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We present the finite difference/element method for a two-dimensional modified fractional diffusion equation. The analysis is carried out first for the time semi-discrete scheme, and then for the full discrete scheme. The time discretization is based on theL1-approximation for the fractional derivative terms and the second-order backward differentiation formula for the classical first order derivative term. We use finite element method for the spatial approximation in full discrete scheme. We show that both the semi-discrete and full discrete schemes are unconditionally stable and convergent. Moreover, the optimal convergence rate is obtained. Finally, some numerical examples are tested in the case of one and two space dimensions and the numerical results confirm our theoretical analysis.

In this paper, we discuss the time-space Caputo-Riesz fractional diffusion equation with variable coefficients on a finite domain. The finite difference schemes for this equation are provided. We theoretically prove and numerically verify that the implicit finite difference scheme is unconditionally stable (the explicit scheme is conditionally stable with the stability condition(∆y) β < C) and 2nd order convergent in space direction, and (2 − γ)-th order convergent in time direction, where γ ∈ (0, 1].

This article discusses the spectral collocation method for numerically solving nonlocal problems: one‐dimensional space fractional advection–diffusion equation; and two‐dimensional linear/nonlinear space fractional advection–diffusion equation. The differentiation matrixes of the left and right Riemann–Liouville and Caputo fractional derivatives are derived for any collocation points within any given bounded interval. Several numerical examples with different boundary conditions are computed to verify the efficiency of the numerical schemes and confirm the exponential convergence; the physical simulations for Lévy–Feller advection–diffusion equation and space fractional Fokker–Planck equation with initial δ‐peak and reflecting boundary conditions are performed; and the eigenvalue distributions of the iterative matrix for a variety of systems are displayed to illustrate the stabilities of the numerical schemes in more general cases. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 514–535, 2014

Three‐dimensional (3D) photonic crystals with face‐centered cubic structure were successfully fabricated on soft polyester fabrics via vertical deposition self‐assembly, with monodisperse P(St‐MAA) microspheres prepared by soap‐free emulsion copolymerization. The resultant polyester fabrics exhibited bright structural colors through the well‐ordered photonic crystal microstructure without any chemical dyes and pigments. The tunable structural colors across the whole visible region confirmed by reflectance spectra could be adjusted by controlling the diameters of the microspheres and viewing angles, and this was consistent with the law of the Bragg diffraction. The resultant polyester fabrics also presented some favorable properties including double‐sided coloration effect, clear fabric texture, and soft fabric handle. The structural coloration by vertical deposition self‐assembly of P(St‐MAA) photonic crystals may provide a new strategy for textile coloration without using chemical colorants, and have a potential to reduce the pollution in the current textile dyeing and printing processes. © 2014 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2015, 132, 41750.

A simple small‐size printed antenna for eight‐band long‐term evolution (LTE)/GSM/UMTS wireless wide area network operation in an internal mobile handset application is proposed. Occupying a compact size of 15 × 40 mm2 on the ungrounded region of the system circuit board and a low profile of 4 mm, the presented antenna is a coupled‐fed configuration. With the longer branch of the feeding strip, a λ/4 resonant mode at around 2.7 GHz can be obtained to cover the desired LTE2500 operation. A fundamental λ/4 resonant mode over the desired lower band of LTE700/GSM850/900 can also be generated at 750 MHz by the long meandered coupling strip (length of about 113 mm). Moreover, two additional higher‐order resonant modes at around 1.75 and 2.25 GHz can also be obtained by the coupling strip. By including a chip inductor L at the feeding point, the impedance matching over the lower and upper bands can be improved in order to achieve the two wide operating bands of 698–960 and 1710–2690 MHz. With the following properties of small size, ease of fabrication and good radiation characteristics, the proposed antenna is a good candidate for the fourth LTE mobile phone applications.

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