Diffusion over tensor fields via Lie group PDE flows: Lagrangian action approach

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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Spacetime symmetries

10.1093/oso/9780198802877.003.0006 ◽

2018 ◽

Author(s):

Michael Kachelriess

Keyword(s):

Riemannian Manifold ◽

Conservation Laws ◽

Vector Fields ◽

Killing Vector ◽

Geodesic Equation ◽

Spacetime Symmetries ◽

Tensor Fields ◽

Killing Vector Fields ◽

Covariant Derivatives ◽

Killing Equation

This chapter introduces tensor fields, covariant derivatives and the geodesic equation on a (pseudo-) Riemannian manifold. It discusses how symmetries of a general space-time can be found from the Killing equation, and how the existence of Killing vector fields is connected to global conservation laws.

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ε‎-Invariance

10.1093/oso/9780198821656.003.0006 ◽

2018 ◽

Author(s):

Ercüment H. Ortaçgil

Keyword(s):

Lie Algebra ◽

Lie Group

The discussions up to Chapter 4 have been concerned with the Lie group. In this chapter, the Lie algebra is constructed by defining the operators ∇ and ∇̃.

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The Centralizer

10.1093/oso/9780198821656.003.0005 ◽

2018 ◽

Author(s):

Ercüment H. Ortaçgil

Keyword(s):

Lie Group ◽

Group Structure ◽

Diffeomorphism Group ◽

Group Diff ◽

Local Solutions

The pseudogroup of local solutions in Chapter 3 defines another pseudogroup by taking its centralizer inside the diffeomorphism group Diff(M) of a manifold M. These two pseudogroups define a Lie group structure on M.

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Some groups with T1 primitive ideal spaces

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s144678870002259x ◽

1985 ◽

Vol 38 (1) ◽

pp. 55-64 ◽

Cited By ~ 2

Author(s):

A. L. Carey ◽

W. Moran

Keyword(s):

Normal Subgroup ◽

Lie Groups ◽

Lie Group ◽

Locally Compact Group ◽

Primitive Ideal ◽

Ideal Space ◽

Solvable Lie Groups ◽

Locally Compact ◽

Simply Connected ◽

Connected Lie Group

AbstractLet G be a second countable locally compact group possessing a normal subgroup N with G/N abelian. We prove that if G/N is discrete then G has T1 primitive ideal space if and only if the G-quasiorbits in Prim N are closed. This condition on G-quasiorbits arose in Pukanzky's work on connected and simply connected solvable Lie groups where it is equivalent to the condition of Auslander and Moore that G be type R on N (-nilradical). Using an abstract version of Pukanzky's arguments due to Green and Pedersen we establish that if G is a connected and simply connected Lie group then Prim G is T1 whenever G-quasiorbits in [G, G] are closed.

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