Related papers: Symbolic Computation of Conservation Laws, General…
In an earlier work by a subset of the present authors, the method of the so-called neural deflation was introduced towards identifying a complete set of functionally independent conservation laws of a nonlinear dynamical system. Here, we…
All three-point and five-point conservation laws for the discrete Korteweg-de Vries equations are found. These conservation laws satisfy a functional equation, which we solve by reducing it to a system of partial differential equations. Our…
A class of generalized nonlinear p-Laplacian evolution equations is studied. These equations model radial diffusion-reaction processes in $n\geq 1$ dimensions, where the diffusivity depends on the gradient of the flow. For this class, all…
Algorithms for the computation of the real zeros of hypergeometric functions which are solutions of second order ODEs are described. The algorithms are based on global fixed point iterations which apply to families of functions satisfying…
The interpretation of numerical methods, such as finite difference methods for differential equations, as point estimators suggests that formal uncertainty quantification can also be performed in this context. Competing statistical…
Taylor series methods show a newfound promise for the solution of non-stiff ordinary differential equations (ODEs) given the rise of new compiler-enhanced techniques for calculating high order derivatives. In this paper we detail a new…
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…
In the present work, we demonstrate how the pseudoinverse concept from linear algebra can be used to represent and analyze the boundary conditions of linear systems of partial differential equations. This approach has theoretical and…
In this paper we consider the spatial semi-discretization of conservative PDEs. Such finite dimensional approximations of infinite dimensional dynamical systems can be described as flows in suitable matrix spaces, which in turn leads to the…
A numerical framework is proposed for identifying partial differential equations (PDEs) governing dynamical systems directly from their observation data using Chebyshev polynomial approximation. In contrast to data-driven approaches such as…
Ordinary differential equations (ODEs) and ordinary difference systems (O$\Delta$Ss) invariant under the actions of the Lie groups $\mathrm{SL}_x(2)$, $\mathrm{SL}_y(2)$ and $\mathrm{SL}_x(2)\times\mathrm{SL}_y(2)$ of projective…
We present an algorithm for determining the Lie point symmetries of differential equations on fixed non transforming lattices, i.e. equations involving both continuous and discrete independent variables. The symmetries of a specific…
Discrete Differential Equations (DDEs) are functional equations that relate polynomially a power series $F(t,u)$ in $t$ with polynomial coefficients in a "catalytic" variable $u$ and the specializations, say at $u=1$, of $F(t,u)$ and of…
Symbolic regression plays a crucial role in modern scientific research thanks to its capability of discovering concise and interpretable mathematical expressions from data. A key challenge lies in the search for parsimonious and…
Differential equations are widely used to describe complex dynamical systems with evolving parameters in nature and engineering. Effectively learning a family of maps from the parameter function to the system dynamics is of great…
All low-order conservation laws are found for a general class of nonlinear wave equations in one dimension with linear damping which is allowed to be time-dependent. Such equations arise in numerous physical applications and have attracted…
We describe a neural-based method for generating exact or approximate solutions to differential equations in the form of mathematical expressions. Unlike other neural methods, our system returns symbolic expressions that can be interpreted…
The reduction operators, i.e., the operators of nonclassical (conditional) symmetry, of (1+1)-dimensional second order linear parabolic partial differential equations and all the possible reductions of these equations to ordinary…
We present a Lohner-type algorithm for rigorous integration of systems of Delay Differential Equations (DDEs) with multiple delays and its application in computation of Poincar\'e maps to study the dynamics of some bounded, eternal…
In this paper we propose a method to couple two or more explicit numerical schemes approximating the same time-dependent PDE, aiming at creating new schemes which inherit advantages of the original ones. We consider both advection equations…