Numerical Analysis and Partial Differential Equations. Volume V. Surveys in Applied Mathematics.

1960 ◽  
Vol 67 (3) ◽  
pp. 306
Author(s):  
Lawrence A. Weller ◽  
George E. Forsythe ◽  
Paul C. Rosenbloom
Keyword(s):  
Numerical Analysis ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Applied Mathematics ◽  
Partial Differential

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 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.

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