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Symmetry
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Conservation Laws, Symmetry Analysis and Variational Principle for Discontinuous Problems
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Conservation Laws, Symmetry Analysis and Variational Principle for Discontinuous Problems

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (30 June 2022) | Viewed by 20976

Special Issue Editors

Prof. Dr. Ji-Huan He

E-Mail Website
Guest Editor
National Engineering Laboratory for Modern Silk, College of Textile and Clothing Engineering, Soochow University, 199 Ren-Ai Road, Suzhou, China
Interests: fractal calculus; fractional calculus; nonlinear science; nanotechnology; biomechanics
Special Issues, Collections and Topics in MDPI journals
Dr. Muhammad Nadeem

E-Mail Website
Guest Editor
Faculty of Science, Yibin University, Yibin 644000, China
Interests: approximate and numerical solution of PDE's; fractal and fractional calculus
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear colleagues,

Lie theory, symmetry analysis, conservation law, and variational principle are useful mathematical tools for various nonlinear differential equations, and their extensions for non-differential problems have increasingly become more promising.

This Special Issue focuses itself on fractal differential equations for various discontinuous problems using the two-scale fractal theory and two-scale mathematics.

The two-scale method considers the same problem by using two different scales—the larger scale always leads to a differential model by the continuum mechanics, while the smaller scale can figure out the discontinuous property of the same problem. For example, water is continuous on a micro scale, and all laws in fluid mechanics can be used to describe its motion. However, when we measure the motion on a molecule’s scale, many uncertainty phenomena arising in the macro observation can be solved certainly.

 This Special Issue welcomes the following main topics:

1)     Two-scale fractal theory for discontinuous problems;

2)     Two-scale fractal calculus for discontinuous problems;

3)     Conservation laws for fractal differential equations;

4)     Symmetry analysis of fractal differential equations;

5)     Variational principles for fractal differential equations;

6)     Lie theory for fractal differential equations;

7)     Other methods for fractal differential equations.

Prof. Ji-huan He
Dr. Muhammad Nadeem
Guest Editors

Manuscript Submission Information

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Published Papers (9 papers)

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Research

18 pages, 1227 KiB  
Article
Analytical Study of Fractional Epidemic Model via Natural Transform Homotopy Analysis Method
by Hamdy R. Abdl-Rahim, Mohra Zayed and Gamal M. Ismail
Symmetry 2022, 14(8), 1695; https://doi.org/10.3390/sym14081695 - 15 Aug 2022
Cited by 7 | Viewed by 1933
Abstract
In this study, we present a new general solution to a rational epidemiological mathematical model via a recent intelligent method called the natural transform homotopy analysis method (NTHAM), which combines two methods: the natural transform method (NTM) and homotopy analysis method (HAM). To [...] Read more.
In this study, we present a new general solution to a rational epidemiological mathematical model via a recent intelligent method called the natural transform homotopy analysis method (NTHAM), which combines two methods: the natural transform method (NTM) and homotopy analysis method (HAM). To assess the precision and the reliability of the present method, we compared the obtained results with those of the Laplace homotopy perturbation method (LHPM) as well as the q-homotopy analysis Sumudu transform method (q-HASTM), which revealed that the NTHAM is more reliable. The Caputo fractional derivative is employed. It not only gives initial conditions with obvious natural interpretation but is also bounded, meaning that there is no derivative of a constant. The results show that the proposed technique is superior in terms of simplicity, quality, accuracy, and stability and demonstrate the effectiveness of the rational technique under consideration. Full article
(This article belongs to the Special Issue Conservation Laws, Symmetry Analysis and Variational Principle for Discontinuous Problems)
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10 pages, 268 KiB  
Article
A Study on Fractional Diffusion—Wave Equation with a Reaction
by Mohammed M. A. Abuomar, Muhammed I. Syam and Amirah Azmi
Symmetry 2022, 14(8), 1537; https://doi.org/10.3390/sym14081537 - 27 Jul 2022
Viewed by 1466
Abstract
An analytical method for solving the fractional diffusion–wave equation with a reaction is investigated. This approach is based on the Laplace transform and fractional series method. An analytical derivation for the proposed method is presented. Examples are given to illustrate the efficiency of [...] Read more.
An analytical method for solving the fractional diffusion–wave equation with a reaction is investigated. This approach is based on the Laplace transform and fractional series method. An analytical derivation for the proposed method is presented. Examples are given to illustrate the efficiency of the method. The obtained solutions are very close to the exact solutions. Based on this study, we think that the obtained method is promising, and we hope that it can be implemented to other physical problems. Full article
(This article belongs to the Special Issue Conservation Laws, Symmetry Analysis and Variational Principle for Discontinuous Problems)
23 pages, 1211 KiB  
Article
Numerical Investigation of the Fredholm Integral Equations with Oscillatory Kernels Based on Compactly Supported Radial Basis Functions
by Suliman Khan, Sharifah E. Alhazmi, Aisha M. Alqahtani, Ahmed EI-Sayed Ahmed, Mansour F. Yaseen, Elsayed M. Tag-Eldin and Dania Qaiser
Symmetry 2022, 14(8), 1527; https://doi.org/10.3390/sym14081527 - 26 Jul 2022
Cited by 5 | Viewed by 1962
Abstract
The integral equations with oscillatory kernels are of great concern in applied sciences and computational engineering, particularly for large-scale data points and high frequencies. Therefore, the interest of this work is to develop an accurate, efficient, and stable algorithm for the computation of [...] Read more.
The integral equations with oscillatory kernels are of great concern in applied sciences and computational engineering, particularly for large-scale data points and high frequencies. Therefore, the interest of this work is to develop an accurate, efficient, and stable algorithm for the computation of the Fredholm integral equations (FIEs) with the oscillatory kernel. The oscillatory part of the FIEs is evaluated by the Levin quadrature coupled with a compactly supported radial basis function (CS-RBF). The algorithm exhibits sparse and well-conditioned matrix even for large-scale data points, as compared to its counterpart, multi-quadric radial basis function (MQ-RBF) coupled with the Levin quadrature. Usually, the RBFs behave with spherical symmetry about the centers, known as radial. The comparison of convergence and stability analysis of both types of RBFs are performed and numerically verified. The proposed algorithm is tested with benchmark problems and compared with the counterpart methods in the literature. It is concluded that the algorithm in this work is accurate, robust, and stable than the existing methods in the literature based on MQ-RBF and the Chebyshev interpolation matrix. Full article
(This article belongs to the Special Issue Conservation Laws, Symmetry Analysis and Variational Principle for Discontinuous Problems)
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10 pages, 265 KiB  
Article
Harmonic Blaschke–Minkowski Homomorphism
by Hongying Xiao, Weidong Wang and Zhaofeng Li
Symmetry 2022, 14(7), 1396; https://doi.org/10.3390/sym14071396 - 7 Jul 2022
Viewed by 1241
Abstract
Centroid bodies are a continuous and GL(n)-contravariant valuation and play critical roles in the solution to the Busemann–Petty problem. In this paper, we introduce the notion of harmonic Blaschke–Minkowski homomorphism and show that such a map is represented [...] Read more.
Centroid bodies are a continuous and GL(n)-contravariant valuation and play critical roles in the solution to the Busemann–Petty problem. In this paper, we introduce the notion of harmonic Blaschke–Minkowski homomorphism and show that such a map is represented by a spherical convolution operator. Furthermore, we consider the Shephard-type problem of whether ΦK⊆ΦL implies V(K)≤V(L), where Φ is a harmonic Blaschke–Minkowski homomorphism. Some important results for centroid bodies are extended to a large class of valuations. Finally, we give two interesting results for even and odd harmonic Blaschke–Minkowski homomorphisms, separately. Full article
(This article belongs to the Special Issue Conservation Laws, Symmetry Analysis and Variational Principle for Discontinuous Problems)
11 pages, 263 KiB  
Article
Lie Symmetries and Conservation Laws of Fokas–Lenells Equation and Two Coupled Fokas–Lenells Equations by the Symmetry/Adjoint Symmetry Pair Method
by Lihua Zhang, Gangwei Wang, Qianqian Zhao and Lingshu Wang
Symmetry 2022, 14(2), 238; https://doi.org/10.3390/sym14020238 - 26 Jan 2022
Cited by 7 | Viewed by 2034
Abstract
The Fokas–Lenells equation and its multi-component coupled forms have attracted the attention of many mathematical physicists. The Fokas–Lenells equation and two coupled Fokas–Lenells equations are investigated from the perspective of Lie symmetries and conservation laws. The three systems have been turned into real [...] Read more.
The Fokas–Lenells equation and its multi-component coupled forms have attracted the attention of many mathematical physicists. The Fokas–Lenells equation and two coupled Fokas–Lenells equations are investigated from the perspective of Lie symmetries and conservation laws. The three systems have been turned into real multi-component coupled systems by appropriate transformations. By procedures of symmetry analysis, Lie symmetries of the three real systems are obtained. Explicit conservation laws are constructed using the symmetry/adjoint symmetry pair method, which depends on Lie symmetries and adjoint symmetries. The relationships between the multiplier and the adjoint symmetry are investigated. Full article
(This article belongs to the Special Issue Conservation Laws, Symmetry Analysis and Variational Principle for Discontinuous Problems)
13 pages, 1432 KiB  
Article
Riemann–Hilbert Approach for Constructing Analytical Solutions and Conservation Laws of a Local Time-Fractional Nonlinear Schrödinger Type Equation
by Bo Xu and Sheng Zhang
Symmetry 2021, 13(9), 1593; https://doi.org/10.3390/sym13091593 - 30 Aug 2021
Cited by 11 | Viewed by 2070
Abstract
Fractal and fractional calculus have important theoretical and practical value. In this paper, analytical solutions, including the N-fractal-soliton solution with fractal characteristics in time and soliton characteristics in space as well as the long-time asymptotic solution of a local time-fractional nonlinear Schrödinger [...] Read more.
Fractal and fractional calculus have important theoretical and practical value. In this paper, analytical solutions, including the N-fractal-soliton solution with fractal characteristics in time and soliton characteristics in space as well as the long-time asymptotic solution of a local time-fractional nonlinear Schrödinger (NLS)-type equation, are obtained by extending the Riemann–Hilbert (RH) approach together with the symmetries of the associated spectral function, jump matrix, and solution of the related RH problem. In addition, infinitely many conservation laws determined by an expression, one end of which is the partial derivative of local fractional-order in time, and the other end is the partial derivative of integral order in space of the local time-fractional NLS-type equation are also obtained. Constraining the time variable to the Cantor set, the obtained one-fractal-soliton solution is simulated, which shows the solution possesses continuous and non-differentiable characteristics in the time direction but keeps the soliton continuous and differentiable in the space direction. The essence of the fractal-soliton feature is that the time and space variables are set into two different dimensions of 0.631 and 1, respectively. This is also a concrete example of the same object showing different geometric characteristics on two scales. Full article
(This article belongs to the Special Issue Conservation Laws, Symmetry Analysis and Variational Principle for Discontinuous Problems)
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17 pages, 1155 KiB  
Article
Symplectic-Structure-Preserving Uncertain Differential Equations
by Xiuling Yin, Xiulian Gao, Yanqin Liu, Yanfeng Shen and Jinchan Wang
Symmetry 2021, 13(8), 1424; https://doi.org/10.3390/sym13081424 - 4 Aug 2021
Viewed by 1849
Abstract
Uncertain differential equations are important mathematical models in uncertain environments. This paper investigates uncertain multi-dimensional and multiple-factor differential equations. First, the solvability of the equations is analyzed. The α-path distributions and expected values of solutions are given. Then, structure preserving uncertain differential [...] Read more.
Uncertain differential equations are important mathematical models in uncertain environments. This paper investigates uncertain multi-dimensional and multiple-factor differential equations. First, the solvability of the equations is analyzed. The α-path distributions and expected values of solutions are given. Then, structure preserving uncertain differential equations, uncertain Hamiltonian systems driven by Liu processes, which possess a kind of uncertain symplectic structures, are presented. A symplectic scheme with six-order accuracy and a Yao-Chen algorithm are applied to design an algorithm to solve uncertain Hamiltonian systems. At last, numerical experiments are given to investigate four uncertain Hamiltonian systems, which highlight the efficiency of our algorithm. Full article
(This article belongs to the Special Issue Conservation Laws, Symmetry Analysis and Variational Principle for Discontinuous Problems)
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13 pages, 3598 KiB  
Article
Huanglongbing Model under the Control Strategy of Discontinuous Removal of Infected Trees
by Weiwei Ling, Pinxia Wu, Xiumei Li and Liangjin Xie
Symmetry 2021, 13(7), 1164; https://doi.org/10.3390/sym13071164 - 28 Jun 2021
Cited by 1 | Viewed by 3435
Abstract
By using differential equations with discontinuous right-hand sides, a dynamic model for vector-borne infectious disease under the discontinuous removal of infected trees was established after understanding the transmission mechanism of Huanglongbing (HLB) disease in citrus trees. Through calculation, the basic reproductive number of [...] Read more.
By using differential equations with discontinuous right-hand sides, a dynamic model for vector-borne infectious disease under the discontinuous removal of infected trees was established after understanding the transmission mechanism of Huanglongbing (HLB) disease in citrus trees. Through calculation, the basic reproductive number of the model can be attained and the properties of the model are discussed. On this basis, the existence and global stability of the calculated equilibria are verified. Moreover, it was found that different I0 in the control strategy cannot change the dynamic properties of HLB disease. However, the lower the value of I0, the fewer HLB-infected citrus trees, which provides a theoretical basis for controlling HLB disease and reducing expenditure. Full article
(This article belongs to the Special Issue Conservation Laws, Symmetry Analysis and Variational Principle for Discontinuous Problems)
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6 pages, 429 KiB  
Article
Generalized Variational Principle for the Fractal (2 + 1)-Dimensional Zakharov–Kuznetsov Equation in Quantum Magneto-Plasmas
by Yan-Hong Liang and Kang-Jia Wang
Symmetry 2021, 13(6), 1022; https://doi.org/10.3390/sym13061022 - 7 Jun 2021
Cited by 4 | Viewed by 1978
Abstract
In this paper, we propose the fractal (2 + 1)-dimensional Zakharov–Kuznetsov equation based on He’s fractal derivative for the first time. The fractal generalized variational formulation is established by using the semi-inverse method and two-scale fractal theory. The obtained fractal variational principle is [...] Read more.
In this paper, we propose the fractal (2 + 1)-dimensional Zakharov–Kuznetsov equation based on He’s fractal derivative for the first time. The fractal generalized variational formulation is established by using the semi-inverse method and two-scale fractal theory. The obtained fractal variational principle is important since it not only reveals the structure of the traveling wave solutions but also helps us study the symmetric theory. The finding of this paper will contribute to the study of symmetry in the fractal space. Full article
(This article belongs to the Special Issue Conservation Laws, Symmetry Analysis and Variational Principle for Discontinuous Problems)
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