A Conservative Scheme with Optimal Error Estimates for a Multidimensional Space-Fractional Gross-Pitaevskii Equation

The present work departs from an extended form of the classical multi-dimensional Gross-Pitaevskii equation, which considers fractional derivatives of the Riesz type in space, a generalized potential function and angular momentum rotation. It is well known that the classical system possesses functionals which are preserved throughout time. It is easy to check that the generalized fractional model considered in this work also possesses conserved quantities, whence the development of conservative and efficient numerical schemes is pragmatically justified. Motivated by these facts, we propose a finite-difference method based on weighted-shifted Grünwald differences to approximate the solutions of the generalized Gross-Pitaevskii system. We provide here a discrete extension of the uniform Sobolev inequality to multiple dimensions, and show that the proposed method is capable of preserving discrete forms of the mass and the energy of the model. Moreover, we establish thoroughly the stability and the convergence of the technique, and provide some illustrative simulations to show that the method is capable of preserving the total mass and the total energy of the generalized system. © 2019 Ahmed S. Hendy et al., published by Sciendo 2019

Discrete monotone method for space-fractional nonlinear reaction–diffusion equations

A discrete monotone iterative method is reported here to solve a space-fractional nonlinear diffusion–reaction equation. More precisely, we propose a Crank–Nicolson discretization of a reaction–diffusion system with fractional spatial derivative of the Riesz type. The finite-difference scheme is based on the use of fractional-order centered differences, and it is solved using a monotone iterative technique. The existence and uniqueness of solutions of the numerical model are analyzed using this approach, along with the technique of upper and lower solutions. This methodology is employed also to prove the main numerical properties of the technique, namely, the consistency, stability, and convergence. As an application, the particular case of the space-fractional Fisher’s equation is theoretically analyzed in full detail. In that case, the monotone iterative method guarantees the preservation of the positivity and the boundedness of the numerical approximations. Various numerical examples are provided to illustrate the validity of the numerical approximations. More precisely, we provide an extensive series of comparisons against other numerical methods available in the literature, we show detailed numerical analyses of convergence in time and in space against fractional and integer-order models, and we provide studies on the robustness and the numerical performance of the discrete monotone method. © 2019, The Author(s).Russian Foundation for Basic Research, RFBR: 19-01-00019Consejo Nacional de Ciencia y Tecnología, CONACYT: A1-S-45928The first author would like to acknowledge the financial support of the National Council for Science and Technology of Mexico (CONACYT). The second (and corresponding) author acknowledges financial support from CONACYT through grant A1-S-45928. ASH is financed by RFBR Grant 19-01-00019

Rigidly Rotating Strings in Stationary Spacetimes

In this paper we study the motion of a rigidly rotating Nambu-Goto test string in a stationary axisymmetric background spacetime. As special examples we consider the rigid rotation of strings in flat spacetime, where explicit analytic solutions can be obtained, and in the Kerr spacetime where we find an interesting new family of test string solutions. We present a detailed classification of these solutions in the Kerr background.Comment: 19 pages, Latex, 9 figures, revised for publication in Classical and Quantum Gravit

Surface-reconstructed Icosahedral Structures for Lead Clusters

We describe a new family of icosahedral structures for lead clusters. In general, structures in this family contain a Mackay icosahedral core with a reconstructed two-shell outer-layer. This family includes the anti-Mackay icosahedra, which have have a Mackay icosahedral core but with most of the surface atoms in hexagonal close-packed positions. Using a many-body glue potential for lead, we identify two icosahedral structures in this family which have the lowest energies of any known structure in the size range from 900 to 15000 lead atoms. We show that these structures are stabilized by a feature of the many-body glue part of the interatomic potential.Comment: 9 pages, 8 figure

Density-functional studies of tungsten trioxide, tungsten bronzes, and related systems

Tungsten trioxide adopts a variety of structures which can be intercalated with charged species to alter the electronic properties, thus forming `tungsten bronzes'. Similar optical effects are observed upon removing oxygen from WO_3, although the electronic properties are slightly different. Here we present a computational study of cubic and hexagonal alkali bronzes and examine the effects on cell size and band structure as the size of the intercalated ion is increased. With the exception of hydrogen (which is predicted to be unstable as an intercalate), the behaviour of the bronzes are relatively consistent. NaWO_3 is the most stable of the cubic systems, although in the hexagonal system the larger ions are more stable. The band structures are identical, with the intercalated atom donating its single electron to the tungsten 5d valence band. Next, this was extended to a study of fractional doping in the Na_xWO_3 system (0 < x < 1). A linear variation in cell parameter, and a systematic change in the position of the Fermi level up into the valence band was observed with increasing x. In the underdoped WO_3-x system however, the Fermi level undergoes a sudden jump into the conduction band at around x = 0.2. Lastly, three compounds of a layered WO_4&#215;a,wdiaminoalkane hybrid series were studied and found to be insulating, with features in the band structure similar to those of the parent WO_3 compound which relate well to experimental UV-visible spectroscopy results.Comment: 12 pages, 16 figure

A discrete Grönwall inequality and energy estimates in the analysis of a discrete model for a nonlinear time-fractional heat equation

In the present work, we investigate the efficiency of a numerical scheme to solve a nonlinear time-fractional heat equation with sufficiently smooth solutions, which was previously reported in the literature [Fract. Calc. Appl. Anal. 16: 892-910 (2013)]. In that article, the authors established the stability and consistency of the discrete model using arguments from Fourier analysis. As opposed to that work, in the present work, we use the method of energy inequalities to show that the scheme is stable and converges to the exact solution with order O(τ2-α + h4), in the case that 0 &lt; α &lt; 1 satisfies 3α ≥ 3/2, which means that 0.369 α ≤ 1. The novelty of the present work lies in the derivation of suitable energy estimates, and a discrete fractional Grönwall inequality, which is consistent with the discrete approximation of the Caputo fractional derivative of order 0 &lt; α &lt; 1 used for that scheme at tk+1/2. © 2020 by the authors.The first author wishes to acknowledge the support of RFBR Grant 19-01-00019. Meanwhile, the second author would like to acknowledge the financial support of the National Council for Science and Technology of Mexico (CONACYT). The second author acknowledges financial support from CONACYT through grant A1-S-45928. Acknowledgments: The authors wish to thank the guest editors for their kind invitation to submit a paper to the special issue of Mathematics MDPI on "Computational Mathematics and Neural Systems". They also wish to thank the anonymous reviewers for their comments and criticisms. All of their comments were taken into account in the revised version of the paper, resulting in a substantial improvement with respect to the original submission

Controlled cardiac reoxygenation does not improve myocardial function following global myocardial ischemia

AbstractBackgroundIt has been shown that abrupt re-exposure of ischemic myocardium to oxygen can lead to increased peroxidative damage to myocytes (oxygen paradox). Controlled cardiac reoxygenation, as an adjunct to substrate-enhanced cardioplegia, has been shown to improve myocardial function and limit reperfusion injury when utilizing standardized hyperoxic cardiopulmonary bypass (CPB). The objective of our study was to evaluate the effect of controlled reoxygenation on myocardial function following global ischemia employing normoxic CPB.Study designNineteen female swine (30–40kg) were placed on vented, normoxic CPB. They were subjected to 45–50min of unprotected global ischemia (aortic cross clamping) followed by 30min of controlled cardiac reperfusion utilizing substrate-enhanced cardioplegia. Group 1 maintained normoxic pO2 (O2 tension of 90–110mmHg). In Group 2, reoxygenation was titrated gradually and increased from venous to arterial levels (O2 tensions from 40 to 110mmHg over 15min). We measured coronary sinus blood samples for CK, CK-MB, nitric oxide, and conjugated dienes at baseline, 5min into the cardioplegic resuscitation, 5min after the cross clamp removal, and just prior to the termination of the study. Hearts were pathologically studied and scored for evidence of tissue peroxidation.ResultsAlthough not significantly different, Group 1 (normoxic reperfusion) animals were more likely to wean from CPB (p=0.141) and had a higher mean arterial pressure (p=0.556). In Group 1, conjugated dienes were significantly higher 5min into the resuscitative protocol (p=0.018) and at the termination of bypass (p=0.035). Five of six animals in Group 1 eventually attained normal sinus rhythm as opposed to three out of 13 in Group 2 (p=0.041). There was no significant difference in histology scoring between the two groups for tissue peroxidation.ConclusionThis study of controlled cardiac reoxygenation in a lethal ischemic swine model failed to demonstrate that the use of controlled reoxygenation on the myocardial function following global ischemia was better with maintained normoxic pO2 (with O2 tensions of 90–110mmHg) than when reoxygenation was titrated gradually and increased from venous to arterial levels (O2 tensions from 40 to 110mmHg over 15min)