In this article we implement an operational matrix of fractional integration for Legendre polynomials. We proposed an algorithm to obtain an approximation solution for fractional differential equations, described in Riemann-Liouville sense, based on shifted Legendre polynomials. This method was applied to solve linear multi-order fractional differential equation with initial conditions, and the exact solutions obtained for some illustrated examples. Numerical results reveal that this method gives ideal approximation for linear multi-order fractional differential equations.

During a survey of Sindh and Punjab provinces of Pakistan, two dragonfly genera were collected and identified for the first time from the country. Anaciaeshna(Family Aeshnidae) is a genus of large dragonflies. Representatives were collectedfrom Gujjo, District Thatta (Sindh) in August 2008 and were identified as Anaciaeshna jaspidea (Burmeister). The second genus was Epophthalmia (Family Corduliidae), medium to large sized, well-built and very fast flying dragonflies. Epophthalmia vittata vittata Burmeister (Family Corduliidae) was collected from Java Dam, Rawalpindi and Dhok Tallian Dam near Chakwal. Individuals of this genus were found maneuvering near the peripheries of small dams. Some taxonomic notes of the said species are provided.

Let be a Hilbert space of functions analytic on a plane domain such that for every in the functional of evaluation at is bounded. Assume further that contains the constants and admits multiplication by the independent variable, , as a bounded operator. We give sufficient conditions for to be reflexive for all positive integers.

The optical spectrum band positions and EPR parameters (g factors g//, g^ and hyperfine structure constants A// and A^) for VO2+ ions in Zn(antipyrine)2(NO3)2 are theoretically studied from the complete diagonalization (of energy matrix) method (CDM) and the perturbation theory method (PTM). In the two methods, the contributions from the spin–orbit (SO) coupling of central 3dn ion and ligand are taken into account. The theoretical results from both methods are not only consistent with the experimental values, but also close to each other. The results are discussed.

The purpose of this paper is the generalization of structure factor for rods by polygon section in two dimensional phononic crystals. If we use the plane wave expansion method (PWE) for the propagation of acoustic waves in 2D phononic crystals, structure factor will be an important quantity. In order to confirm the obtained relations, we have calculated the band structure for XY and Z vibration modes in 2D phononic crystals and the propagation of bulk acoustic waves (BAW) are considered. In addition, the effect of sides’ number on the band structure and the complete band gaps width are investigated. Phononic crystals studied in this paper are composites medium of a square lattice consisting of parallel nickel rods embedded in epoxy. The frequency is calculated by PWE in the condition of elastic rigidity to the solid inclusions. The results showed that, when the section of rods have 2n+2 (n is even) and 2n+1 by increasing sides number of the rod sections, the bands of XY mode shift to lower-frequency, the bands are smoother and the width of the band gap increases, but the band of Z mode has not changed by n variations. Moreover, when the section of rods have 2n+2 (n is odd) by increasing sides number of rod sections, the band structure of XY mode changes slightly and the width of the complete band gap is decreased. This confirms the effect of lattice symmetry on the complete band gap width. But, the band structure of Z mode has not changed.

The theoretical calculation of the energies of projectile like fragments (PLFs) using the heavy-ion reactions between 76Ge and 198Pt are reported in this article. The incident beam energy was 635 MeV. The calculated values of PLFs are compared with the previous experimental results and it is shown that the theoretical calculations of PLFs are consistent with the experimental values. The elastic peak of the projectile is compared theoretically and experimentally. Moreover, the Q-value and binding energy of PLFs were also calculated.

The Zagros Fold-and-Thrust Belt is tectonically active and often has active faults with insensible and slow motions. Determining the rate of movement and displacement in these faults requires very precise measurements. One of the measurement methods of fault movements is using geodetic and micro-geodetic studies. This research is focused on one of the active faults in the north of Shiraz city, Fars province called Darvazeh Quran fault in this study. In order to determine the deformation matrix, the local networks are preferred. Deforming area is normally covered by four control points. These points constitute a geodetic network and their location or structure is defined by the topographic and geological parameters. The results show that the obtained displacement vector is from SE to NW with a dextral strike-slip creep. Deformation matrix indicated 4mm±6ppm displacement per year and elongation changes of network have an ascending trend into time.

The numerical methods are of great importance for approximating the solutions of nonlinear ordinary or partial differential equations, especially when the nonlinear differential equation under consideration faces difficulties in obtaining its exact solution. In this latter case, we usually resort to one of the efficient numerical methods. In this paper, the Chebyshev collocation method is suggested to deal numerically with some nonlinear partial differential equations in mathematical physics.

This paper is concerned with global analysis of a delay SVEIR epidemiological model in a population of varying size. By using Lyapunov stability method and LaSalle’s invariance principle for delay systems, we prove that when there is no endemic equilibrium, the disease free equilibrium is globally asymptotically stable, otherwise the endemic equilibrium is globally stable.

The aim of this study was to find any structural differences between the digital pads of forelimbs and hind limbs as well as more careful investigation of the internal and external structures of the toe-pad. In this study, pad morphology and cytology in Litoria caerulea is described using SEM, TEM and light microscopy. At the gross anatomical level, toe-pads in hind limbs were subdivided into medial and lateral parts by two large grooves. Semi-thin sections also showed that the toe-pad epidermis in hind limbs consisted of four layers with a cuboidal outermost layer, while the epidermis of forelimbs consisted of 3 layers with a columnar outermost layer. SEM study revealed two basic shapes of epidermal cells arranged very regularly across the surface of the pad: pentagonal and hexagonal. The pentagonal mainly occupied the most distal part of the toe. Three types of mucous-secreting pores were also seen in between the epithelial cells.

The purpose of this study is to implement a new modification of the variational iteration method (H-S-VIM), which is a combination of spectral method and variational iteration method for heat transfer problems with high nonlinearity order. The merit of this method is that it does not require the solution of any linear or nonlinear system of equations unlike spectral method. Furthermore the proposed method is easy to implement and computationally very attractive. Here, H-S-VIM is used to solve an unsteady nonlinear convective-radiative equation containing two small parameters, and. It is observed that H-S-VIM may be implemented on other strongly nonlinear models of physical nature.

On the basis of a reproducing kernel space, an iterative algorithm for solving the one-dimensional linear and nonlinear Schrödinger equations is presented. The analytical solution is shown in a series form in the reproducing kernel space and the approximate solution is constructed by truncating the series. The convergence of the approximate solution to the analytical solution is also proved. The method is examined for the single soliton solution and interaction of two solitons. Numerical experiments show that the proposed method is of satisfactory accuracy and preserves the conservation laws of charge and energy. The numerical results are compared with both the analytical and numerical solutions of some earlier papers in the literature.

We consider the fundamental relation on a semihypergroup to interpret the lower and upper approximations as subsets of the fundamental semigroup and we give some results in this connection. Also, we introduce the notion of a bi-hyperideal to study the relationship between approximations and bi-hyperideals.

In this paper, it was shown that, where and, and, where is not prime and, are OD-characterizable.