scholarly journals Geometrical Formulation for Adjoint-Symmetries of Partial Differential Equations

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1547
Author(s):  
Stephen C. Anco ◽  
Bao Wang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Vector Fields ◽  
Evolution Equations ◽  
Applied Mathematics ◽  
Tangent Vector ◽  
Solution Space ◽  
Partial Differential ◽  
Geometrical Meaning ◽  
Geometrical Formulation

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.

scholarly journals On the relations between some well-known methods and the projective Riccati equations

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 613-618
Author(s):  
Şamil Akçağıl
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Riccati Equations ◽  
Nonlinear Evolution Equations ◽  
Alternative Methods ◽  
Computational Techniques ◽  
Partial Differential ◽  
Traditional Methods

AbstractSolving nonlinear evolution equations is an important issue in the mathematical and physical sciences. Therefore, traditional methods, such as the method of characteristics, are used to solve nonlinear partial differential equations. A general method for determining analytical solutions for partial differential equations has not been found among traditional methods. Due to the development of symbolic computational techniques many alternative methods, such as hyperbolic tangent function methods, have been introduced in the last 50 years. Although all of them were introduced as a new method, some of them are similar to each other. In this study, we examine the following four important methods intensively used in the literature: the tanh–coth method, the modified Kudryashov method, the F-expansion method and the generalized Riccati equation mapping method. The similarities of these methods attracted our attention, and we give a link between the methods and a system of projective Riccati equations. It is possible to derive new solution methods for nonlinear evolution equations by using this connection.

scholarly journals ON SOME ASPECTS OF THE GEOMETRY OF DIFFERENTIAL EQUATIONS IN PHYSICS

2004 ◽  
Vol 01 (03) ◽  
pp. 265-284 ◽  
Author(s):  
XAVIER GRÀCIA ◽  
MIGUEL C. MUÑOZ-LECANDA ◽  
NARCISO ROMÁN-ROY
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Control Theory ◽  
Review Paper ◽  
Applied Mathematics ◽  
Field Theories ◽  
Partial Differential ◽  
Singular Differential Equations ◽  
Systems Of Differential Equations ◽  
Hamilton Equations

In this review paper, we consider three kinds of systems of differential equations, which are relevant in physics, control theory and other applications in engineering and applied mathematics; namely: Hamilton equations, singular differential equations, and partial differential equations in field theories. The geometric structures underlying these systems are presented and commented on. The main results concerning these structures are stated and discussed, as well as their influence on the study of the differential equations with which they are related. In addition, research to be developed in these areas is also commented on.

scholarly journals PROLONGATIONS OF GEOMETRIC OVERDETERMINED SYSTEMS

2006 ◽  
Vol 17 (06) ◽  
pp. 641-664 ◽  
Author(s):  
THOMAS BRANSON ◽  
ANDREAS ČAP ◽  
MICHAEL EASTWOOD ◽  
A. ROD GOVER
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Wide Class ◽  
Linear Equations ◽  
Solution Space ◽  
Partial Differential ◽  
Sharp Bounds ◽  
Overdetermined Systems ◽  
Closed Systems

We show that a wide class of geometrically defined overdetermined semilinear partial differential equations may be explicitly prolonged to obtain closed systems. As a consequence, in the case of linear equations we extract sharp bounds on the dimension of the solution space.

scholarly journals Introduction: Big data and partial differential equations

2017 ◽  
Vol 28 (6) ◽  
pp. 877-885 ◽  
Author(s):  
YVES VAN GENNIP ◽  
CAROLA-BIBIANE SCHÖNLIEB
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Large Scale ◽  
Data Science ◽  
Applied Mathematics ◽  
Mathematical Finance ◽  
Energy Functional ◽  
Continuous Change ◽  
Opinion Formation ◽  
Partial Differential

Partial differential equations (PDEs) are expressions involving an unknown function in many independent variables and their partial derivatives up to a certain order. Since PDEs express continuous change, they have long been used to formulate a myriad of dynamical physical and biological phenomena: heat flow, optics, electrostatics and -dynamics, elasticity, fluid flow and many more. Many of these PDEs can be derived in a variational way, i.e. via minimization of an ‘energy’ functional. In this globalised and technologically advanced age, PDEs are also extensively used for modelling social situations (e.g. models for opinion formation, mathematical finance, crowd motion) and tasks in engineering (such as models for semiconductors, networks, and signal and image processing tasks). In particular, in recent years, there has been increasing interest from applied analysts in applying the models and techniques from variational methods and PDEs to tackle problems in data science. This issue of the European Journal of Applied Mathematics highlights some recent developments in this young and growing area. It gives a taste of endeavours in this realm in two exemplary contributions on PDEs on graphs [1, 2] and one on probabilistic domain decomposition for numerically solving large-scale PDEs [3].

Sign in / Sign up

Export Citation Format

Share Document