Abstract In this work, the Mellin transform method was used to obtain solutions for the stress field components in two dimensional (2D) elasticity problems in terms of plane polar coordinates. the Mellin transformation was applied to the biharmonic stress compatibility equation expressed in terms of the Airy stress potential function, and the boundary value problem transformed to an algebraic  ...

The classical problem of finding conditions on the entire coefficients A(z) and B(z) guaranteeing that all nontrivial solutions of f ′′+A(z)f ′+ B(z)f = 0 are of infinite order is discussed. Two distinct approaches are used. In the first approach the coefficient A(z) itself is a solution of a differential equation w′′ + P (z)w = 0, where P (z) is a polynomial. This assumption yields stability o...

In the case of the equidistant discretization of the Airy differential equation (“discrete Airy equation”) the exact solution can be found explicitly. This fact is used to derive a discrete transparent boundary condition (TBC) for a Schrödinger–type equation with linear varying potential, which can be used in “parabolic equation” simulations in (underwater) acoustics and for radar propagation i...

We introduce the generalized Airy-Gauss (AiG) beams and analyze their propagation through optical systems described by ABCD matrices with complex elements in general. The transverse mathematical structure of the AiG beams is form-invariant under paraxial transformations. The conditions for square integrability of the beams are studied in detail. The model of the AiG beam describes in a more rea...

We consider the distribution of the maximum MT of branching Brownian motion with time-inhomogeneous variance of the form σ(t/T ), where σ(·) is a strictly decreasing function. This corresponds to the study of the time-inhomogeneous Fisher–KolmogorovPetrovskii-Piskunov (FKPP) equation Ft(x, t) = σ (1− t/T )Fxx(x, t)/2 + g(F (x, t)), for appropriate nonlinearities g(·). Fang and Zeitouni (2012) s...