Related papers: Theory of characteristics for first order partial …
Using irreducible representations of semi simple Lie algebras, we construct Klein geometries of arbitrarily high order.
Some differential equations are considered in the context of Synthetic Differential Geometry. Here, this means that not only nilpotent infinitesimals, but also the formation of function spaces, is exploited. In particular, we utilize…
This article reviews the use of differential forms and Lie derivatives to find symmetries of differential equations, as originally presented by Harrison and Estabrook, J. Math. Phys., 12 (1971), 653. An outline of the method is given,…
We give simple upper and lower bounds for the order of a Klein geometry
The linearization of complex ordinary differential equations is studied by extending Lie's criteria for linearizability to complex functions of complex variables. It is shown that the linearization of complex ordinary differential equations…
In this paper, we further consider the symmetry-based method for seeking nonlocally related systems for partial differential equations. In particular, we show that the symmetry-based method for partial differential equations is the natural…
Symmetry is a powerful tool for finding analytical solutions to differential equations, both partial and ordinary, via the similarity variables or via the invariance of the equation under group transformations. It is the largest group of…
We extend two of the methods previously introduced to find discrete symmetries of differential equations to the case of difference and differential-difference equations. As an example of the application of the methods, we construct the…
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…
The theory of Lie remarkable equations, i.e. differential equations characterized by their Lie point symmetries, is reviewed and applied to ordinary differential equations. In particular, we consider some relevant Lie algebras of vector…
In this dissertation we study basic local differential geometry, projective differential geometry, and prolongations of overdetermined geometric partial differential equations. It is simple to prolong an n-th order linear ordinary…
First, we use the theory of characteristics of first order partial differential equations to derive the guiding equation directly from the Quantum Evolution Equation (QEE). After obtaining the general result, we apply it to a set of…
Lie symmetry analysis is an established method for generating symmetries of differential equations. We apply this method together the generalized fundamental theorem of double reduction. In particular, Noether symmetries and some associated…
Using geometric methods for linearizing systems of second order cubically semi-linear ordinary differential equations and third order quintically semi-linear ordinary differential equations, we extend to the fourth order by differentiating…
The notion of the characteristic Lie algebra of the discrete hyperbolic type equation is introduced. An effective algorithm to compute the algebra for the equation given is suggested. Examples and further applications are discussed.
An algorithmic method to exploit a general class of infinitesimal symmetries for reducing stochastic differential equations is presented and a natural definition of reconstruction, inspired by the classical reconstruction by quadratures, is…
The geometrical theory of partial differential equations in the absolute sense, without any additional structures, is developed. In particular the symmetries need not preserve the hierarchy of independent and dependent variables. The order…
This paper gives a first step towards developing synthetic differential geometry within homotopy type theory. Its model theory will be discussed in a subsequent paper.
In this article two implementations of a symmetric finite difference algorithm for a first-order partial differential equation are discussed. The considered partial differential equation discribes the time evolution of the crack length…
We give accurate and ergodic numerical methods for semilinear, second-order Langevin stochastic partial differential equations (SPDE). As a byproduct, we also give good geometric numerical methods for their infinite-dimensional Hamiltonian…