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An algorithm to compute polynomial conserved densities of polynomial nonlinear lattices is presented. The algorithm is implemented in Mathematica and can be used as an automated integrability test. With the code diffdens.m, conserved…
In this paper we shall introduce a simple, effective numerical method for finding differential operators for scalar and vector-valued functions on surfaces. The key idea of our algorithm is to develop an intrinsic and unified way to compute…
We provide linearizability criteria for a class of systems of third-order ordinary differential equations (ODEs) that is cubically semi-linear in the first derivative, by differentiating a system of second-order quadratically semi-linear…
Scalar conservation laws in one space variable allow a Lagrangian (particle path) formulation. The Lagrangian trajectory in the infinite-dimensional group of diffeomorphisms on the physical space can be written as a system of conservation…
We study a class of degenerate parabolic equations with boundary point degeneracy in dimensions N>=2 and investigate the associated boundary observability problem by means of shape design. While one-dimensional degenerate models have been…
The explicit integrability of second order ordinary differential equations invariant under time-translation and rescaling is investigated. Quadratic systems generated from the linearisable version of this class of equations are analysed to…
Recent advancements in machine learning have transformed the discovery of physical laws, moving from manual derivation to data-driven methods that simultaneously learn both the structure and parameters of governing equations. This shift…
Invariants of general linear system of two hyperbolic partial differential equations (PDEs) are derived under transformations of the dependent and independent variables by real infinitesimal method earlier. Here a subclass of the general…
Motivated by important applications in image processing, we study a class of second-order geometric quasilinear hyperbolic partial differential equations (PDEs). This is inspired by the recent development of second-order damping systems…
One way of constructing explicit expressions of solutions of integrable systems of Partial Differential Equations (PDEs) goes via the Darboux method. This requires the construction of Darboux matrices. Here we introduce a novel algorithm to…
A unification of characteristic mode decomposition for all method-of-moment formulations of field integral equations describing free-space scattering is derived. The work is based on an algebraic link between impedance and transition…
In this paper, approximate nonlinear self-adjointness for perturbed PDEs is introduced and its properties are studied. Consequently, approximate conservation laws which cannot be obtained by the approximate Noether theorem are constructed…
We show that the so-called hidden potential symmetries considered in a recent paper [Gandarias M., Physica A, 2008, V.387, 2234-2242] are ordinary potential symmetries that can be obtained using the method introduced by Bluman and…
This paper develops methods for simplifying systems of partial differential equations that have families of conservation laws which depend on functions of the independent or dependent variables. In some cases, such methods can be combined…
A new method for the computation of conserved densities of nonlinear differential-difference equations is applied to Toda lattices and discretizations of the Korteweg-de Vries and nonlinear Schrodinger equations. The algorithm, which can be…
We develop a new approach to the classification of integrable equations of the form $$ u_{xy}=f(u, u_x, u_y, \triangle_z u \triangle_{\bar z}u, \triangle_{z\bar z}u), $$ where $\triangle_{ z}$ and $\triangle_{\bar z}$ are the…
The classical problem of construction of Gardner's deformations for infinite-dimensional completely integrable systems of evolutionary partial differential equations (PDE) amounts essentially to finding the recurrence relations between the…
English version of abstract: The dynamic optimization problems treated by the calculus of variations are usually solved with the help of the 2nd order Euler-Lagrange differential equations. These equations are, generally speaking,…
Recent work in deep learning focuses on solving physical systems in the Ordinary Differential Equation or Partial Differential Equation. This current work proposed a variant of Convolutional Neural Networks (CNNs) that can learn the hidden…
In this paper, we present the Partial Integral Equation (PIE) representation of linear Partial Differential Equations (PDEs) in one spatial dimension, where the PDE has spatial integral terms appearing in the dynamics and the boundary…