Related papers: Derivative Corrections to the Trapezoidal Rule
A randomised trapezoidal quadrature rule is proposed for continuous functions which enjoys less regularity than commonly required. Indeed, we consider functions in some fractional Sobolev space. Various error bounds for this randomised rule…
A systematic theory is introduced for calculating the derivatives of quaternion matrix function with respect to quaternion matrix variables. The proposed methodology is equipped with the matrix product rule and chain rule and it is able to…
Several techniques were proposed to model the Piecewise linear (PWL) functions, including convex combination, incremental and multiple choice methods. Although the incremental method was proved to be very efficient, the attention of the…
Approximations to the integral $\int_a^b\int_c^d f(x,y)\,dy\,dx$ are obtained under the assumption that the partial derivatives of the integrand are in an $L^p$ space, for some $1\leq p\leq\infty$. We assume ${\lVert f_{xy}\rVert}_p$ is…
This work is an extension of previous work by Alazah et al. [M. Alazah, S. N. Chandler-Wilde, and S. La Porte, Numerische Mathematik, 128(4):635-661, 2014]. We split the computation of the Fresnel Integrals into 3 cases: a truncated Taylor…
The linearization of complex ordinary differential equations is studied by extending Lie's criteria for linearizability to complex functions of complex variables. It is shown that the linearization of complex ordinary differential equations…
Transcendental Liouvillian extensions are differential fields, in which one can model poly-logarithmic, hyperexponential, and trigonometric functions, logarithmic integrals, and their (nested) rational expressions. For such an extension…
We present a family of high order trapezoidal rule-based quadratures for a class of singular integrals, where the integrand has a point singularity. The singular part of the integrand is expanded in a Taylor series involving terms of…
Variational data assimilation technique applied to identification of optimal approximations of derivatives near boundary is discussed in frames of one-dimensional wave equation. Simplicity of the equation and of its numerical scheme allows…
New techniques for evaluating the closed time path action for non-equilibrium quantum fields are presented. A derivative expansion is performed using a proper time kernel. Applications relevant to the scalar field theory of warm inflation…
The implicit boundary integral method (IBIM) provides a framework to construct quadrature rules on regular lattices for integrals over irregular domain boundaries. This work provides a systematic error analysis for IBIMs on uniform…
In the 1990's exponential-type error bounds appeared in the theory of radial basis functions. This kind of error bounds is very powerful. However it only measures the difference between the approximant and approximand. Mathematicians and…
A straightforward 3-point quadrature formula of closed type is derived that improves on Simpson's rule. Just using the additional information of the integrand's derivative at the two endpoints we show the error is sixth order in grid…
The construction of modified equations is an important step in the backward error analysis of symplectic integrator for Hamiltonian systems. In the context of partial differential equations, the standard construction leads to modified…
A general procedure for constructing conservative numerical integrators for time dependent partial differential equations is presented. In particular, linearly implicit methods preserving a time discretised version of the invariant is…
Fractional derivatives are generalization to classical integer-order derivatives. The rules which are true for classical derivative need not hold for the fractional derivatives, for example, we cannot simply add the fractional orders…
This article provides an accessible introduction to fractional derivatives, a concept that extends classical calculus by allowing derivatives of non-integer order. It explores both the fundamental definitions and some of the most relevant…
Connections between Lie derivatives and the deviation equation has been investigated in spaces with affine connection. The deviation equations of the geodesics as well as deviation equations of non-geodesics trajectories have been obtained…
Rough paths techniques give the ability to define solutions of stochastic differential equations driven by signals $X$ which are not semimartingales and whose $p$-variation is finite only for large values of $p$. In this context, rough…
This paper surveys recent analytical and numerical research on linear problems for the integral fractional Laplacian, fractional obstacle problems, and fractional minimal graphs. The emphasis is on the interplay between regularity,…