Related papers: A note on higher-order differential operations
In this paper we present a recurrent relation for counting meaningful compositions of the higher-order differential operations on the space $R^{n}$ (n=3,4,...) and extract the non-trivial compositions of order higher than two.
This paper deals with the enumeration of the higher order non-trivial compositions of the differential operations and the directional derivative in the space $R^n$ ($n \geq 3$). We present the recurrences for a counting the higher order…
A variational equation of the third order in three-dimensional space is proposed which describes autoparallel curves of some connection.
This paper presents an algebraic approach to characterizing higher-order differential operators. While the foundational Leibniz rule addresses first-order derivatives, its extension to higher orders typically involves identities relating…
In this paper we determine the number of the meaningful compositions of higher order of the differential operations and Gateaux directional derivative.
First-order automatic differentiation is a ubiquitous tool across statistics, machine learning, and computer science. Higher-order implementations of automatic differentiation, however, have yet to realize the same utility. In this paper I…
The nature of so-called differential-algebraic operators and their approximations is constitutive for the direct treatment of higher-index differential-algebraic equations. We treat first-order differential-algebraic operators in detail and…
A generalization of the already studied transformations of the linear differential equation into a system of the first order equations is given. The proposed transformation gives possibility to get new forms of the N-dimensional system of…
We motivate and study an infinite sequence of binary operations on the ordinal numbers, extending the standard arithmetic on the ordinals to higher degrees of iteration. Connections to the hyperoperations on the natural numbers are…
In this paper we discuss some results related to commuting ordinary differential operators of rank greater than one.
In this paper we construct high order numerical methods for solving third and fourth orders nonlinear functional differential equations (FDE). They are based on the discretization of iterative methods on continuous level with the use of the…
In the past decades, the finite difference methods for space fractional operators develop rapidly; to the best of our knowledge, all the existing finite difference schemes, including the first and high order ones, just work on uniform…
We present new examples of complexes of differential operators of order $k$ (any given positive integer) that satisfy div-curl and/or $L^1$-duality estimates.
The function spaces of continuously differentiable functions are extensively studied and appear in various mathematical settings. In this context, we investigate the spaces of continuously fractional differentiable functions of order…
The problem of equivalency for linear differential operators of the first order is discussed.
We give a rigorous formulation of the intuitive idea that a differentiable map should be thesame thing as a locally, or infinitesimally, linear map: just as a linear map respects the operations of addition and multiplication by scalars ina…
Motivated by the problem of solving the Einstein equations, we discuss high order finite difference discretizations of first order in time, second order in space hyperbolic systems.Particular attention is paid to the case when first order…
In this work we study the solutions to some fractional higher-order equations. Special cases in which time-fractional derivatives take integer values are also examined and the explicit solutions are presented. Such solutions can be…
The present paper develops two concepts of pointwise differentiability of higher order for arbitrary subsets of Euclidean space defined by comparing their distance functions to those of smooth submanifolds. Results include that…
We perform conformal perturbation theory by marginal operators to first order. A suitable renormalization method is needed that makes the conformal invariance of the deformed correlation functions manifest. Combining the embedding space…