For wave propagation in a slowly varying waveguide, it is necessary to solve the Helmholtz equation in a domain that is much larger than the typical wavelength. Standard finite difference and finite element methods must resolve the small oscillatory behavior of the wave field and are prohibitively expensive for practical applications. A popular method is to approximate the waveguide by segments...

as x →∞, where x is the independent variable and the prime denotes d/dx. The functions pi(x) (0 ≤ i ≤ 2) and qi(x) (i = 1,2) are defined on an interval [a,∞), are not necessarily real-valued, and are all nowhere zero in this interval. Our aims are to identify relations between q0, q1, p0, p1, and p2 that represents an Euler case for (1.1) and to obtain the asymptotic forms of four linearly inde...

We investigate the existence of solutions of the fourth order nonlinear elliptic boundary value problem under Dirichlet boundary condition A2u + CAU = bu+ + f in 0, where R is a bounded open set in Rn with smooth boundary and the nonlinearity bu+ crosses eigenvalues of A2 + CA. We also investigate a relation between multiplicity of solutions and source terms of the equation with the nonlinearit...

The rule of trajectory structure for fourth-order nonlinear difference equation xn 1 x a n−2 xn−3 / x a n−2xn−3 1 , n 0, 1, 2, . . . , where a ∈ 0, 1 and the initial values x−3, x−2, x−1, x0 ∈ 0,∞ , is described clearly out in this paper. Mainly, the lengths of positive and negative semicycles of its nontrivial solutions are found to occur periodically with prime period 15. The rule is 4 , 3−, ...

This survey paper reports on the properties of the fourth-order Bessel-type linear ordinary differential equation, on the generated self-adjoint differential operators in two associated Hilbert function spaces, and on the generalisation of the classical Hankel integral transform. These results are based upon the properties of the classical Bessel and Laguerre second-order differential equations...

Fourth order analogue to the second Painlevé equation is studied. This equation has its origin in the modified Korteveg de Vries equation of the fifth order when we look for its self similar solution. All power and non power expansions of the solutions for the fouth order analogue to the second Painlevé equation near points z = 0 and z = ∞ are found by means of the power geometry method. The ex...

The m-order connectivity index (G) m of a graph G is     1 2 1 1 2 1 ... ... 1 ( ) i i im m v v v i i i m d d d  G where 1 2 1 ... i i im d d d  runs over all paths of length m in G and i d denotes the degree of vertex i v . Also,        1 2 1 1 2 1 ... ... 1 ( ) i i im m v v v i i i ms d d d X G is its m-sum connectivity index. A dendrimer is an artificially manufactured or synth...

In this paper we make a comparison between the two-scale asymptotic expansion method for periodic homogenization and the so-called Bloch wave method. It is wellknown that the homogenized tensor coincides with the Hessian matrix of the first Bloch eigenvalue when the Bloch parameter vanishes. In the context of the two-scale asymptotic expansion method, there is the notion of high order homogeniz...