Related papers: Some remarks on the solution of linearisable secon…
We consider the one-dimensional porous medium equation $u_t=\left (u^nu_x \right )_x+\frac{\mu}{x}u^nu_x$. We derive point transformations of a general class that map this equation into itself or into equations of a similar class. In some…
We tersely review a recently introduced technique to identify systems of two nonlinearly-coupled Ordinary Di{\S}erential Equations (ODEs) solvable by algebraic operations; and we report some specifc examples of this kind, namely systems of…
A general method for solving linear differential equations of arbitrary order, is used to arrive at new representations for the solutions of the known differential equations, both without and with a source term. A new quasi-solvable…
Solving initial value problems and boundary value problems of Linear Ordinary Differential Equations (ODEs) plays an important role in many applications. There are various numerical methods and solvers to obtain approximate solutions…
It is shown, how to generate infinite sequences of differential equations of the second order based on some standard equations, using Euler-Imshenetsky-Darboux (EID) transformation. For all this, factorizations of differential operators and…
We describe a set of Gaussian Process based approaches that can be used to solve non-linear Ordinary Differential Equations. We suggest an explicit probabilistic solver and two implicit methods, one analogous to Picard iteration and the…
A new method of solving third-order ordinary complex differential equations (OCDEs) by generalizing Prelle-Singer. The idea which is a procedure for finding the solution for second-order differential equations in the real domain. We have…
The method and the formula of variation of constants for ordinary differential equations (ODEs) is a fundamental tool to analyze the dynamics of an ODE near an equilibrium. It is natural to expect that such a formula works for delay…
We introduce a new theory of generalised solutions which applies to fully nonlinear PDE systems of any order and allows for merely measurable maps as solutions. This approach bypasses the standard problems arising by the application of…
In a recent work [Manucci, Unger, ArXiv e-print 2404.10511, 2024], the authors propose using two generalized Lyapunov equations (GLEs) to derive a balancing-based model order reduction~(MOR) method for a general class of switched…
Welcome to a beautiful subject in scientific computing: numerical solution of ordinary differential equations (ODEs) with initial conditions.
A system of inhomogeneous second-order difference equations with linear parts given by noncommutative matrix coefficients are considered. Closed form of its solution is derived by means of newly defined delayed matrix sine/cosine using the…
The linearizability of differential equations was first considered by Lie for scalar second order semi-linear ordinary differential equations. Since then there has been considerable work done on the algebraic classification of linearizable…
In this paper we show that an arbitrary solution of one ordinary difference equation is also a solution for a hierarchy of integrable difference equations. We also provide an example of such a solution that is related to sequence generated…
Natural laws are often described through differential equations yet finding a differential equation that describes the governing law underlying observed data is a challenging and still mostly manual task. In this paper we make a step…
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…
The equivalence group is determined for systems of linear ordinary differential equations in both the standard form and the normal form. It is then shown that the normal form of linear systems reducible by an invertible point transformation…
Here we present a new approach to compute symmetries of rational second order ordinary differential equations (rational 2ODEs). This method can compute Lie symmetries (point symmetries, dynamical symmetries and non-local symmetries)…
We describe a way of solving a partial differential equation using the differential invariants of its point symmetries. By first solving its quotient PDE, which is given by the differential syzygies in the algebra of differential…
In this short note we discuss ordinary differential equations which linearize upon one (or more) differentiations. Although the subject is fairly elementary, equations of this type arise naturally in the context of integrable systems.