This research article introduces a novel operator termed q-convolution, strategically integrated with foundational principles of q-calculus. Leveraging this innovative operator alongside q-Bernoulli ...polynomials, a distinctive class of functions emerges, characterized by both analyticity and bi-univalence. The determination of initial coefficients within the Taylor-Maclaurin series for this function class is accomplished, showcasing precise bounds. Additionally, explicit computation of the second Hankel determinant is provided. These pivotal findings, accompanied by their corollaries and implications, not only enrich but also extend previously published results.
Numerical methods for fractional diffusion Bonito, Andrea; Borthagaray, Juan Pablo; Nochetto, Ricardo H. ...
Computing and visualization in science,
15/12, Letnik:
19, Številka:
5-6
Journal Article
Recenzirano
Odprti dostop
We present three schemes for the numerical approximation of fractional diffusion, which build on different definitions of such a non-local process. The first method is a PDE approach that applies to ...the spectral definition and exploits the extension to one higher dimension. The second method is the integral formulation and deals with singular non-integrable kernels. The third method is a discretization of the Dunford–Taylor formula. We discuss pros and cons of each method, error estimates, and document their performance with a few numerical experiments.
The aim of this paper is to derive a refined first-order expansion formula in Rn , the goal being to get an optimal reduced remainder, compared to the one obtained by usual Taylor's formula. For a ...given function, the formula we derived is obtained by introducing a linear combination of the first derivatives, computed at n + 1 equally spaced points. We show how this formula can be applied to two important applications: the interpolation error and the finite elements error estimates. In both cases, we illustrate under which conditions a significant improvement of the errors can be obtained, namely how the use of the refined expansion can reduce the upper bound of error estimates.
In the framework of the European project CORTEX, included in the H2020 program, the experimental campaign carried out with the CROCUS reactor, at the École Polytechnique Fédérale de Lausanne (EPFL) ...in Switzerland, aims at setting up methodologies and tools for the analysis and the interpretation of neutron noise in view of their industrial applications. An important part of the CORTEX project deals with code development and validation. In this paper, a new methodology for the simulation of neutron noise has been formulated and implemented in the APOLLO3 code. In the first part of this paper, the experimental setup and the methodology for the experimental data analysis developed by CEA are presented. The experimental outcomes are reported in terms of Cross-Power Spectral Densities (CPSD) of detector responses. The results produced with CEA analysis are compared to those obtained by EPFL. The second part of the work presents the noise simulator based on an improved point-kinetics model employed in a fully heterogeneous 2D transport calculation including its theoretical derivation. The computed results, i.e. the temporal signals of the detector responses, are treated to recover the CPSDs to be compared to the experimental ones. In the third part of the work, the paper presents a series of interpretive exercises performed with the IPK noise model aiming at showing its simulation capabilities and at trying to address some of the discrepancies observed in the direct comparison. Beside the promising outcomes of the simulations, some work is still being done from the experimental and computation sides to better comprehend the problem before the extension of the model to full 3D problems and industrial application.
We investigate the problem of approximating the matrix function $f(A)$ by $r(A)$, with $f$ a Markov function, $r$ a rational interpolant of $f$, and $A$ a symmetric Toeplitz matrix. In a first step, ...we obtain a new upper bound for the relative interpolation error $1-r/f$ on the spectral interval of $A$. By minimizing this upper bound over all interpolation points, we obtain a new, simple and sharp a priori bound for the relative interpolation error. We then consider three different approaches of representing and computing the rational interpolant $r$. Theoretical and numerical evidence is given that any of these methods for a scalar argument allows to achieve high precision, even in the presence of finite precision arithmetic. We finally investigate the problem of efficiently evaluating $r(A)$, where it turns out that the relative error for a matrix argument is only small if we use a partial fraction decomposition for $r$ following Antoulas and Mayo. An important role is played by a new stopping criterion which ensures to automatically find the degree of $r$ leading to a small error, even in presence of finite precision arithmetic.
The supercritical carbon dioxide (S-COsub.2) Brayton cycle efficiency increases as the compressor inlet condition approaches the critical point. However, the thermodynamic properties of COsub.2 vary ...dramatically near the critical point, and phase change is most likely to happen. Both cavitation and condensation bring about significant adverse effects on the performance of compressors. In this paper, the quantitative effects of nonequilibrium condensation and cavitation on the performance of an S-COsub.2 centrifugal compressor with different inlet-relative entropy values are investigated. The properties of COsub.2 were provided by the real-gas property table, and the nonequilibrium phase-change model was adopted. The numerical simulation method with the nonequilibrium phase-change model was validated in the Lettieri nozzle and Sandia compressor. Furthermore, simulations were carried out in a two-stage centrifugal compressor under conditions of various inlet-relative entropy values. The type of nonequilibrium phase change can be distinguished by inlet-relative entropy. Cavitation makes the choke mass flow rate decrease due to the drop in the speed of sound. Condensation mainly occurs on the leading edge of the main blade at a large mass flow rate, but cavitation occurs on the splitter. The condensation is more evenly distributed on the main blade, but the cavitation is mainly centered on the leading edge.
•Experimental and numerical results on heat transfer in PCM/foam are presented.•Morphology, orientation effects, and IR-camera melting front tracking, are shown.•Melting front and fraction ...comparisons show a good agreement for all cases.•Liquid phase velocities are shown to appreciate morphology and orientation effects.•Porosity has a higher impact on melting compared to PPI and orientation.
Phase Change Materials (PCM) are promising materials for thermal energy storage systems. Since they present a relatively low thermal conductivity, they are often embedded in an open cell a metallic foam to enhance the overall thermal conductivity. In this paper, both experimental and numerical results on PCMs coupled with aluminum foams under different heat fluxes, porosities, number of Pores Per Inch (PPIs) and orientation are presented. The test cell is equipped with a Zincum Selenide window that allows to capture the whole temperature distribution by means of a IR camera. The melting front position in time is tracked by means of a MATLAB® algorithm based on IR camera images that are useful for a more robust tracking of melting front. Numerical simulations are performed with references to the porous media volume-averaged approach, under the assumption of local thermal non-equilibrium between the two phases. The most updated correlations for the porous media closing coefficients are taken from the literature. All the experiments are compared with numerical simulations, showing a very good agreement. After showing the effects of the different input parameters on melting front evolution, an analysis in terms of different convective heat losses to the environment and melting temperature range is presented to appreciate how these two variables affect the melting front position. Finally, total melting front evolution has been compared between experiments and simulations, showing a good agreement. This has been evaluated for different conditions, showing that a decrease in the porosity drastically reduces the melting time, while PPI has no relevant effect and small effects can be observed from orientation.
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