Related papers: Lie groups and numerical solutions of differential…
In this survey we discuss a wide variety of aspects related to Lie group integrators. These numerical integration schemes for differential equations on manifolds have been studied in a general and systematic manner since the 1990s and the…
The main purpose of this article is to show how symmetry structures in partial differential equations can be preserved in a discrete world and reflected in difference schemes. Three different structure preserving discretizations of the…
In these lectures we review two procedures for constructing finite difference numerical schemes that preserve symmetries of differential equations. The first approach is based on Lie's infinitesimal symmetry generators, while the second…
We present a generalization of Lie's method for finding the group invariant solutions to a system of partial differential equations. Our generalization relaxes the standard transversality assumption and encompasses the common situation…
A set of linear second-order differential equations is converted into a semigroup, whose algebraic structure is used to generate many novel equations. Two independent methods that can be used to derive the equations of the semigroup are…
A procedure for obtaining a "minimal" discretization of a partial differential equation, preserving all of its Lie point symmetries is presented. "Minimal" in this case means that the differential equation is replaced by a partial…
In this paper, we study invariants of linear differential operators with respect to algebraic Lie pseudogroups. Then we use these invariants and the principle of n-invariants to get normal forms (or models) of the differential operators and…
A method is introduced for the construction of meshless discretization schemes which preserve Lie symmetries of the differential equations that these schemes approximate. The method exploits the fact that equivariant moving frames provide a…
Singular perturbation theory plays a central role in the approximate solution of nonlinear differential equations. However, applying these methods is a subtle art owing to the lack of globally applicable algorithms. Inspired by the fact…
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…
A renormalization group method with the Lie symmetry is presented for the singular perturbation problems. Asymptotic solutions are obtained as group-invariant solutions under approximate Lie group admitted by perturbed differential…
Given a differential equation with infinite-dimensional symmetry pseudo-group it is shown, using an example, that it is generally not possible to construct enough joint invariants to form an invariant numerical scheme of the equation. To…
The Lie linearizability criteria are extended to complex functions for complex ordinary differential equations. The linearizability of complex ordinary differential equations is used to study the linearizability of corresponding systems of…
We consider difference equations of order four and determine the one parameter Lie group of transformations (Lie symmetries) that leave them invariant. We introduce a technique for finding their first integrals and discuss the association…
Lie symmetry transformations that leave a differential equation invariant play a fundamental role in science and mathematics. Such Lie symmetry groups uniquely determine their Lie symmetry algebras. Exact differential elimination algorithms…
We review studies on the application of Lie group methods to delay ordinary differential equations (DODEs). For first- and second-order DODEs with a single delay parameter that depends on independent and dependent variables, the group…
Lie group theory states that knowledge of a $m$-parameters solvable group of symmetries of a system of ordinary differential equations allows to reduce by $m$ the number of equations. We apply this principle by finding some \emph{affine…
Nonlinear ODEs invariant under the group SL(2,R) are solved numerically. We show that solution methods incorporating the Lie point symmetries provide better results than standard methods.
The interpretation of numerical methods, such as finite difference methods for differential equations, as point estimators allows for formal statistical quantification of the error due to discretisation in the numerical context. Competing…
To combine a feedforward neural network (FNN) and Lie group (symmetry) theory of differential equations (DEs), an alternative artificial NN approach is proposed to solve the initial value problems (IVPs) of ordinary DEs (ODEs). Introducing…