High precision numerical approach for Davey–Stewartson II type equations for Schwartz class initial data

We present an efficient high-precision numerical approach for Davey–Stewartson (DS) II type equa- tions, treating initial data from the Schwartz class of smooth, rapidly decreasing functions. As with previous approaches, the presented code uses discrete Fourier transforms for the spatial dependence and Driscoll’s composite Runge–Kutta method for the time dependence. Since DS equations are non-local, nonlinear Schrödinger equations with a singular symbol for the non-locality, standard Fourier methods in practice only reach accuracy of the order of 10 −6 or less for typical examples. This was previously demonstrated for the defocusing integrable case by comparison with a numerical approach for DS II via inverse scattering. By applying a regularization to the singular symbol, originally developed for D-bar problems, the presented code is shown to reach machine precision. The code can treat integrable and non-integrable DS II equations. Moreover, it has the same numerical complexity as existing codes for DS II. Several examples for the integrable defocusing DS II equation are discussed as test cases. In an appendix by C. Kalla, a doubly periodic solution to the defocusing DS II equation is presented, providing a test for direct DS codes based on Fourier methods.

On the Soliton Solution and Jacobi Doubly Periodic Solution of the Fractional Coupled Schrödinger–KdV Equation by a Novel Approach

In this paper, fractional coupled Schrödinger–Korteweg–de Vries equation (or Sch–KdV) equation with appropriate initial values has been solved by using a new novel method. The fractional derivatives are described in the Caputo sense. By using the present method, we can solve many linear and nonlinear coupled fractional differential equations. Basically, the present method originated from generalized Taylor’s formula [1]. The results reveal that the proposed method is very effective and simple for obtaining approximate solutions of fractional coupled Schrödinger–KdV equation. Numerical solutions are presented graphically to show the reliability and efficiency of the method. The method does not need linearization, weak nonlinearity assumptions or perturbation theory. The convergence of the method as applied to Sch–KdV is illustrated numerically as well as derived analytically. Moreover, the results are compared with those obtained by the Adomian decomposition method (ADM).

Refined Theory for Bending and Torsion of Perforated Plates

An asymptotic solution is given for the effective elastic constants and the stresses in a perforated plate which is loaded in bending and torsion. In this solution terms 0(h/R)2 are neglected with respect to unity; h being the plate thickness and R the hole radius. In addition to the doubly periodic solution of the classical plate problem another bi-potential problem and two auxiliary problems, viz., a plane strain and a torsion problem for a half-infinite strip, have to be solved. The asymptotic solution together with an approximate solution for an infinitely thick plate permits us often to construct a solution which covers the entire range of h/R; viz., 0≤h/R<∞. In a number of cases accurate interpolation requires additional finite-element calculations. The numerical data presented here apply to a square or an equilateral triangular hole pattern.