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.     

Observation of enhanced room temperature multiferroicity in nanoparticles BiFeO3

Impurity free monophasic rhombohedral BiFeO₃ (BFO) nanoparticles are synthesized by sol?Cgel method. Effect of processing technique and particle size are found to influence the dielectric, ferroelectric, magnetic and leakage behavior of BFO ceramic prepared by sol?Cgel as well as conventional solid state reaction route. From XRD analysis it is observed that bulk BFO sample showed rhombohedral structure (R3c) along with other impurity phases, which become suppressed with the decrease of particles size to few nanometers. The dielectric behavior and leakage current characteristic of the samples were improved significantly in nanoparticles of BFO. Ferroelectric hysteresis loops of sintered bulk BFO ceramic is found to change its shape from semi elliptical lossy P?CE features to a typical ferroelectric feature with improved remnant and saturation polarization value for the particle size down to nanometer scale. Furthermore, BFO nanoparticles also showed a good ferromagnetic M?CH hysteresis loop with enhanced saturation magnetization value of 0.138?emu/mg.     

A High-Order <Emphasis Type="Italic">B</Emphasis>-Spline Collocation Method for Solving Nonlinear Singular Boundary Value Problems Arising in Engineering and Applied Science

In this paper, a high-order B-spline collocation method on a uniform mesh is presented for solving nonlinear singular two-point boundary value problems with Neumann and Robin boundary conditions:

$$\begin{aligned} (p(x)y')'= & {} p(x)f(x,y), \quad 0<x\le 1, \\ y'(0)= & {} 0,\quad ay(1)+by'(1)=c, \end{aligned}$$

where \(p(x)=x^{\alpha }g(x),\alpha \ge 0\) is a general class of non-negative function. The error analysis for the quartic B-spline interpolation is discussed. To demonstrate the applicability and efficiency of our method we consider eight numerical examples, seven of which arise in various branches of applied science and engineering: (1) equilibrium of isothermal gas sphere; (2) thermal explosion; (3) thermal distribution in the human head; (4) oxygen diffusion in a spherical cell; (5) stress distribution on shallow membrane cap; (6) reaction diffusion process in a spherical permeable catalyst; (7) heat and mass transfer in a spherical catalyst. It is shown that our method has fourth-order convergence and is more accurate than finite difference methods (Chawla et al., in BIT 28:88–97, 1988; Pandey et al. in J Comput Appl Math 224:734–742, 2009) and B-spline collocation methods (Abukhaled et al. in Int J Numer Anal Model 8:353–363, 2011; Khuri and Sayfy in Int J Comput Methods 11(1):1350052, 2014).

     

A comparative study of phase-shifting algorithms in digital speckle pattern interferometry

Digital speckle pattern interferometry (DSPI) is a tool for making qualitative as well as quantitative measurements of deformation of objects. Phase-shifting algorithms in DSPI are useful for extracting quantitative deformation data from the system. Comparative studies of the different phase-shifting algorithms in DSPI for object deformation measurement are presented. Static and quasi-dynamic deformation of the object can be measured using these algorithms. Error compensating five-step phase-shifting method is used for the algorithms.     

Curvature measurement using three-aperture digital shearography and fast Fourier transform

Curvature measurement using a three-aperture digital shearography (DS) system is reported in this paper. The outer apertures are covered with wedge plates for introducing shear. Four images by sequentially blocking the outer apertures are used for quantitative measurement. Fourier transform technique is used to determine two sheared slope phase maps from two images at a time representing initial and deformed states. Subtraction of these two-phase maps yields the curvature phase map. Experimental results are presented for a circular diaphragm clamped along the edges and loaded at the center.     

Measurement of surface figure of plane optical surfaces with polarization phase-shifting Fizeau interferometer

A Fizeau interferometer based set up for measurement of surface forms of plane optical surfaces has been discussed. Phase shifting interferometry has been applied using polarization phase shifter. A linearly polarized (632.8 nm) He–Ne laser has been used as the source. Light reflected from the object and the reference/master surfaces are made circularly polarized in opposite senses by means of two properly oriented quarter wave retardation plates placed at appropriate positions, one inside and other outside the interference cavity of the interferometer, and phase shifts are introduced between the object and the reference/master waves by varying angular orientation of a polarizer/analyzer. Final result is made free from any residual wave-front aberrations introduced by the (intra-cavity) wave plate by subtracting phase values obtained by PSI technique between a high optical quality master surface and the reference surface from that obtained for the test object surface with respect to the same reference surface for each point of the interference field. Results are shown for a plane surface.Advantages of the technique presented are linearity and high accuracy in phase stepping, no perturbation of the interference cavity during the phase shifting and possibility of real time or dynamic interferometry.     

Two wavelength simultaneous DSPI and DSP for 3D displacement field measurements

A novel optical system is proposed that implements digital speckle pattern interferometry (DSPI) and digital speckle photography (DSP) simultaneously using two wavelength illumination of an object for simultaneous measurement of all three components of the displacement vector field. A collimated red light illuminates both the object and a reference surface in a DSPI configuration which is sensitive to out-of-plane displacement field while a blue light illuminates the object in a DSP configuration which is sensitive to in-plane displacement fields. A color 3-CCD camera records the red and blue lights individually through its red and blue channels, respectively. Two reference images and one image in the deformed state of the object are required for quantitative measurement. Experimental results are presented to validate the system.     

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 difusion 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 efcient technique for solving the fractional diferential equations without requiring linearization or restrictive assumptions.The basis ideas presented in the paper can be further applied to solve other similar fractional partial diferential equations.     

Room temperature ferroelectricity in multiferroic HoMnO3 ceramics

We have observed good ferroelectric loops at room temperature in hexagonal HoMnO₃ (HMO) bulk ceramics sintered at 1250 °C. Microstructure and ferroelectric hysteresis loops are observed to be dependent on sintering temperature. It is observed that with an increase of sintering temperature, hexagonal a-axis is compressed and c-axis is marginally dilated with overall contraction of the hexagonal unit-cell volume. We explain the origin of ferroelectricity through relative movement of Ho ion and Mn ion within its unit-cell volume, resulting in the distortion of unit-cell volume. We argue that ferroelectric origin of hexagonal HMO at room temperature may be due to the marginal displacement of Ho ion along a-axis, that influences the exchange interaction among the Mn³⁺ and Ho³⁺ ions, causing magnetically induced ferroelectricity at room temperature.