Analogues to Lie Method and Noether’s Theorem in Fractal Calculus

In this manuscript, we study symmetries of fractal differential equations. We show that using symmetry properties, one of the solutions can map to another solution. We obtain canonical coordinate systems for differential equations on fractal sets, which makes them simpler to solve. An analogue for Noether’s Theorem on fractal sets is given, and a corresponding conservative quantity is suggested. Several examples are solved to illustrate the results.

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Fractal Logistic Equation

Fractal and Fractional ◽

10.3390/fractalfract3030041 ◽

2019 ◽

Vol 3 (3) ◽

pp. 41 ◽

Cited By ~ 3

Author(s):

Alireza Khalili Golmankhaneh ◽

Carlo Cattani

Keyword(s):

Differential Equations ◽

Analytical Solutions ◽

Euler Method ◽

Logistic Equation ◽

Fractal Sets ◽

Logistic Equations ◽

Fractal Calculus ◽

Approximate Analytical Solutions ◽

Fractal Differential Equations ◽

Modeling Growth

In this paper, we give difference equations on fractal sets and their corresponding fractal differential equations. An analogue of the classical Euler method in fractal calculus is defined. This fractal Euler method presets a numerical method for solving fractal differential equations and finding approximate analytical solutions. Fractal differential equations are solved by using the fractal Euler method. Furthermore, fractal logistic equations and functions are given, which are useful in modeling growth of elements in sciences including biology and economics.

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SYMMETRIES, NEWTONOID VECTOR FIELDS AND CONSERVATION LAWS IN THE LAGRANGIAN k-SYMPLECTIC FORMALISM

Reviews in Mathematical Physics ◽

10.1142/s0129055x12500304 ◽

2012 ◽

Vol 24 (10) ◽

pp. 1250030 ◽

Cited By ~ 5

Author(s):

LUCÍA BUA ◽

IOAN BUCATARU ◽

MODESTO SALGADO

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Conservation Laws ◽

Conservation Law ◽

Vector Fields ◽

Partial Differential ◽

Noether’S Theorem ◽

Dynamical Symmetries ◽

Noether's Theorem ◽

Configuration Manifold

In this paper, we study symmetries, Newtonoid vector fields, conservation laws, Noether's theorem and its converse, in the framework of the k-symplectic formalism, using the Frölicher–Nijenhuis formalism on the space of k1-velocities of the configuration manifold.For the case k = 1, it is well known that Cartan symmetries induce and are induced by constants of motions, and these results are known as Noether's theorem and its converse. For the case k > 1, we provide a new proof for Noether's theorem, which shows that, in the k-symplectic formalism, each Cartan symmetry induces a conservation law. We prove that, under some assumptions, the converse of Noether's theorem is also true and we provide examples when this is not the case. We also study the relations between dynamical symmetries, Newtonoid vector fields, Cartan symmetries and conservation laws, showing when one of them will imply the others. We use several examples of partial differential equations to illustrate when these concepts are related and when they are not.

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Hamiltonian Fluid Dynamics

Lectures on Geophysical Fluid Dynamics ◽

10.1093/oso/9780195108088.003.0010 ◽

1998 ◽

Author(s):

Rick Salmon

Keyword(s):

Variational Principle ◽

Conservation Laws ◽

Variational Principles ◽

Symmetry And Conservation Laws ◽

The Body ◽

Hamilton’S Principle ◽

Hamilton's Principle ◽

Noether’S Theorem ◽

Noether's Theorem ◽

Symmetry Properties

In this final chapter, we return to the subject of the first: the fundamental principles of fluid mechanics. In chapter 1, we derived the equations of fluid motion from Hamilton’s principle of stationary action, emphasizing its logical simplicity and the resulting close correspondence between mechanics and thermodynamics. Now we explore the Hamiltonian approach more fully, discovering its other advantages. The most important of these advantages arise from the correspondence between the symmetry properties of the Lagrangian and the conservation laws of the resulting dynamical equations. Therefore, we begin with a very brief introduction to symmetry and conservation laws. Noether’s theorem applies to the equations that arise from variational principles like Hamilton’s principle. According to Noether’s theorem : If a variational principle is invariant to a continuous transformation of its dependent and independent variables, then the equations arising from the variational principle possess a divergence-form conservation law. The invariance property is also called a symmetry property. Thus Noether’s theorem connects symmetry properties and conservation laws. We shall neither state nor prove the general form of Noether’s theorem; to do so would require a lengthy digression on continuous groups. Instead we illustrate the connection between symmetry and conservation laws with a series of increasingly complex and important examples. These examples convey the flavor of the general theory. Our first example is very simple. Consider a body of mass m moving in one dimension. The body is attached to the end of a spring with spring-constant K. Let x(t) be the displacement of the body from its location when the spring is unstretched.

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Generalization of Noether’s Theorem in Modern Form to Non-variational Partial Differential Equations

Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science - Fields Institute Communications ◽

10.1007/978-1-4939-6969-2_5 ◽

2017 ◽

pp. 119-182 ◽

Cited By ~ 27

Author(s):

Stephen C. Anco

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Partial Differential ◽

Noether’S Theorem ◽

Noether's Theorem ◽

Modern Form

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Comparison of Different Approaches to Construct First Integrals for Ordinary Differential Equations

Abstract and Applied Analysis ◽

10.1155/2014/978636 ◽

2014 ◽

Vol 2014 ◽

pp. 1-15 ◽

Cited By ~ 7

Author(s):

Rehana Naz ◽

Igor Leite Freire ◽

Imran Naeem

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Direct Method ◽

Direct Methods ◽

First Integrals ◽

Physical Models ◽

Noether’S Theorem ◽

Direct Construction ◽

Noether's Theorem ◽

Partial Lagrangian

Different approaches to construct first integrals for ordinary differential equations and systems of ordinary differential equations are studied here. These approaches can be grouped into three categories: direct methods, Lagrangian or partial Lagrangian formulations, and characteristic (multipliers) approaches. The direct method and symmetry conditions on the first integrals correspond to first category. The Lagrangian and partial Lagrangian include three approaches: Noether’s theorem, the partial Noether approach, and the Noether approach for the equation and its adjoint as a system. The characteristic method, the multiplier approaches, and the direct construction formula approach require the integrating factors or characteristics or multipliers. The Hamiltonian version of Noether’s theorem is presented to derive first integrals. We apply these different approaches to derive the first integrals of the harmonic oscillator equation. We also study first integrals for some physical models. The first integrals for nonlinear jerk equation and the free oscillations of a two-degree-of-freedom gyroscopic system with quadratic nonlinearities are derived. Moreover, solutions via first integrals are also constructed.

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Non-local Integrals and Derivatives on Fractal Sets with Applications

Open Physics ◽

10.1515/phys-2016-0062 ◽

2016 ◽

Vol 14 (1) ◽

pp. 542-548 ◽

Cited By ~ 12

Author(s):

Alireza K. Golmankhaneh ◽

D. Baleanu

Keyword(s):

Differential Equations ◽

Cantor Sets ◽

Physical Models ◽

Scaling Properties ◽

Fractal Sets ◽

Fractal Differential Equations ◽

Non Local

AbstractIn this paper, we discuss non-local derivatives on fractal Cantor sets. The scaling properties are given for both local and non-local fractal derivatives. The local and non-local fractal differential equations are solved and compared. Related physical models are also suggested.

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Symmetry properties of fractional order transport equations

Proceedings of the Mavlyutov Institute of Mechanics ◽

10.21662/uim2012.1.010 ◽

2012 ◽

Vol 9 (1) ◽

pp. 59-64

Author(s):

R.K. Gazizov ◽

A.A. Kasatkin ◽

S.Yu. Lukashchuk

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Lie Group ◽

Initial Conditions ◽

Group Analysis ◽

Equivalence Transformation ◽

Point Change ◽

Infinitesimal Operator ◽

Symmetry Properties ◽

Fractional Differential

In the paper some features of applying Lie group analysis methods to fractional differential equations are considered. The problem related to point change of variables in the fractional differentiation operator is discussed and some general form of transformation that conserves the form of Riemann-Liouville fractional operator is obtained. The prolongation formula for extending an infinitesimal operator of a group to fractional derivative with respect to arbitrary function is presented. Provided simple example illustrates the necessity of considering both local and non-local symmetries for fractional differential equations in particular cases including the initial conditions. The equivalence transformation forms for some fractional differential equations are discussed and results of group classification of the wave-diffusion equation are presented. Some examples of constructing particular exact solutions of fractional transport equation are given, based on the Lie group methods and the method of invariant subspaces.

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Introduction

10.1093/oso/9780198788393.003.0001 ◽

2017 ◽

Author(s):

Laurent Baulieu ◽

John Iliopoulos ◽

Roland Sénéor

Keyword(s):

Conservation Laws ◽

Lagrangian Mechanics ◽

General Introduction ◽

Noether’S Theorem ◽

Noether's Theorem

General introduction with a review of the principles of Hamiltonian and Lagrangian mechanics. The connection between symmetries and conservation laws, with a presentation of Noether’s theorem, is included.

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