The Hamilton formalism with fractional derivatives

Recently the traditional calculus of variations has been extended to be applicable for systems containing fractional derivatives. In this paper the passage from the Lagrangian containing fractional derivatives to the Hamiltonian is achieved. The Hamilton's equations of motion are obtained in a similar manner to the usual mechanics. In addition, the classical fields with fractional derivatives are investigated using Hamiltonian formalism. Two discrete problems and one continuous are considered to demonstrate the application of the formalism, the results are obtained to be in exact agreement with Agrawal's formalism.     

On streamline diffusion arising in Galerkin FEM with predictor/multi‐corrector time integration

In the present paper, the author shows that the predictor/multi‐corrector (PMC) time integration for the advection–diffusion equations induces numerical diffusivity acting only in the streamline direction, even though the equations are spatially discretized by the conventional Galerkin finite element method (GFEM). The transient 2‐D and 3‐D advection problems are solved with the PMC scheme using both the GFEM and the streamline upwind/Petrov Galerkin (SUPG) as the spatial discretization methods for comparison. The solutions of the SUPG‐PMC turned out to be overly diffusive due to the additional PMC streamline diffusion, while the solutions of the GFEM‐PMC were comparatively accurate without significant damping and phase error. A similar tendency was seen also in the quasi‐steady solutions to the incompressible viscous flow problems: 2‐D driven cavity flow and natural convection in a square cavity. Copyright © 2002 John Wiley & Sons, Ltd.     

Cohomological aspects of gauge theories: superfield formalism

Some of the key cohomological features of the two (1 + 1)-dimensional (2D) free Abelian- and self-interacting non-Abelian gauge theories (having no interaction with matter fields) are briefly discussed first in the language of symmetry properties of the Lagrangian densities and the same issues are subsequently addressed in the framework of superfield formulation on the four (2 + 2)-dimensional supermanifold. Special emphasis is laid on the on-shell- and off-shell nilpotent (co-)BRST symmetries that emerge after the application of (dual) horizontality conditions on the supermanifold. The (anti-)chiral superfields play a very decisive role in the derivation of the on-shell nilpotent symmetries. The study of the present superfield formulation leads to the derivation of some new symmetries for the Lagrangian density and the symmetric energy-momentum tensor. The topological nature of the above theories is captured in the framework of superfield formulation and the geometrical interpretations are provided for some of the topologically interesting quantities.     

On stress rate and plasticity constitutive equations referred to a body-fixed coordinate system

Investigators have viewed the stress rate in two different ways: the material (body-fixed) point of view and the Eulerian point of view. We discuss the Zaremba–Jaumann rate and Oldroyd’s rate from the material viewpoint and apply them to the material formulation of a theory of plasticity for materials undergoing anisotropic plastic deformation. Significant advantages of the material formulation are that the derivation of equations is straight forward, the distortion of yield surface can be easily accounted for, and the issue of self-consistent elastic equation does not arise.     

Error analysis of the L2 least‐squares finite element method for incompressible inviscid rotational flows

In this article we analyze the L² least‐squares finite element approximations to the incompressible inviscid rotational flow problem, which is recast into the velocity‐vorticity‐pressure formulation. The least‐squares functional is defined in terms of the sum of the squared L² norms of the residual equations over a suitable product function space. We first derive a coercivity type a priori estimate for the first‐order system problem that will play the crucial role in the error analysis. We then show that the method exhibits an optimal rate of convergence in the H¹ norm for velocity and pressure and a suboptimal rate of convergence in the L² norm for vorticity. A numerical example in two dimensions is presented, which confirms the theoretical error estimates. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

Navier-Stokes equations with imposed pressure and velocity fluxes

Boundary value problems for Stokes and Navier-Stokes equations with non-standard boundary conditions are studied. Included is the case where the pressure or its normal derivative is given on some part of the boundary or the pressure is given up to a constant but given velocity flux. First, a variational formulation is introduced which is shown to be equivalent to the Stokes equations with the non-standard boundary conditions under consideration. The existence and uniqueness of the solution of the variational problem are studied. Secondly, most of the results obtained for the Stokes equations are extended to the case of the Navier-Stokes equations. The final section is devoted to numerical experiments, flows in pipes and physiological flows.     

Advanced Materials Processing by Adsorption Control

Precise control and knowledge of surface structures are essential inorder to meet the requirements of today's and future materials. One possiblegrowth technique capable of meeting the requirements is atomic layer epitaxy(ALE). ALE is based on sequentially applied saturated gas-solid reactions,which provide the means for adsorption controlled material deposition atatomic layer level. In this paper the potentiality of the use of porousmaterials in a detailed study of adsorption controlled growth is discussed.At the same time the study promotes the application of adsorption controlledmaterials processing for advanced catalysts manufacturing.     

Solvability of the fractional hyperbolic Keller–Segel system

We study a new nonlocal approach to the mathematical modelling of the chemotaxis problem, which describes the random motion of a certain population due to a substance concentration. Considering the initial–boundary value problem for the fractional hyperbolic Keller–Segel model, we prove the solvability of the problem. The solvability result relies mostly on fractional calculus and kinetic formulation of scalar conservation laws.     

On Non-Symmetric Deformations of Neo-Hookean Solids

Deformations possible (i.e., those satisfying the governing three-dimensional equations of equilibrium and the incompressibility constraint) within a class of non-symmetric deformations for a neo-Hookean nonlinearly elastic body were determined in [1], where it was found that only three special cases of the class of deformation fields considered could be solutions. One of these is the trivial solution, one the solution describing radially symmetric deformation, and the other a (non-symmetric, non-homogeneous) deformation contained within a family of universal deformations. In this paper, the results reported in [1] are shown to hold for a substantially broadened deformation field. This revised version was published online in July 2006 with corrections to the Cover Date.     

Global structural behaviour of thin and moderately thick monoclinic spherical shells using a mixed shear deformation model

Summary The static and dynamic responses of anisotropic spherical shells under a uniformly distributed transverse load are investigated. Analytical solutions using the mixed variational formulation are presented for spherical shells subjected to various boundary conditions. Numerical results of a refined mixed first-order shear deformation theory for natural frequencies, critical buckling, center deflections and stresses are compared with those obtained using the classical shell theory. A variety of simply-supported and clamped boundary conditions are considered and comparisons with the existing literature are made. The sample numerical results presented herein for global structural behaviour of monoclinic spherical shells should serve as references for future comparisons.