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37 pages, 371 KiB

Open AccessArticle

Analytical Computation of Hyper-Ellipsoidal Harmonics

by George Dassios and George Fragoyiannis

Mathematics 2024, 12(15), 2433; https://doi.org/10.3390/math12152433 - 5 Aug 2024

Viewed by 314

The four-dimensional ellipsoid of an anisotropic hyper-structure corresponds to the four-dimensional sphere of an isotropic hyper-structure. In three dimensions, both theories for spherical and ellipsoidal harmonics have been developed by Laplace and Lamé, respectively. Nevertheless, in four dimensions, only the theory of hyper-spherical [...] Read more.

The four-dimensional ellipsoid of an anisotropic hyper-structure corresponds to the four-dimensional sphere of an isotropic hyper-structure. In three dimensions, both theories for spherical and ellipsoidal harmonics have been developed by Laplace and Lamé, respectively. Nevertheless, in four dimensions, only the theory of hyper-spherical harmonics is hitherto known. This void in the literature is expected to be filled up by the present work. In fact, it is well known that the spectral decomposition of the Laplace equation in three-dimensional ellipsoidal geometry leads to the Lamé equation. This Lamé equation governs each one of the spectral functions corresponding to the three ellipsoidal coordinates, which, however, live in non-overlapping intervals. The analysis of the Lamé equation leads to four classes of Lamé functions, giving a total of 2n + 1 functions of degree n. In four dimensions, a much more elaborate procedure leads to similar results for the hyper-ellipsoidal structure. Actually, we demonstrate here that there are eight classes of the spectral hyper-Lamé equation and we provide a complete analysis for each one of them. The number of hyper-Lamé functions of degree n is (n + 1)²; that is, n² more functions than the three-dimensional case. However, the main difficulty in the four-dimensional analysis concerns the evaluation of the three separation constants appearing during the separation process. One of them can be extracted from the corresponding theory of the hyper-sphero-conal system, but the other two constants are obtained via a much more complicated procedure than the three-dimensional case. In fact, the solution process exhibits specific nonlinearities of polynomial type, itemized for every class and every degree. An example of this procedure is demonstrated in detail in order to make the process clear. Finally, the hyper-ellipsoidal harmonics are given as the product of four identical hyper-Lamé functions, each one defined in its own domain, which are explicitly calculated and tabulated for every degree less than five. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 2nd Edition)

29 pages, 412 KiB

Open AccessArticle

Unsteady Magnetohydrodynamics PDE of Monge–Ampère Type: Symmetries, Closed-Form Solutions, and Reductions

by Andrei D. Polyanin and Alexander V. Aksenov

Mathematics 2024, 12(13), 2127; https://doi.org/10.3390/math12132127 - 6 Jul 2024

Viewed by 454

Abstract

The paper studies an unsteady equation with quadratic nonlinearity in second derivatives, that occurs in electron magnetohydrodynamics. In mathematics, such PDEs are referred to as parabolic Monge–Ampère equations. An overview of the Monge–Ampère type equations is given, in which their unusual qualitative features [...] Read more.

The paper studies an unsteady equation with quadratic nonlinearity in second derivatives, that occurs in electron magnetohydrodynamics. In mathematics, such PDEs are referred to as parabolic Monge–Ampère equations. An overview of the Monge–Ampère type equations is given, in which their unusual qualitative features are noted. For the first time, the Lie group analysis of the considered highly nonlinear PDE with three independent variables is carried out. An eleven-parameter transformation is found that preserves the form of the equation. Some one-dimensional reductions allowing to obtain self-similar and other invariant solutions that satisfy ordinary differential equations are described. A large number of new additive, multiplicative, generalized, and functional separable solutions are obtained. Special attention is paid to the construction of exact closed-form solutions, including solutions in elementary functions (in total, more than 30 solutions in elementary functions were obtained). Two-dimensional symmetry and non-symmetry reductions leading to simpler partial differential equations with two independent variables are considered (including stationary Monge–Ampère type equations, linear and nonlinear heat type equations, and nonlinear filtration equations). The obtained results and exact solutions can be used to evaluate the accuracy and analyze the adequacy of numerical methods for solving initial boundary value problems described by highly nonlinear partial differential equations. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 2nd Edition)

14 pages, 300 KiB

Open AccessArticle

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://doi.org/10.3390/math12030434 - 29 Jan 2024

Viewed by 922

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

Open AccessArticle

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://doi.org/10.3390/math12030383 - 24 Jan 2024

Cited by 1 | Viewed by 768

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

3 D

representation and a

2 D

representation, as well as a

2 D

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

Open AccessArticle

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://doi.org/10.3390/math11204362 - 20 Oct 2023

Viewed by 723

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

Open AccessArticle

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://doi.org/10.3390/math11173699 - 28 Aug 2023

Viewed by 822

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

Open AccessArticle

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://doi.org/10.3390/math11163506 - 14 Aug 2023

Cited by 10 | Viewed by 891

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

Open AccessArticle

Exact Solutions of Reaction–Diffusion PDEs with Anisotropic Time Delay

by Andrei D. Polyanin and Vsevolod G. Sorokin

Mathematics 2023, 11(14), 3111; https://doi.org/10.3390/math11143111 - 14 Jul 2023

Cited by 2 | Viewed by 1718

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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