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8 pages, 235 KiB

Open AccessArticle

Variational Principle of the Unstable Nonlinear Schrödinger Equation with Fractal Derivatives

by Kang-Jia Wang and Ming Li

Axioms 2025, 14(5), 376; https://doi.org/10.3390/axioms14050376 - 16 May 2025

Viewed by 101

Abstract

The well-known nonlinear Schrödinger equation (NLSE) plays a crucial role in describing the temporal evolution of disturbances in marginally stable or unstable media. However, when the media is a fractal form, it becomes ineffective. Thus, the fractal modification to the NLSE is presented [...] Read more.

The well-known nonlinear Schrödinger equation (NLSE) plays a crucial role in describing the temporal evolution of disturbances in marginally stable or unstable media. However, when the media is a fractal form, it becomes ineffective. Thus, the fractal modification to the NLSE is presented based on the fractal derivative in this work for the first time. The semi-inverse method is employed to establish the fractal variational principle. The entire process of deriving the fractal variational principle is presented in detail. To our knowledge, the fractal variational principle mentioned in this article is the first exploration and report to date. The fractal variational principle established in this paper is expected to deepen our understanding of the essence of physical phenomena in the fractal space and offer new ideas for the application and exploration of the variational approaches. Full article

(This article belongs to the Special Issue Principles of Variational Methods in Mathematical Physics)

21 pages, 322 KiB

Open AccessArticle

A Quasi-Explicit Method Applied to Missing Boundary Data Reconstruction for the Stokes System

by Abdulaziz H. Alharbi, Fadhel Jday, Abdulrahman B. Albidah and Ali M. Alhartomi

Axioms 2025, 14(3), 177; https://doi.org/10.3390/axioms14030177 - 27 Feb 2025

Viewed by 392

Abstract

In this paper, we study the data completion problem for the Cauchy–Stokes equation in a cylindrical domain,

Ω

. Neumann and Dirichlet boundary conditions are prescribed on part of the overdetermined boundary,

Γ_{0}

, and the goal is to complete the data [...] Read more.

In this paper, we study the data completion problem for the Cauchy–Stokes equation in a cylindrical domain,

Ω

. Neumann and Dirichlet boundary conditions are prescribed on part of the overdetermined boundary,

Γ_{0}

, and the goal is to complete the data on the other part of the boundary,

Γ_{a}

. Here,

Γ_{0}

and

Γ_{a}

represent the side faces of the cylinder

Ω

. This problem is known to be ill-posed and is formulated as an optimal control problem with a regularized cost function. To directly approximate the missing data on

Γ_{a}

, we employ the method of factorization of elliptic boundary value problems. This technique allows the factorization of a boundary value problem into a product of parabolic problems. It is successfully applied to the optimality system in this work, yielding new and significant results. Full article

(This article belongs to the Special Issue Principles of Variational Methods in Mathematical Physics)

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15 pages, 459 KiB

Open AccessArticle

Elasticity Problem with a Cusp between Thin Inclusion and Boundary

by Alexander Khludnev

Axioms 2023, 12(12), 1081; https://doi.org/10.3390/axioms12121081 - 27 Nov 2023

Cited by 1 | Viewed by 1143

Abstract

This paper concerns an equilibrium problem for an an elastic body with a thin rigid inclusion crossing an external boundary of the body at zero angle. The inclusion is assumed to be exfoliated from the surrounding elastic material that provides an interfacial crack. [...] Read more.

This paper concerns an equilibrium problem for an an elastic body with a thin rigid inclusion crossing an external boundary of the body at zero angle. The inclusion is assumed to be exfoliated from the surrounding elastic material that provides an interfacial crack. To avoid nonphysical interpenetration of the opposite crack faces, we impose inequality type constraints. Moreover, boundary conditions at the crack faces depend on a positive parameter describing a cohesion. A solution existence of the problem with different conditions on the external boundary is proved. Passages to the limit are analyzed as the damage parameter tends to infinity and to zero. Finally, an optimal control problem with a suitable cost functional is investigated. In this case, a part of the rigid inclusion is located outside of the elastic body, and a control function is a shape of the inclusion. Full article

(This article belongs to the Special Issue Principles of Variational Methods in Mathematical Physics)

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17 pages, 376 KiB

Open AccessArticle

Solutions for Some Specific Mathematical Physics Problems Issued from Modeling Real Phenomena: Part 2

by Irina Meghea

Axioms 2023, 12(8), 726; https://doi.org/10.3390/axioms12080726 - 26 Jul 2023

Cited by 1 | Viewed by 1242

Abstract

This paper brings together methods to solve and/or characterize solutions of some problems of mathematical physics equations involving p-Laplacian and p-pseudo-Laplacian. Using the widely debated results of surjectivity or variational approaches, one may obtain or characterize weak solutions for Dirichlet or [...] Read more.

This paper brings together methods to solve and/or characterize solutions of some problems of mathematical physics equations involving p-Laplacian and p-pseudo-Laplacian. Using the widely debated results of surjectivity or variational approaches, one may obtain or characterize weak solutions for Dirichlet or Newmann problems for these important operators. The relevance of these operators and the possibility to be involved in the modeling of an important class of real phenomena is once again revealed by their applications. The use of certain variational methods facilitates the complete solution of the problem using appropriate numerical methods and computational algorithms. Some theoretical results are involved to complete the solutions for a sequence of models issued from real phenomena drawing. Full article

(This article belongs to the Special Issue Principles of Variational Methods in Mathematical Physics)

33 pages, 653 KiB

Open AccessArticle

A Systematic Approach to Standard Dissipative Continua

by Sebastian Stark

Axioms 2023, 12(3), 267; https://doi.org/10.3390/axioms12030267 - 4 Mar 2023

Cited by 4 | Viewed by 1589

Abstract

Many isothermal dissipative continuum problems can be formulated in a variational setting using the concept of “standard dissipative continua”. A major advantage of this approach is that complex problems can be cast into a compact, thermodynamically consistent formulation based on a single space–time [...] Read more.

Many isothermal dissipative continuum problems can be formulated in a variational setting using the concept of “standard dissipative continua”. A major advantage of this approach is that complex problems can be cast into a compact, thermodynamically consistent formulation based on a single space–time continuous functional together with a corresponding variational principle. Formulating the problem in terms of a functional provides an immediate avenue for performing spatial and temporal discretization, which are the prerequisites for a numerical solution. Within the present contribution, a novel systematic approach to standard dissipative formulations is proposed, with the main goal being the development and implementation of generic procedures and algorithms for the formulation as well as the computational solution of a subset of isothermal dissipative continuum problems. In order to demonstrate the capabilities of the approach, its application to example problems is discussed. Full article

(This article belongs to the Special Issue Principles of Variational Methods in Mathematical Physics)

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15 pages, 302 KiB

Open AccessArticle

Ekeland Variational Principle and Some of Its Equivalents on a Weighted Graph, Completeness and the OSC Property

by Basit Ali, Ştefan Cobzaş and Mokhwetha Daniel Mabula

Axioms 2023, 12(3), 247; https://doi.org/10.3390/axioms12030247 - 28 Feb 2023

Cited by 1 | Viewed by 2215

Abstract

We prove a version of the Ekeland Variational Principle (EkVP) in a weighted graph G and its equivalence to Caristi fixed point theorem and to the Takahashi minimization principle. The usual completeness and topological notions are replaced with some weaker versions expressed in [...] Read more.

We prove a version of the Ekeland Variational Principle (EkVP) in a weighted graph G and its equivalence to Caristi fixed point theorem and to the Takahashi minimization principle. The usual completeness and topological notions are replaced with some weaker versions expressed in terms of the graph G. The main tool used in the proof is the OSC property for sequences in a graph. Converse results, meaning the completeness of weighted graphs for which one of these principles holds, are also considered. Full article

(This article belongs to the Special Issue Principles of Variational Methods in Mathematical Physics)

7 pages, 235 KiB

Open AccessArticle

Unfolding a Hidden Lagrangian Structure of a Class of Evolution Equations

by Philip Rosenau

Axioms 2023, 12(1), 2; https://doi.org/10.3390/axioms12010002 - 20 Dec 2022

Cited by 1 | Viewed by 1416

Abstract

It is shown that a simple modification of the standard Lagrangian underlying the dynamics of Newtonian lattices enables one to infer the hidden Lagrangian structure of certain classes of first order in time evolution equations which lack the conventional Lagrangian structure. Implication to [...] Read more.

It is shown that a simple modification of the standard Lagrangian underlying the dynamics of Newtonian lattices enables one to infer the hidden Lagrangian structure of certain classes of first order in time evolution equations which lack the conventional Lagrangian structure. Implication to other setups is outlined and exemplified. Full article

(This article belongs to the Special Issue Principles of Variational Methods in Mathematical Physics)

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Jump to: Research

66 pages, 6454 KiB

Open AccessReview

Solutions for Some Mathematical Physics Problems Issued from Modeling Real Phenomena: Part 1

by Irina Meghea

Axioms 2023, 12(6), 532; https://doi.org/10.3390/axioms12060532 - 29 May 2023

Cited by 2 | Viewed by 1568

Abstract

This paper brings together methods to solve and/or characterize solutions of some problems of mathematical physics equations involving p-Laplacian and p-pseudo-Laplacian. Using surjectivity or variational approaches, one may obtain or characterize weak solutions for Dirichlet or Newmann problems for these important [...] Read more.

This paper brings together methods to solve and/or characterize solutions of some problems of mathematical physics equations involving p-Laplacian and p-pseudo-Laplacian. Using surjectivity or variational approaches, one may obtain or characterize weak solutions for Dirichlet or Newmann problems for these important operators. This article details three ways to use surjectivity results for a special type of operator involving the duality mapping and a Nemytskii operator, three methods starting from Ekeland’s variational principle and, lastly, one with a generalized variational principle to solve or describe the above-mentioned solutions. The relevance of these operators and the possibility of their involvement in the modeling of an important class of real phenomena determined the author to group these seven procedures together, presented in detail, followed by many applications, accompanied by a general overview of specialty domains. The use of certain variational methods facilitates the complete solution of the problem via appropriate numerical methods and computational algorithms. The exposure of the sequence of theoretical results, together with their demonstration in as much detail as possible has been fulfilled as an opportunity for the complete development of these topics. Full article

(This article belongs to the Special Issue Principles of Variational Methods in Mathematical Physics)

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