Research

20 pages, 1648 KiB

Open AccessArticle

Accurate Spectral Collocation Solutions to 2nd-Order Sturm–Liouville Problems

by Călin-Ioan Gheorghiu

Symmetry 2021, 13(3), 385; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13030385 - 27 Feb 2021

Cited by 2 | Viewed by 1631

This work is about the use of some classical spectral collocation methods as well as with the new software system Chebfun in order to compute the eigenpairs of some high order Sturm–Liouville eigenproblems. The analysis is divided into two distinct directions. For problems [...] Read more.

This work is about the use of some classical spectral collocation methods as well as with the new software system Chebfun in order to compute the eigenpairs of some high order Sturm–Liouville eigenproblems. The analysis is divided into two distinct directions. For problems with clamped boundary conditions, we use the preconditioning of the spectral collocation differentiation matrices and for hinged end boundary conditions the equation is transformed into a second order system and then the conventional ChC is applied. A challenging set of “hard” benchmark problems, for which usual numerical methods (FD, FE, shooting, etc.) encounter difficulties or even fail, are analyzed in order to evaluate the qualities and drawbacks of spectral methods. In order to separate “good” and “bad” (spurious) eigenvalues, we estimate the drift of the set of eigenvalues of interest with respect to the order of approximation N. This drift gives us a very precise indication of the accuracy with which the eigenvalues are computed, i.e., an automatic estimation and error control of the eigenvalue error. Two MATLAB codes models for spectral collocation (ChC and SiC) and another for Chebfun are provided. They outperform the old codes used so far and can be easily modified to solve other problems. Full article

(This article belongs to the Special Issue Geometric Analysis of Nonlinear Partial Differential Equations)

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38 pages, 501 KiB

Open AccessArticle

Lagrangian Curve Flows on Symplectic Spaces

by Chuu-Lian Terng and Zhiwei Wu

Symmetry 2021, 13(2), 298; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13020298 - 09 Feb 2021

Cited by 2 | Viewed by 1283

Abstract

A smooth map

γ

in the symplectic space

R^{2 n}

is Lagrangian if

γ, γ_{x}, \dots

,

γ_{x}^{(2 n - 1)}

are linearly independent and the span of [...] Read more.

A smooth map

γ

in the symplectic space

R^{2 n}

is Lagrangian if

γ, γ_{x}, \dots

,

γ_{x}^{(2 n - 1)}

are linearly independent and the span of

γ, γ_{x}, \dots, γ_{x}^{(n - 1)}

is a Lagrangian subspace of

R^{2 n}

. In this paper, we (i) construct a complete set of differential invariants for Lagrangian curves in

R^{2 n}

with respect to the symplectic group

S p (2 n)

, (ii) construct two hierarchies of commuting Hamiltonian Lagrangian curve flows of C-type and A-type, (iii) show that the differential invariants of solutions of Lagrangian curve flows of C-type and A-type are solutions of the Drinfeld-Sokolov’s

{\hat{C}}_{n}^{(1)}

-KdV flows and

{\hat{A}}_{2 n - 1}^{(2)}

-KdV flows respectively, (iv) construct Darboux transforms, Permutability formulas, and scaling transforms, and give an algorithm to construct explicit soliton solutions, (v) give bi-Hamiltonian structures and commuting conservation laws for these curve flows. Full article

(This article belongs to the Special Issue Geometric Analysis of Nonlinear Partial Differential Equations)

10 pages, 256 KiB

Open AccessArticle

Generalized Navier–Stokes Equations and Dynamics of Plane Molecular Media

by Alexei Kushner and Valentin Lychagin

Symmetry 2021, 13(2), 288; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13020288 - 08 Feb 2021

Cited by 1 | Viewed by 1291

Abstract

The first analysis of media with internal structure were done by the Cosserat brothers. Birkhoff noted that the classical Navier–Stokes equation does not fully describe the motion of water. In this article, we propose an approach to the dynamics of media formed by [...] Read more.

The first analysis of media with internal structure were done by the Cosserat brothers. Birkhoff noted that the classical Navier–Stokes equation does not fully describe the motion of water. In this article, we propose an approach to the dynamics of media formed by chiral, planar and rigid molecules and propose some kind of Navier–Stokes equations for their description. Examples of such media are water, ozone, carbon dioxide and hydrogen cyanide. Full article

(This article belongs to the Special Issue Geometric Analysis of Nonlinear Partial Differential Equations)

10 pages, 503 KiB

Open AccessArticle

On Monotonic Pattern in Periodic Boundary Solutions of Cylindrical and Spherical Kortweg–De Vries–Burgers Equations

by Alexey Samokhin

Symmetry 2021, 13(2), 220; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13020220 - 29 Jan 2021

Cited by 1 | Viewed by 1266

Abstract

We studied, for the Kortweg–de Vries–Burgers equations on cylindrical and spherical waves, the development of a regular profile starting from an equilibrium under a periodic perturbation at the boundary. The regular profile at the vicinity of perturbation looks like a periodical chain of [...] Read more.

We studied, for the Kortweg–de Vries–Burgers equations on cylindrical and spherical waves, the development of a regular profile starting from an equilibrium under a periodic perturbation at the boundary. The regular profile at the vicinity of perturbation looks like a periodical chain of shock fronts with decreasing amplitudes. Further on, shock fronts become decaying smooth quasi-periodic oscillations. After the oscillations cease, the wave develops as a monotonic convex wave, terminated by a head shock of a constant height and equal velocity. This velocity depends on integral characteristics of a boundary condition and on spatial dimensions. In this paper the explicit asymptotic formulas for the monotonic part, the head shock and a median of the oscillating part are found. Full article

(This article belongs to the Special Issue Geometric Analysis of Nonlinear Partial Differential Equations)

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7 pages, 227 KiB

Open AccessArticle

Contact Symmetries of a Model in Optimal Investment Theory

by Daniel J. Arrigo and Joseph A. Van de Grift

Symmetry 2021, 13(2), 217; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13020217 - 28 Jan 2021

Cited by 2 | Viewed by 1407

Abstract

It is generally known that Lie symmetries of differential equations can lead to a reduction of the governing equation(s), lead to exact solutions of these equations and, in the best case scenario, lead to a linearization of the original equation. In this paper, [...] Read more.

It is generally known that Lie symmetries of differential equations can lead to a reduction of the governing equation(s), lead to exact solutions of these equations and, in the best case scenario, lead to a linearization of the original equation. In this paper, we consider a model from optimal investment theory where we show the governing equation possesses an extensive contact symmetry and, through this, we show it is linearizable. Several exact solutions are provided including a solution to a particular terminal value problem. Full article

(This article belongs to the Special Issue Geometric Analysis of Nonlinear Partial Differential Equations)

9 pages, 258 KiB

Open AccessArticle

Quotients of Euler Equations on Space Curves

by Anna Duyunova, Valentin Lychagin and Sergey Tychkov

Symmetry 2021, 13(2), 186; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13020186 - 25 Jan 2021

Cited by 3 | Viewed by 1115

Abstract

Quotients of partial differential equations are discussed. The quotient equation for the Euler system describing a one-dimensional gas flow on a space curve is found. An example of using the quotient to solve the Euler system is given. Using virial expansion of the [...] Read more.

Quotients of partial differential equations are discussed. The quotient equation for the Euler system describing a one-dimensional gas flow on a space curve is found. An example of using the quotient to solve the Euler system is given. Using virial expansion of the Planck potential, we reduce the quotient equation to a series of systems of ordinary differential equations (ODEs). Possible solutions of the ODE system are discussed. Full article

(This article belongs to the Special Issue Geometric Analysis of Nonlinear Partial Differential Equations)

12 pages, 287 KiB

Open AccessFeature PaperArticle

Iterated Darboux Transformation for Isothermic Surfaces in Terms of Clifford Numbers

by Jan L. Cieśliński and Zbigniew Hasiewicz

Symmetry 2021, 13(1), 148; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13010148 - 17 Jan 2021

Cited by 2 | Viewed by 1621

Abstract

Isothermic surfaces are defined as immersions with the curvture lines admitting conformal parameterization. We present and discuss the reconstruction of the iterated Darboux transformation using Clifford numbers instead of matrices. In particulalr, we derive a symmetric formula for the two-fold Darboux transformation, explicitly [...] Read more.

Isothermic surfaces are defined as immersions with the curvture lines admitting conformal parameterization. We present and discuss the reconstruction of the iterated Darboux transformation using Clifford numbers instead of matrices. In particulalr, we derive a symmetric formula for the two-fold Darboux transformation, explicitly showing Bianchi’s permutability theorem. In algebraic calculations an important role is played by the main anti-automorphism (reversion) of the Clifford algebra

C (4, 1)

and the spinorial norm in the corresponding Spin group. Full article

(This article belongs to the Special Issue Geometric Analysis of Nonlinear Partial Differential Equations)

11 pages, 869 KiB

Open AccessFeature PaperArticle

Singularities in Euler Flows: Multivalued Solutions, Shockwaves, and Phase Transitions

by Valentin Lychagin and Mikhail Roop

Symmetry 2021, 13(1), 54; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13010054 - 31 Dec 2020

Cited by 1 | Viewed by 1422

Abstract

In this paper, we analyze various types of critical phenomena in one-dimensional gas flows described by Euler equations. We give a geometrical interpretation of thermodynamics with a special emphasis on phase transitions. We use ideas from the geometrical theory of partial differential equations [...] Read more.

In this paper, we analyze various types of critical phenomena in one-dimensional gas flows described by Euler equations. We give a geometrical interpretation of thermodynamics with a special emphasis on phase transitions. We use ideas from the geometrical theory of partial differential equations (PDEs), in particular symmetries and differential constraints, to find solutions to the Euler system. Solutions obtained are multivalued and have singularities of projection to the plane of independent variables. We analyze the propagation of the shockwave front along with phase transitions. Full article

(This article belongs to the Special Issue Geometric Analysis of Nonlinear Partial Differential Equations)

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25 pages, 401 KiB

Open AccessFeature PaperArticle

Differential Invariants of Linear Symplectic Actions

by Jørn Olav Jensen and Boris Kruglikov

Symmetry 2020, 12(12), 2023; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12122023 - 07 Dec 2020

Cited by 2 | Viewed by 1252

Abstract

We consider the equivalence problem for symplectic and conformal symplectic group actions on submanifolds and functions of symplectic and contact linear spaces. This is solved by computing differential invariants via the Lie-Tresse theorem. Full article

(This article belongs to the Special Issue Geometric Analysis of Nonlinear Partial Differential Equations)

15 pages, 321 KiB

Open AccessArticle

Joint Invariants of Linear Symplectic Actions

by Fredrik Andreassen and Boris Kruglikov

Symmetry 2020, 12(12), 2020; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12122020 - 07 Dec 2020

Cited by 3 | Viewed by 1188

Abstract

We review computations of joint invariants on a linear symplectic space, discuss variations for an extension of group and space and relate this to other equivalence problems and approaches, most importantly to differential invariants. Full article

(This article belongs to the Special Issue Geometric Analysis of Nonlinear Partial Differential Equations)

12 pages, 295 KiB

Open AccessArticle

Nonlocal Conservation Laws of PDEs Possessing Differential Coverings

by Iosif Krasil’shchik

Symmetry 2020, 12(11), 1760; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12111760 - 23 Oct 2020

Cited by 2 | Viewed by 1214

Abstract

In his 1892 paper, L. Bianchi noticed, among other things, that quite simple transformations of the formulas that describe the Bäcklund transformation of the sine-Gordon equation lead to what is called a nonlocal conservation law in modern language. Using the techniques of differential [...] Read more.

In his 1892 paper, L. Bianchi noticed, among other things, that quite simple transformations of the formulas that describe the Bäcklund transformation of the sine-Gordon equation lead to what is called a nonlocal conservation law in modern language. Using the techniques of differential coverings, we show that this observation is of a quite general nature. We describe the procedures to construct such conservation laws and present a number of illustrative examples. Full article

(This article belongs to the Special Issue Geometric Analysis of Nonlinear Partial Differential Equations)

17 pages, 831 KiB

Open AccessFeature PaperArticle

Geometrical Formulation for Adjoint-Symmetries of Partial Differential Equations

by Stephen C. Anco and Bao Wang

Symmetry 2020, 12(9), 1547; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12091547 - 19 Sep 2020

Cited by 6 | Viewed by 1447

Abstract

A geometrical formulation for adjoint-symmetries as one-forms is studied for general partial differential equations (PDEs), which provides a dual counterpart of the geometrical meaning of symmetries as tangent vector fields on the solution space of a PDE. Two applications of this formulation are [...] Read more.

A geometrical formulation for adjoint-symmetries as one-forms is studied for general partial differential equations (PDEs), which provides a dual counterpart of the geometrical meaning of symmetries as tangent vector fields on the solution space of a PDE. Two applications of this formulation are presented. Additionally, for systems of evolution equations, adjoint-symmetries are shown to have another geometrical formulation given by one-forms that are invariant under the flow generated by the system on the solution space. This result is generalized to systems of evolution equations with spatial constraints, where adjoint-symmetry one-forms are shown to be invariant up to a functional multiplier of a normal one-form associated with the constraint equations. All of the results are applicable to the PDE systems of interest in applied mathematics and mathematical physics. Full article

(This article belongs to the Special Issue Geometric Analysis of Nonlinear Partial Differential Equations)

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