Similarity Reductions for a Nonlinear Diffusion Equation

Abstract

Similarity reductions and new exact solutions are obtained for a nonlinear diffusion equation. These are obtained by using the classical symmetry group and reducing the partial differential equation to various ordinary differential equations. For the equations so obtained, first integrals are deduced which consequently give rise to explicit solutions. Potential symmetries, which are realized as local symmetries of a related auxiliary system, are obtained. For some special nonlinearities new symmetry reductions and exact solutions are derived by using the nonclassical method.     

Constraints on new interactions from neutron scattering experiments

Constraints for the constants of hypothetical Yukawa-type corrections to the Newtonian gravitational potential are obtained from analysis of neutron scattering experiments. Restrictions are obtained for the interaction range between 10⁻¹² and 10⁻⁷ cm, where Casimir force experiments and atomic-force microscopy are not sensitive. Experimental limits are obtained also for nonelectromagnetic inverse-power-law neutron-nucleus potentials. Some possibilities are discussed to strengthen these constraints. The text was submitted by the author in English.     

Dirac equation with multiparameter-potential and Hulthén tensor interaction under relativistic symmetries

The pseudospin and spin symmetric solutions of the Dirac equation with Hulthén-type tensor interaction are obtained under multi-parameter-exponential potential (MEP) for arbitrary κ states. The energy eigenvalues and the corresponding eigenfunctions are also obtained using the parametric Nikiforov-Uvarov (NU) method. Some numerical results are also obtained for pseudospin and spin symmetry limits.     

Recent developments in multiple beam interferometry in the ultraviolet region 1700–2500 Å

New results are given for Fabry-Pérot interferometry in the ultraviolet below 2500 Å down to 1700 Å. Better performances are obtained with improved Al?MgF₂ double layers. Reflective finesse values in the range of 20–30 are obtained for various wavelengths. Interferograms are shown for 1702, 2062 and 2537 Å.     

Application of the Green's function method to ferrimagnetism

The Green function theory, which has been used hitherto for ferro- and antiferromagnetism, has been extended for the case of a two-sublattice ferrimagnet. Two parametrized Green functions are used corresponding to the two sublattices. The equations of motion are set up and the higher order Green functions are decoupled according toCallen's approximation. The functions are obtained by solving the simultaneous equations. The quasi-particle energies are evaluated from the singularities of the Green functions and the magnetization at low temperatures is found to obey theT ^3/2-law. These results obtained for the energy and magnetization agree well with those obtained by the conventional spin wave method.     

Dynamic characteristics of composite laminates

An exact solution for the vibration of elastic composite laminates in cylindrical bending is presented. Dispersion curves for multi-layer symmetrical and unsymmetrical laminates with materials possessing high and low degrees of anisotropy at various fiber orientations are compared with those obtained from an approximate shear deformation theory. Mode shapes are also drawn for different wavelengths and their variation with fiber orientation is studied. Equations are developed for the wave propagation in an infinite medium consisting of a repeated pair of anisotropic layers by extending the “continuum” theory of Sun, Achenbach and Herrmann. Dispersion characteristics for 0–90° fiber orientations obtained by the “continuum” approach are also compared with those obtained by the exact method. The range of validity of each approximate theory is then assessed.     

Four component electromagnetic fields and electrodynamics

Dirac-Maxwell equations with magnetic monopoles are generalized to electromagnetic fields by introducing fourth components to the fields and their solutions are obtained. The formalism is presented into tensor, dyonic as well as quaternionic forms and conservation theorems for the field energy and momenta are obtained involving the new contribution from the mutual interaction of the fields and currents. The generation of the standard modeste, tm andtem ofem waves is also obtained in the formalism.     

Solutions of the Lane–Emden and Thomas–Fermi Equations

A solution of the Lane–Emden equation is obtained based on the approach developed for the Thomas–Fermi equation. The solution is obtained for small values of the independent variable. Based on the solution obtained, analytical expressions are suggested for arbitrary values of the independent variable.     

DKP equation under a vector Yukawa-type potential

Approximate analytical solutions of the Duffin-Kemmer-Petiau equation are obtained for a vector Yukawa potential. The solutions are reported for any J state. The results are obtained based on an exponential approximation as well as the NU analytical technique. Energy solutions are numerically discussed for various quantum numbers and different η, α.     

Radiation transfer with synthetic scattering phase function

Equations connecting the relation between the reflection and transmission functions for a finite slab and those of an infinite one are obtained in terms of an operator which satisfies a semigroup. In trying to calculate the infinite medium reflection function, we use a synthetic scattering function to approximate the Henyey-Greenstein scattering law. Numerical calculations are done and compared with the results obtained from different scattering functions used by other authors. Good results are obtained, especially for |c?1| ?i1 and for τ>1. The deviation in some results is due to the effect of back-scattering.     

The bubbles of matter from multiskyrmions

Approximate analytical solutions describing the skyrmions given by rational-map ansatz are obtained. At large baryon numbers, these solutions are similar to the domain wall or to spherical bubbles with energy and baryon number density concentrated at their boundary. A rigorous upper bound is obtained for the masses of rational map multiskyrmions, which is close to known masses, especially at large B. The main properties of bubbles of matter are obtained for an arbitrary number of flavors.     

Stress and mixed boundary conditions for two-dimensional dodecagonal quasi-crystal plates

For plate bending and stretching problems in two-dimensional (2D) dodecagonal quasi-crystal (QC) media, the reciprocal theorem and the general solution for QCs are applied in a novel way to obtain the appropriate stress and mixed boundary conditions accurate to all order. The method developed by Gregory and Wan is used to generate necessary conditions which the prescribed data on the edge of the plate must satisfy in order that it should generate a decaying state within the plate; these decaying state conditions are obtained explicitly for axisymmetric bending and stretching of a circular plate when stress or mixed conditions are imposed on the plate edge. They are then used for the correct formulation of boundary conditions for the interior solution. For the stress data, our boundary conditions coincide with those obtained in conventional forms of plate theories. More importantly, appropriate boundary conditions with a set of mixed edge-data are obtained for the first time. Furthermore, the corresponding necessary conditions for transversely isotropic elastic plate are obtained directly, and their isotropic elastic counterparts are also obtained.      

On the Regge-pole description of nucleus-nucleus scattering

The properties of the Regge poles of theS-matrix for scattering of strongly-absorbed nuclear particles are considered. Simple formulae are obtained for describing the Regge trajectories in terms of the nuclear radius, the quasi stationary levels in the combined nuclear-Coulomb-potential and the widths of these levels. The predictions of these formulae are compared with the Regge trajectories obtained previously, for a Woods-Saxon potential, and with those required to fit¹⁶O-¹²C backward scattering.     

<Emphasis Type="Italic">T</Emphasis>-solutions of different types and their relations to <Emphasis Type="Italic">R</Emphasis>-solutions

Different types of T-solutions are investigated. The conditions at which one region of space-time can be described only by a statistic R-solution and another region can be described only by a T-solution are obtained. The common metric for R- and T-solutions is obtained where the spatial and temporal variables are interchanged. The T-solution which has the same mass function as for the internal static Schwarzschild solution is obtained.     

A new method for determining the relaxation times,T 1 andT 2, from progressive saturation of inhomogeneously broadened lines

An analytical method to determine the spin-lattice (T ₁), and spin-spin (T ₂), relaxation times for inhomogeneously broadened lines obtained from electron paramagnetic resonance (EPR) experiments is presented in this work. To apply this method, the knowledge of the lineshape of the saturation curve is not necesssary, only the Lorentzian and Gaussian widths are required and these are obtained from a non-saturated line. The relaxation times are calculated by using continuous saturation under slow passage conditions. An explicit algebraic equation for the direct calculation ofT ₁, and the general form of the saturation curve for a single line in an EPR signal are given. The equation given to calculateT ₁ can also be used for substances in which the full saturation can not be obtained experimentally. A comparison of the results obtained for some substances by using the present method with respect to other existent ones, is carried out to show its reliability.     

Thermodynamic properties and the approximate solutions of the Schrödinger equation with the shifted Deng–Fan potential model

With the introduction of a new improved approximation scheme (Pekeris-type approximation) to deal with the centrifugal term, the energy eigenvalues and the wave functions of the Schrödinger equation of the shifted Deng–Fan molecular potential are obtained, via the asymptotic iteration method. Rotational–vibrational energy eigenvalues of some diatomic molecules are presented, these results are in good agreement with other results in the literature. For these selected diatomic molecules, energy eigenvalues obtained are in much better agreement with the results obtained from the rotating Morse potential model for moderate values of rotational and vibrational quantum numbers. Furthermore, thermodynamic properties such as the vibrational mean U, specific heat C, free energy F and entropy S for the pure vibrational state in the classical limit for these energy eigenvalues are studied.     

Multi-band spectra of odd and even nuclei of the s-d shell

The energy levels of^{21, 23}Na,^{22, 23}Ne,^{24, 28}Mg and^{25, 29}Al are obtained by mixing various bands using the projected deformed Hartree-Fock (DHF) method. Solutions having minimum energies are found to be prolate for all the nuclei considered here. Higher bands are obtained either by considering particle-hole excitations or oblate solutions. These various bands are mixed using the projection method and care has been taken to orthogonalize the bands. The interactions used in this study are those given by Kuo, Preedom-Wildenthal (PW) and WHMK interactions. The last one seems to give good results for most of the nuclei considered here. Not only are the lowest bands well-reproduced but the second lowest bands agree reasonably well in most nuclei. The third lowest ones obtained in some nuclei are not yet observed as complete bands. However, K obtained for the third band seems to be correct. A comparison with shell model calculations—which are numerically exhausting—shows similar results for the lowest band. However, the agreement of the second band varies from nucleus to nucleus. A comparison between matrix elements of the interactions is made to analyze the results.     

Subgroups of Hypercubic Group and Many Electron States in Crystals

The point subgroups of index 2 of hypercubic group and their irreducible representations are obtained. The elements of the hypercubic group are represented as rotation about two axis. Possible physical meaning of hypercubic group for electron states is investigated. The reduction relations for the representations of orthogonal group O ₄ on hypercubic group are obtained. These relations are used for additional classification of electron states in crystals.     

The Bayes' optimal receiver for digital fibre optic communication systems

A rigorous treatment of the optimal receiver for digital fibre optic systems is presented. Starting from physical and theoretical considerations, a signal-plus-noise model is obtained for the input stage of an optical receiver. The model is used to set up a hypothesis-testing problem for a transmitted sequence of binary independent and identically distributed information symbols. The Bayes optimal receiver is next derived using the Karhunen-Loève expansion, and upper bounds of the Chernoff type are obtained for its performance. The derived formulae are applied to a typical system and numerical results are obtained which show an improvement of at least 4 db in favour of the optimal receiver when compared to its conventional counterpart at a 10^–9 error rate.