Graceful labeling is one of the best known labeling methods of graphs. Despite the large number of papers published on the subject of graph labeling, there are few particular techniques to be used by researchers to gracefully label graphs. In this paper, first a new approach based on the mathematical programming technique is presented to model the graceful labeling problem. Then a “branching method” is developed to solve the problem for special classes of graphs. Computational results show the efficiency of the proposed algorithm for different classes of graphs. One of the interesting results of our model is in the class of trees. The largest tree known to be graceful has at most 27 vertices but our model can easily solve the graceful labeling for trees with 40 vertices.

The present work is concerned with unsteady two-dimensional laminar flow of an incompressible, viscous, perfectly electrically conducting fluid past a nonisothermal stretching sheet in the presence of a transverse magnetic field acting perpendicularly to the direction of fluid. By means of the successive approximation method, the governing equations for momentum and energy have been solved. The effects of surface mass transfer $f\omega$, Alfven velocity $\alpha$, Prandtl number $P$, and relaxation time parameter ${\tau }_0$ on the velocity and temperature have been discussed. Numerical results are given and illustrated graphically for the problem considered.

Investigation of the blow-up solutions of the problem in finite time of the first mixed-value problem with a homogeneous boundary condition on a bounded domain of $n$-dimensional Euclidean space for a class of nonlinear Ginzburg-Landau-Schrödinger evolution equation is continued. New simple sufficient conditions have been obtained for a wide class of initial data under which collapse happens for the given new values of parameters.

We study the strong asymptotics of orthogonal polynomials with respect to a measure of the type $d\mu /2\pi +{\sum }_{j=1}^{\infty }{A}_j\delta \left(z-z_k\right)$, where $\mu$ is a positive measure on the unit circle $\Gamma$ satisfying the Szegö condition and ${\left\{z_j\right\}}_{j=1}^{\infty }$ are fixed points outside $\Gamma$. The masses ${\left\{A_j\right\}}_{j=1}^{\infty }$ are positive numbers such that . Our main result is the explicit strong asymptotic formulas for the corresponding orthogonal polynomials.

The problem of crack propagation along the interface of two bonded dissimilar orthotropic plates is considered. Using Galilean transformation, the problem is reduced to a quasistatic one. Then, using Fourier transforms and asymptotic analysis, the problem is reduced to a pair of singular integral equations with Cauchy-type singularity. These equations are solved using Gauss-Chebyshev quadrature formulae. The dynamic stress intensity factors are obtained in closed form expressions. Furthermore, a parametric study is introduced to investigate the effect of crack growth rate and geometric and elastic characteristics of the plates on values of dynamic stress intensity factors.

Analysis for the propagation of plane harmonic thermoelastic waves in an infinite homogeneous orthotropic plate of finite thickness in the generalized theory of thermoelasticity with two thermal relaxation times is studied. The frequency equations corresponding to the extensional (symmetric) and flexural (antisymmetric) thermoelastic modes of vibration are obtained and discussed. Special cases of the frequency equations are also discussed. Numerical solution of the frequency equations for orthotropic plate is carried out, and the dispersion curves for the first six modes are presented for a representative orthotropic plate. The three motions, namely, longitudinal, transverse, and thermal, of the medium are found dispersive and coupled with each other due to the thermal and anisotropic effects. The phase velocity of the waves gets modified due to the thermal and anisotropic effects and is also influenced by the thermal relaxation time. Relevant results of previous investigations are deduced as special cases.

The object of this paper is to give new expressions for the wave field produced when a time harmonic point source is diffracted by a wedge with Dirichlet or Neumann boundary conditions on its faces. The representation of the total field is expressed in terms of quadratures of elementary functions, rather than Bessel functions, which is usual in the literature. An analogous expression is given for the three-dimensional free-space Green's function.