A note on non-separable solutions of linear partial differential equation

A method has been presented for constructing non-separable solutions of homogeneous linear partial differential equations of the type F(D, D′)W = 0, where D = ∂/∂x, D′ = ∂/∂y, Image where crs are constants and n stands for the order of the equation. The method has also been extended for equations of the form Φ(D, D′, D″)W = 0, where D = ∂/∂x, D′ = ∂/∂y, D″ = ∂/∂z and Image As illustration, the method has been applied to obtain nonseparable solutions of the two and three dimensional Helmholtz equations.

Boundary value problems for third-order nonlinear ordinary differential equations

In this paper, we describe how to analyze boundary value problems for third-order nonlinear ordinary differential equations over an infinite interval. Several physical problems of interest are governed by such systems. The seminumerical schemes described here offer some advantages over solutions obtained by using traditional methods such as finite differences, shooting method, etc. These techniques also reveal the analytic structure of the solution function. For illustrative purposes, several physical problems, mainly drawn from fluid mechanics, are considered; they clearly demonstrate the efficiency of the techniques presented here.

Boundary value problems for third-order nonlinear ordinary differential equations

In this paper, we describe how to analyze boundary value problems for third-order nonlinear ordinary differential equations over an infinite interval. Several physical problems of interest are governed by such systems. The seminumerical schemes described here offer some advantages over solutions obtained by using traditional methods such as finite differences, shooting method, etc. These techniques also reveal the analytic structure of the solution function. For illustrative purposes, several physical problems, mainly drawn from fluid mechanics, are considered; they clearly demonstrate the efficiency of the techniques presented here.

A parallel multistep predictor-corrector algorithm for solving ordinary differential equations

In this paper we give a generalized predictor-corrector algorithm for solving ordinary differential equations with specified initial values. The method uses multiple correction steps which can be carried out in parallel with a prediction step. The proposed method gives a larger stability interval compared to the existing parallel predictor-corrector methods. A method has been suggested to implement the algorithm in multiple processor systems with efficient utilization of all the processors.

Analysis of an LMS linear equalizer for fading channels in decision directed mode

We consider a time varying wireless fading channel, equalized by an LMS linear equalizer in decision directed mode (DD-LMS-LE). We study how well this equalizer tracks the optimal Wiener equalizer. Initially we study a fixed channel.For a fixed channel, we obtain the existence of DD attractors near the Wiener filter at high SNRs using an ODE (Ordinary Differential Equation) approximating the DD-LMS-LE. We also show, via examples, that the DD attractors may not be close to the Wiener filters at low SNRs. Next we study a time varying fading channel modeled by an Auto-regressive (AR) process of order 2. The DD-LMS equalizer and the AR process are jointly approximated by the solution of a system of ODEs. We show via examples that the LMS equalizer ODE show tracks the ODE corresponding to the instantaneous Wiener filter when the SNR is high. This may not happen at low SNRs.

Computation of stress intensity factors by the sub-region mixed finite element method of lines

Based on the sub-region generalized variational principle, a sub-region mixed version of the newly-developed semi-analytical 'finite element method of lines' (FEMOL) is proposed in this paper for accurate and efficient computation of stress intensity factors (SIFs) of two-dimensional notches/cracks. The circular regions surrounding notch/crack tips are taken as the complementary energy region in which a number of leading terms of singular solutions for stresses are used, with the sought SIFs being among the unknown coefficients. The rest of the arbitrary domain is taken as the potential energy region in which FEMOL is applied to obtain approximate displacements. A mixed system of ordinary differential equations (ODEs) and algebraic equations is derived via the sub-region generalized variational principle. A singularity removal technique that eliminates the stress parameters from the mixed equation system eventually yields a standard FEMOL ODE system, the solution of which is no longer singular and is simply and efficiently obtained using a standard general-purpose ODE solver. A number of numerical examples, including bi-material notches/cracks in anti-plane and plane elasticity, are given to show the generally excellent performance of the proposed method.