction law dependent on the slip rate, assuming that the foundation is electrically conductive. The mechanical problem is represented by a system of partial differential equations, and establishing its solution involves [...] Read more.
We investigate a quasi-static-antiplane contact problem, examining a thermo-electro-visco-elastic material with a friction law dependent on the slip rate, assuming that the foundation is electrically conductive. The mechanical problem is represented by a system of partial differential equations, and establishing its solution involves several key steps. Initially, we obtain a variational formulation of the model, which comprises three systems: a hemivariational inequality, an elliptic equation, and a parabolic equation. Subsequently, we demonstrate the existence of a unique weak solution to the model. The proof relies on various arguments, including those related to evolutionary inequalities, techniques for decoupling unknowns, and certain results from differential equations. Full article
(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 2nd Edition)
15 pages, 1545 KiB  
Article
The Integrability and Modification to an Auxiliary Function Method for Solving the Strain Wave Equation of a Flexible Rod with a Finite Deformation
by Adel Elmandouh, Aqilah Aljuaidan and Mamdouh Elbrolosy
Mathematics 2024, 12(3), 383; https://0-doi-org.brum.beds.ac.uk/10.3390/math12030383 - 24 Jan 2024
Viewed by 561
Abstract
Our study focuses on the governing equation of a finitely deformed flexible rod with strain waves. By utilizing the well-known Ablowita–Ramani–Segur (ARS) algorithm, we prove that the equation is non-integrable in the Painlevé sense. Based on the bifurcation theory for planar dynamical systems, [...] Read more.
Our study focuses on the governing equation of a finitely deformed flexible rod with strain waves. By utilizing the well-known Ablowita–Ramani–Segur (ARS) algorithm, we prove that the equation is non-integrable in the Painlevé sense. Based on the bifurcation theory for planar dynamical systems, we modify an auxiliary equation method to obtain a new systematic and effective method that can be used for a wide class of non-linear evolution equations. This method is summed up in an algorithm that explains and clarifies the ease of its applicability. The proposed method is successfully applied to construct wave solutions. The developed solutions are grouped as periodic, solitary, super periodic, kink, and unbounded solutions. A graphic representation of these solutions is presented using a 3D representation and a 2D representation, as well as a 2D contour plot. Full article
(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 2nd Edition)
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12 pages, 573 KiB  
Article
A New Modified Helmholtz Equation for the Expression of the Gravity Gradient and the Intensity of an Electrostatic Field in Spherical Harmonics
by Gerassimos Manoussakis
Mathematics 2023, 11(20), 4362; https://0-doi-org.brum.beds.ac.uk/10.3390/math11204362 - 20 Oct 2023
Viewed by 589
Abstract
In this work, it is shown that the geometry of a gravity field generated by a spheroid with low eccentricity can be described with the help of a newly modified Helmholtz equation. To distinguish this equation from the modified Helmholtz equation, we refer [...] Read more.
In this work, it is shown that the geometry of a gravity field generated by a spheroid with low eccentricity can be described with the help of a newly modified Helmholtz equation. To distinguish this equation from the modified Helmholtz equation, we refer to it as the G-modified Helmholtz equation. The use of this new equation to study the spheroid’s gravity field is advantageous in expressing the gravity vector as a vector series of spherical harmonics. The solution of the G-modified Helmholtz equation involves both the gravity intensity g (or simply gravity g) and the intensity E of an electrostatic field as shown in sequel. An electrostatic field generated by an oblate spheroid charged with l electrons (uniform ellipsoidal charge distribution) is demonstrated to be a special case. Both gravity intensity g and intensity E are governed by the same law and can be expressed as a series of spherical harmonics, and thus the G-modified Helmholtz equation is useful for describing the gravity and electrostatic fields. Full article
(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 2nd Edition)
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17 pages, 4220 KiB  
Article
Identification and Control of Rehabilitation Robots with Unknown Dynamics: A New Probabilistic Algorithm Based on a Finite-Time Estimator
by Naif D. Alotaibi, Hadi Jahanshahi, Qijia Yao, Jun Mou and Stelios Bekiros
Mathematics 2023, 11(17), 3699; https://0-doi-org.brum.beds.ac.uk/10.3390/math11173699 - 28 Aug 2023
Viewed by 622
Abstract
The control of rehabilitation robots presents a formidable challenge owing to the myriad of uncharted disturbances encountered in real-world applications. Despite the existence of several techniques proposed for controlling and identifying such systems, many cutting-edge approaches have yet to be implemented in the [...] Read more.
The control of rehabilitation robots presents a formidable challenge owing to the myriad of uncharted disturbances encountered in real-world applications. Despite the existence of several techniques proposed for controlling and identifying such systems, many cutting-edge approaches have yet to be implemented in the context of rehabilitation robots. This highlights the necessity for further investigation and exploration in this field. In light of this motivation, we introduce a pioneering algorithm that employs a finite estimator and Gaussian process to identify and forecast the uncharted dynamics of a 2-DoF knee rehabilitation robot. The proposed algorithm harnesses the probabilistic nature of Gaussian processes, while also guaranteeing finite-time convergence through the utilization of the Lyapunov theorem. This dual advantage allows for the effective exploitation of the Gaussian process’s probabilistic capabilities while ensuring reliable and timely convergence of the algorithm. The algorithm is delineated and the finite time convergence is proven. Subsequently, its performance is investigated through numerical simulations for estimating complex unknown and time-varying dynamics. The results obtained from the proposed algorithm are then employed for controlling the rehabilitation robot, highlighting its remarkable capability to provide precise estimates while effectively handling uncertainty. Full article
(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 2nd Edition)
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17 pages, 1633 KiB  
Article
On the Dynamical Behavior of Solitary Waves for Coupled Stochastic Korteweg–De Vries Equations
by Wael W. Mohammed, Farah M. Al-Askar and Clemente Cesarano
Mathematics 2023, 11(16), 3506; https://0-doi-org.brum.beds.ac.uk/10.3390/math11163506 - 14 Aug 2023
Cited by 7 | Viewed by 697
Abstract
In this paper, we take into account the coupled stochastic Korteweg–De Vries (CSKdV) equations in the Itô sense. Using the mapping method, new trigonometric, rational, hyperbolic, and elliptic stochastic solutions are obtained. These obtained solutions can be applied to the analysis of a [...] Read more.
In this paper, we take into account the coupled stochastic Korteweg–De Vries (CSKdV) equations in the Itô sense. Using the mapping method, new trigonometric, rational, hyperbolic, and elliptic stochastic solutions are obtained. These obtained solutions can be applied to the analysis of a wide variety of crucial physical phenomena because the coupled KdV equations have important applications in various fields of physics and engineering. Also, it is used in the design of optical fiber communication systems, which transmit information using soliton-like waves. The dynamic performance of the various obtained solutions are depicted using 3D and 2D curves in order to interpret the effects of multiplicative noise. We conclude that multiplicative noise influences the behavior of the solutions of CSKdV equations and stabilizes them. Full article
(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 2nd Edition)
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19 pages, 389 KiB  
Article
Exact Solutions of Reaction–Diffusion PDEs with Anisotropic Time Delay
by Andrei D. Polyanin and Vsevolod G. Sorokin
Mathematics 2023, 11(14), 3111; https://0-doi-org.brum.beds.ac.uk/10.3390/math11143111 - 14 Jul 2023
Cited by 1 | Viewed by 1355
Abstract
This study is devoted to reaction–diffusion equations with spatially anisotropic time delay. Reaction–diffusion PDEs with either constant or variable transfer coefficients are considered. Nonlinear equations of a fairly general form containing one, two, or more arbitrary functions and free parameters are analyzed. For [...] Read more.
This study is devoted to reaction–diffusion equations with spatially anisotropic time delay. Reaction–diffusion PDEs with either constant or variable transfer coefficients are considered. Nonlinear equations of a fairly general form containing one, two, or more arbitrary functions and free parameters are analyzed. For the first time, reductions and exact solutions for such complex delay PDEs are constructed. Additive, multiplicative, generalized, and functional separable solutions and some other exact solutions are presented. In addition to reaction–diffusion equations, wave-type PDEs with spatially anisotropic time delay are considered. Overall, more than twenty new exact solutions to reaction–diffusion and wave-type equations with anisotropic time delay are found. The described nonlinear delay PDEs and their solutions can be used to formulate test problems applicable to the verification of approximate analytical and numerical methods for solving complex PDEs with variable delay. Full article
(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 2nd Edition)
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