Symmetries of equations with functional arguments (original) (raw)

Symmetries of integro-differential equations

Reports on Mathematical Physics, 2001

A new general method is presented for the determination of Lie symmetry groups of integro-differential equations. The suggested method is a natural extension of the Ovsiannikov method developed for differential equations. The method leads to important applications for instance to the Vlasov Maxwell equations.

Symmetries of Integro-Differential Equations: A Survey of Methods Illustrated by the Benny Equations

2002

Classical Lie group theory provides a universal tool for calculating symmetry groups for systems of differential equations. However Lie's method is not as much effective in the case of integral or integro-differential equations as well as in the case of infinite systems of differential equations. This paper is aimed to survey the modern approaches to symmetries of integrodifferential equations. As an illustration, an infinite symmetry Lie algebra is calculated for a system of integro-differential equations, namely the well-known Benney equations. The crucial idea is to look for symmetry generators in the form of canonical Lie-Bäcklund operators.

A remark on nonlocal symmetries for the Calogero-Degasperis-Ibragimov-Shabat equation

We consider the Calogero-Degasperis-Ibragimov-Shabat (CDIS) equation and find the complete set of its nonlocal symmetries depending on the local variables and on the integral of the only local conserved density of the equation in question. The Lie algebra of these symmetries turns out to be a central extension of that of local generalized symmetries.

General Criterion of Invariance for Integro-Differential Equations

2010

General criterion of invariance of integro-differential equations under Lie symmetry group of point transformations is derived. It is a generalization of the previous form of the criterion to the case of a moving range of integration. This is the situation when a region of integration depends on external, with respect to integration, variables which leads to its explicit dependence on a group parameter.

Nonlocal Symmetries for Time-Dependent Order Differential Equations

2018

A new type of ordinary differential equation is introduced and discussed, namely, the time-dependent order ordinary differential equations. These equations can be solved via fractional calculus and are mapped into Volterra integral equations of second kind with singular integrable kernel. The solutions of the time-dependent order differential equations smoothly deforms solutions of the classical integer order ordinary differential equations into one-another, and can generate or remove singularities. An interesting symmetry of the solution in relation to the Riemann zeta function and Harmonic numbers was also proved.

Lie symmetries of first order neutral differential equations

Journal of Applied Mathematics and Computational Mechanics

In this paper we extend the method of obtaining symmetries of ordinary differential equations to first order non-homogeneous neutral differential equations with variable coefficients. The existing method for delay differential equations uses a Lie-Bäcklund operator and an Invariant Manifold Theorem to define the operators which are used to obtain the infinitesimal generators of the Lie group. In this paper, we adopt a different approach and use Taylor's theorem to obtain a Lie type invariance condition and the determining equations for a neutral differential equation. We then split this equation in a manner similar to that of ordinary differential equations to obtain an over-determined system of partial differential equations. These equations are then solved to obtain corresponding infinitesimals, and hence desired equivalent symmetries. We then obtain the symmetry algebra admitted by this neutral differential equation.

Lie Symmetry Analysis of the Hopf Functional-Differential Equation

Symmetry, 2015

In this paper, we extend the classical Lie symmetry analysis from partial differential equations to integro-differential equations with functional derivatives. We continue the work of Oberlack and Wacławczyk (2006, Arch. Mech. 58, 597), (2013, J. Math. Phys. 54, 072901), where the extended Lie symmetry analysis is performed in the Fourier space. Here, we introduce a method to perform the extended Lie symmetry analysis in the physical space where we have to deal with the transformation of the integration variable in the appearing integral terms. The method is based on the transformation of the product y(x)dx appearing in the integral terms and applied to the functional formulation of the viscous Burgers equation. The extended Lie symmetry analysis furnishes all known symmetries of the viscous Burgers equation and is able to provide new symmetries associated with the Hopf formulation of the viscous Burgers equation. Hence, it can be employed as an important tool for applications in continuum mechanics.

Symmetries and Exact Solutions of Some Partial Differential Equations

International Journal of Mechanics and Applications, 2021

Sophus Lie developed technique to obtain solutions of differential equations using continuous symmetries. Using these continuous symmetries, Peter E. Hydon developed technique to obtain discrete symmetries which led to finding further new solutions of the underlying equations. In this paper continuous and discrete symmetries of Korteweg de Vries and nonlinear filtration equations are analyzed. Using these symmetries group invariant solutions and the exact solutions of these equations are obtained.

Lie Symmetry Analysis for the General Classes of Generalized Modified Kuramoto-Sivashinsky Equation

Journal of Function Spaces

Lie symmetry analysis of differential equations proves to be a powerful tool to solve or at least reduce the order and nonlinearity of the equation. Symmetries of differential equations is the most significant concept in the study of DE’s and other branches of science like physics and chemistry. In this present work, we focus on Lie symmetry analysis to find symmetries of some general classes of KS-type equation. We also compute transformed equivalent equations and some invariant solutions of this equation.