Analytical Approach for Nonlinear Partial Differential Equations of Fractional Order

The purpose of the paper is to present analytical and numerical solutions of a degenerate parabolic equation with time-fractional derivatives arising in the spatial diffusion of biological populations. The homotopy-perturbation method is employed for solving this class of equations, and the time-fractional derivatives are described in the sense of Caputo. Comparisons are made with those derived by Adomian's decomposition method, revealing that the homotopy perturbation method is more accurate and convenient than the Adomian's decomposition method. Furthermore, the results reveal that the approximate solution continuously depends on the time-fractional derivative and the proposed method incorporating the Caputo derivatives is a powerful and efficient technique for solving the fractional differential equations without requiring linearization or restrictive assumptions. The basis ideas presented in the paper can be further applied to solve other similar fractional partial differential equations.     

Structure of high-Tc superconducting Bi-Sr-Ca-Cu-O films prepared by spray pyrolysis

High-T_c superconducting films of Bi-Sr-Ca-Cu-O have been synthesized by spray pyrolysis of nitrate precursors onto yttria stabilized zirconia (YSZ), followed by rapid annealing at 840°C in air/oxygen. The films are preferentially oriented with c-axis parallel to the plane of the substrate surface. The diffraction patterns of the films correspond to a mixed phase orthorhombic structure, with different c-values of 24.49 Å, 30.78 Å and 37.25 Å. These films exhibit superconductivity at 91 K and zero resistance at 77 K.     

Growth temperature induced effects in non-polar a-plane GaN on r-plane sapphire substrate by RF-MBE

Non-polar a-plane GaN films were grown on an r-plane sapphire substrate by plasma assisted molecular beam epitaxy (PAMBE). The effect of growth temperature on structural, morphological and optical properties has been studied. The growth of non-polar a-plane (1 1 ?2 0) orientation of the GaN epilayers were confirmed by high resolution X-ray diffraction (HRXRD) study. The X-ray rocking curve (XRC) full width at half maximum of the (1 1 ?2 0) reflection shows in-plane anisotropic behavior and found to decrease with increase in growth temperature. The atomic force micrograph (AFM) shows island-like growth for the film grown at a lower temperature. Surface roughness has been decreased with increase in growth temperature. Room temperature photoluminescence shows near band edge emission at 3.434–3.442 eV. The film grown at 800 °C shows emission at 2.2 eV, which is attributed to yellow luminescence along with near band edge emission.     

Structural and magnetic property of Mn:ZnO bulk ceramic doped with rare earth (Gd/Sm) atoms

This paper reports the paramagnetic behavior of Mn doped ZnO co-doped with rare earth (Gd and Sm) atoms. The formation of secondary rare earth oxides (Gd₂O₃ and Sm₂O₃) is confirmed from the X-ray diffraction patterns. The rare earth oxides in the system forbids the grain growth and interconnection between the grains. The weak link between the grains suppresses the long range exchange interaction between the Mn ions and hence, reduces the ferromagnetic ordering. Owing to the large mismatch between ionic radii of rare earth and transition metal atoms inside the matrix, the rare earth element cannot contribute to promote ferromagnetic behavior in Mn doped ZnO, irrespective of their high individual magnetic moments.     

Multifragmentation and the phase transition: A systematic study of the multifragmentation of 1A GeV Au, La and Kr

A systematic analysis of the multifragmentation (MF) in fully reconstructed events from 1A GeV Au, La and Kr collisions with C has been performed. Detailed comparisons of the various fragment properties are presented as a function of excitation energy, E*_th. The charged particle multiplicity from MF stage shows a saturation beyond E*_th ∼ 8 MeV/nucleon for Kr. The universal behavior of intermediate mass fragment yields and of the size of the largest fragment is observed only for Au and La when scaled with size of the system. The Kr data are found to lack this property. Moments of the fragment size distribution show that the Kr MF is different than the MF of Au and La. A power law behavior is observed for Au and La with exponent τ>2, while for Kr τ<2. The results are compared with the statistical multifragmentation model (SMM). A single value of all the parameters of the model fits the data for all the three systems. The breakup of Au and La is consistent with a continuous phase transition. The data indicate that both E*_th and the isotope ratio temperature T _Hc-DT decrease with increase in system size at the critical point. The breakup temperature obtained from SMM also shows the same trend as seen in data. This trend is attributed primarily to the increasing Coulomb energy with finite size effects playing a smaller role. The percolation and Ising model studies for finite size neutral matter show behavior which is opposite to the one seen in the present work. EOS Collaboration     

Effect of Gd doping on structural, electrical and magnetic properties of BiFeO3 electroceramic

Room temperature multiferroic electroceramics of Gd doped BiFeO₃ monophasic materials have been synthesized adopting a slow step sintering schedule. Incorporation of Gd nucleates the development of orthorhombic grain growth habit without the appearance of any significant impurity phases with respect to original rhombohedral (R3c) phase of un-doped BiFeO₃. It is observed that, the materials showed room temperature enhanced electric polarization as well as ferromagnetism when rare earth ions like Gd doping is critically optimized (x=0.15) in the composition formula of Bi_1+2xGd_2x/2Fe_1−2xO₃. We believe that magnetic moment of Gd⁺³ ions in Gd doped BiFeO₃ tends to align in the same direction with respect to ferromagnetic component associated with the iron sub lattice. The dielectric constant as well as loss factor shows strong dispersion at lower frequencies and the value of leakage current is greatly suppressed with the increase in concentration of x in the above composition. Addition of excess bismuth and Gd (x=0.1 and 0.15) caused structural transformation as well as compensated bismuth loss during high temperature sintering. Doping of Gd in BiFeO₃ also suppresses spiral spin modulation structure, which can change Fe-O-Fe bond angle or spin order resulting in enhanced ferromagnetic property.     

A high‐order B‐spline collocation scheme for solving a nonhomogeneous time‐fractional diffusion equation

A high‐accuracy numerical approach for a nonhomogeneous time‐fractional diffusion equation with Neumann and Dirichlet boundary conditions is described in this paper. The time‐fractional derivative is described in the sense of Riemann‐Liouville and discretized by the backward Euler scheme. A fourth‐order optimal cubic B‐spline collocation (OCBSC) method is used to discretize the space variable. The stability analysis with respect to time discretization is carried out, and it is shown that the method is unconditionally stable. Convergence analysis of the method is performed. Two numerical examples are considered to demonstrate the performance of the method and validate the theoretical results. It is shown that the proposed method is of order O(Δx⁴ + Δt^{2 ? α}) convergence, where α ∈ (0,1) . Moreover, the impact of fractional‐order derivative on the solution profile is investigated. Numerical results obtained by the present method are compared with those obtained by the method based on standard cubic B‐spline collocation method. The CPU time for present numerical method and the method based on cubic B‐spline collocation method are provided.