Related papers: Integration of an arbitrary linear ODE
We introduce the Optimizing a Discrete Loss (ODIL) framework for the numerical solution of Partial Differential Equations (PDE) using machine learning tools. The framework formulates numerical methods as a minimization of discrete residuals…
For a system of linear partial differential equations (LPDEs) we introduce an operator equation for auxiliary operators. These operators are used to construct a kernel of an integral transformation leading the LPDE to the separation of…
We describe a Lohner-type algorithm for the computation of rigorous upper bounds for reachable set for control systems, solutions of ordinary differential inclusions and perturbations of ODEs.
We are concerned with the arithmetic of solutions to ordinary or partial nonlinear differential equations which are algebraic in the indeterminates and their derivatives. We call these solutions D-algebraic functions, and their equations…
In this paper we develop a general concept of Lax operators on algebraic curves introduced in [1]. We observe that the space of Lax operators is closed with respect to their usual multiplication as matrix-valued functions. We construct the…
Using the theory of Hilbert direct integrals, we introduce and study a monotonicity-preserving operation, termed the integral resolvent mixture. It combines arbitrary families of monotone operators acting on different spaces and linear…
An integrable system is often formulated as a flat connection, satisfying a Lax equation. It is given in terms of compatible systems having a common solution called the ``wave function" $\Psi$ living in a Lie group $G$, which satisfies some…
An explicit formula to find symmetry recursion operators for partial differential equations (PDEs) is obtained from new results connecting variational integrating factors and non-variational symmetries. The formula is special case of a…
The Lindblad master equation can always be transformed into a first-order linear ordinary differential equation (1ODE) for the coherence vector. We pose the inverse problem: given a finite-dimensional, non-homogeneous 1ODE, does a…
This short, self-contained article seeks to introduce and survey continuous-time deep learning approaches that are based on neural ordinary differential equations (neural ODEs). It primarily targets readers familiar with ordinary and…
Several applied problems are characterized by the need to numerically solve equations with an operator function (matrix function). In particular, in the last decade, mathematical models with a fractional power of an elliptic operator and…
In the present paper authors introduce the L_n-integral transform and the inverse integral transform for n = 2^k, k=0,1,2,..., as a generalization of the classical Laplace transform and the inverse Laplace transform, respectively.…
In a recent article, the class of functions from the integers to the integers computable in polynomial time has been characterized using discrete ordinary differential equations (ODE), also known as finite differences. Doing so, we pointed…
The Fundamental Theorem of Integral Calculus links the integrand and its antiderivative via a simple first order differential equation. A numerical solution of this ode yields the antiderivative and hence the required integral. This…
We present two algorithms for computing what we call the absolute factorization of a difference operator. We also give an algorithm to solve third order difference equations in terms of second order equations, together with applications to…
In this paper we introduce and investigate a new kind of functional (including ordinary and evolutionary partial) differential equations. The main goal of this paper is to explore our new philosophy by some examples on functional ODEs and…
It is a classical fact that the exponential function is solution of the integral equation $ \int_0^X f(x)dx + f(0) =f(X)$. If we slightly modify this equation to $ \int_0^X f(x)dx+f(0)=f(\alpha X)$ with $\alpha\in ]0,1[$, it seems that no…
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…
We consider the algebra of mixed multidimensional integral operators. In particular, Fredholm integral operators of the first and second kind belongs to this algebra. For the piecewise constant kernels we provide an explicit representation…
Meleshko presented a new method for reducing third order autonomous ordinary differential equations (ODEs) to Lie linearizable second order ODEs. We extended his work by reducing fourth order autonomous ODEs to second and third order…