Related papers: Derivative Corrections to the Trapezoidal Rule
A method is proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra. As an example the generalized quantum plane is studied. It is found that there is a strong correlation, but not a…
Two approximations, derived from continuous expansions of Riemann-Liouville fractional derivatives into series involving integer order derivatives, are studied. Using those series, one can formally transform any problem that contains…
We consider a system of differential equations and obtain its solutions with exponential asymptotics and analyticity with respect to the spectral parameter. Solutions of such type have importance in studying spectral properties of…
In this paper we consider extensions of the gradient elasticity models proposed earlier by the second author to describe materials with fractional non-locality and fractality using the techniques developed recently by the first author. We…
When training predictive models on data with missing entries, the most widely used and versatile approach is a pipeline technique where we first impute missing entries and then compute predictions. In this paper, we view prediction with…
In backward error analysis, an approximate solution to an equation is compared to the exact solution to a nearby modified equation. In numerical ordinary differential equations, the two agree up to any power of the step size. If the…
This article presents a novel approach to enhance the accuracy of classical quadrature rules by incorporating correction terms. The proposed method is particularly effective when the position of an isolated discontinuity in the function and…
Structure learning algorithms that learn the graph of a Bayesian network from observational data often do so by assuming the data correctly reflect the true distribution of the variables. However, this assumption does not hold in the…
This work presents an analytical and computational study of fractional-order delay differential equations formulated using both the conformable and Caputo derivatives. For the conformable case, we develop the associated integral,…
We propose a new integral based on Taylor measures, study its properties extensively, and we illustrate that it includes many concepts from mathematics as special cases. In particular, the new integral emerges as a generalization of the…
We have recently presented an extension of the standard variational calculus to include the presence of deformed derivatives in the Lagrangian of a system of particles and in the Lagrangian density of field-theoretic models. Classical…
The task of finding an extension to a given partial drawing of a graph while adhering to constraints on the representation has been extensively studied in the literature, with well-known results providing efficient algorithms for…
In this paper, we systematically review a series of effective methods for studying the qualitative properties of solutions to fractional equations. Beginning with the pioneering extension method and the method of moving planes in integral…
Coefficient inverse problems related to identifying the right-hand side of an equation with use of additional information is of interest among inverse problems for partial differential equations. When considering non-stationary problems,…
In this brief note we critically examine the process of partial and of total differentiation, showing some of the problems that arise when we relate both concepts. A way to solve all the problems is proposed.
Inspired by applications in optimal control of semilinear elliptic partial differential equations and physics-integrated imaging, differential equation constrained optimization problems with constituents that are only accessible through…
We study the convergence of the derivative expansion for flow equations. The convergence strongly depends on the choice for the infrared regularisation. Based on the structure of the flow, we explain why optimised regulators lead to better…
In this paper, we are interested in the study of a problem with fractional derivatives having boundary conditions of integral types. The problem represents a Caputo type advection-diffusion equation where the fractional order derivative…
Calculating the long term solution of ordinary differential equations, such as those of the $N$-body problem, is central to understanding a wide range of dynamics in astrophysics, from galaxy formation to planetary chaos. Because generally…
This paper is concerned with a posteriori error bounds for linear transport equations and related questions of contriving corresponding adaptive solution strategies in the context of Discontinuous-Petrov-Galerkin schemes. After indicating…