In this paper, we derive a novel numerical method to find out the numerical solution of fractional partial differential equations (PDEs) involving Caputo-Fabrizio (C-F) fractional derivatives. We first find out the approximation formula of C-F derivative of function tk. We approximate the C-F derivative in time with the help of the Legendre spectral method and approximation formula o...

We dispute the claim that Hermite functions (similar to derivatives of Gaussians) minimize a joint uncertainty relation in space and spatial frequency. These functions are found to maximize rather than minimize the uncertainty of the class of functions consisting of an mth-order polynomial times a Gaussian.

Using a recent classification of local symmetries of the vacuum Einstein equations, it is shown that there can be no observables for the vacuum gravitational field (in a closed universe) built as spatial integrals of local functions of Cauchy data and their derivatives.

Light as an external trigger is a valuable and easily controllable tool for directing chemical reactions with high spatial and temporal accuracy. Two o-nitrobenzyl derivatives, benzoyl- and thiophenyl-NPPOC, undergo photo-deprotection with significantly improved efficiency over that of the commonly used NPPOC group. The two- and twelvefold increase in photo-deprotection efficiency was proven us...

We consider uniformly elliptic and parabolic second-order equations with bounded zeroth-order and bounded VMO leading coefficients and possibly growing first-order coefficients. We look for solutions which are summable to the p-th power with respect to the usual Lebesgue measure along with their first and second-order derivatives with respect to the spatial variable.

We consider the strong solution of an initial boundary value problem for a system of evolution equations describing the flow of a generalized Newtonian fluid of power law type. For a rather large scale of growth rates we prove local initial regularity results such as higher integrability of the pressure function or the existence of the second spatial derivatives of the velocity field.

Twisted curves in R' are those which have independent derivatives up to order m . For plane and spatial curves the twisted condition is equivalent to never vanishing curvature and torsion respectively. We give a necessary and sufficient condition for a (q, p)-curve on a torus to be twisted, and use those curves to construct closed twisted curves in R'" for all m .

We consider divergence form uniformly parabolic SPDEs with bounded and measurable leading coefficients and possibly growing lower-order coefficients in the deterministic part of the equations. We look for solutions which are summable to the second power with respect to the usual Lebesgue measure along with their first derivatives with respect to the spatial variable.

A 3π algorithm is obtained in which all the derivatives are confined to a detector array. Distance weighting of backprojection coefficients of the algorithm is studied. A numerical experiment indicates that avoiding differentiation along the source trajectory improves spatial resolution. Another numerical experiment shows that the terms depending on the non-standard distance weighting 1/|x − y(...