Analytical solution and semi-analytical solution for anisotropic functionally graded beam subject to arbitrary loading

Analytical and semi-analytical solutions are presented for anisotropic functionally graded beams subject to an arbitrary load, which can be expanded in terms of sinusoidal series. For plane stress problems, the stress function is assumed to consist of two parts, one being a product of a trigonometric function of the longitudinal coordinate (x) and an undetermined function of the thickness coordinate (y), and the other a linear polynomial of x with unknown coefficients depending on y. The governing equations satisfied by these y-dependent functions are derived. The expressions for stresses, resultant forces and displacements are then deduced, with integral constants determinable from the boundary conditions. While the analytical solution is derived for the beam with material coefficients varying exponentially or in a power law along the thickness, the semi-analytical solution is sought by making use of the sub-layer approximation for the beam with an arbitrary variation of material parameters along the thickness. The present analysis is applicable to beams with various boundary conditions at the two ends. Three numerical examples are presented for validation of the theory and illustration of the effects of certain parameters. Supported by the National Natural Science Foundation of China (Grant Nos. 10472102, 10432030, and 10725210)     

Analytical solution for the one-dimensional scattering problem

We introduce the scattering problem in the case of one-dimensional space. The transmission factor is calculated for the whole domain of the potential energy. Connection with a field in interaction with a two-level quantum system is established for which the coupling parameter is considered to be explicitly time-dependent.     

Analytical solution to the problem of a temperature jump in a metal

An analytical solution to the problem of a temperature jump in a metal is obtained. The jump is caused by a heat flow to the surface. An exact expression for the heat-induced electric field is derived. Based on the exact expressions, the electric field profiles are plotted for different ratios of the electron free path to the Debye length. It is shown that the field profile near the surface may differ substantially from the Debye profile.     

Analytical solution of the fundamental space filling mode of photonic crystal fibers

Analytical solution of the fundamental space filling mode of photonic crystal fibers is revisited based on previous results by Midrio et al. [J Lightwave Technol 2000;18(7):1031–7]. The fundamental space filling mode is designated HE₁₁ mode following the conventional mode classification. A comparison is made between the vectorial method and the scalar method when identical parameters are used in the analytical solution. A more accurate radius of the equivalent circular unit cell is determined.     

Analytical solution to the skin-effect problem for an arbitrary accommodation coefficient of electron tangential momentum

An analytical solution to the boundary-value problem of an electric field and electrons in a metal-filled half-space is obtained for arbitrary values of the tangential-momentum accommodation coefficient. The frequency of an external electromagnetic field directed tangentially to the surface is allowed to take on complex values. Both the normal and anomalous skin effects are considered. In the latter case, the low-and high-frequency limits are examined.     

A valence space solution to the B(E2) saturation problem in deformed nuclei

The empirically observed saturation in B(E2: 0⁺₁→2⁺₁) values in deformed nuclei near mid shell is interpreted in terms of a reduced integrated valence proton-neutron interaction arising from reduced spatial overlaps. This analysis, based on very general properties of the deformed field, removes a serious difficulty encountered by simple valence space models and may provide a simpler interpretation of other observables as well.     

Analytical solution of the generalized smoluchowski problem using a new model kinetic equation

A new kinetic equation whose solution yields a correct Prandtl number is suggested. A technique for analytically solving half-space boundary-value problems is demonstrated with the classical Smoluchowski problem of finding temperature and concentration steps. Numerical results indicating the succession of the equation.     

Small-phase solution to the phase-retrieval problem

A solution to the phase-retrieval problem when the unknown phase is small is presented. The solution specifies the even and odd parts of the unknown phase in two separate equations. The odd part requires a single intensity measurement, and the even part requires two measurements. Phase diversity is used for the second measurement, and computer simulations are given.     

Analytical approximate solution for nonlinear space-time fractional Klein Gordon equation

The fractional derivatives in the sense of Caputo and the homotopy analysis method are used to construct an approximate solution for the nonlinear space-time fractional derivatives Klein-Gordon equation. The numerical results show that the approaches are easy to implement and accurate when applied to the nonlinear space-time fractional derivatives KleinGordon equation. This method introduces a promising tool for solving many space-time fractional partial differential equations. This method is efficient and powerful in solving wide classes of nonlinear evolution fractional order equations.     

Regularized variable metric method versus the conjugate gradient method in solution of radiative boundary design problem

In this paper, an appropriate distribution of the heating elements’ strengths in a radiation furnace is estimated using inverse methods so that a pre-specified temperature and heat flux distribution is attained on the design surface. Minimization of the sum of the squares of the error function is performed using the variable metric method (VMM), and the results are compared with those obtained by the conjugate gradient method (CGM) established previously in the literature. It is shown via test cases and a well-founded validation procedure that the VMM, when using a “regularized” estimator, is more accurate and is able to reach at a higher quality final solution as compared to the CGM. The test cases used in this study were two-dimensional furnaces filled with an absorbing, emitting, and scattering gas.     

Exact solution to the averaging problem in cosmology

The exact solution of a two-scale Buchert average of the Einstein equations is derived for an inhomogeneous universe that represents a close approximation to the observed universe. The two scales represent voids, and the bubble walls surrounding them within which clusters of galaxies are located. As described elsewhere [New J. Phys. 9, 377 (2007)10.1088/1367-2630/9/10/377], apparent cosmic acceleration can be recognized as a consequence of quasilocal gravitational energy gradients between observers in bound systems and the volume-average position in freely expanding space. With this interpretation, the new solution presented here replaces the Friedmann solutions, in representing the average evolution of a matter-dominated universe without exotic dark energy, while being observationally viable.     

M theory solution to the hierarchy problem

An old idea for explaining the hierarchy is strong gauge dynamics. We show that such dynamics also stabilizes the moduli in M theory compactifications on manifolds of G2 holonomy without fluxes. This gives stable vacua with softly broken supersymmetry, grand unification, and a distinctive spectrum of TeV and sub-TeV sparticle masses.     

A solution to the non-Abelian duality problem

Dualities uniquely excel at resolving non-perturbative aspects of complex phase diagrams of interacting, Landau or topologically ordered, systems. However, traditional duality transformations fail for systems like the Heisenberg model and non-Abelian gauge theories. The bond-algebraic theory of quantum and classical dualities provides a solution to this long-standing conundrum, the so-called non-Abelian duality problem, by embedding traditional dualities into a more general transformation scheme that always preserves locality in any number of dimensions. Remarkably, it turns out to be unimportant whether a model?s group of symmetries is Abelian or non-Abelian. The capability of the bond-algebraic approach to handle finite and infinite systems with arbitrary boundary conditions has recently led to the discovery of holographic symmetries  , relating topological order, edge states, and generalized order parameters. We discuss the interplay between these distinguished boundary symmetries and our solution to the non-Abelian duality problem. To illustrate our technique we present, among others, novel dualities for the SU(2)$\mathit{SU}(2)$ principal chiral field and both U(1)$U(1)$ and SU(2)$\mathit{SU}(2)$ generalizations of the planar quantum compass model of orbital ordering.