Numerical procedure for second order non-linear ordinary differential equations and application to heat transfer problem

In this paper, we have first given a numerical procedure for the solution of second order non-linear ordinary differential equations of the type y″ = f (x;y, y′) with given initial conditions. The method is based on geometrical interpretation of the equation, which suggests a simple geometrical construction of the integral curve. We then translate this geometrical method to the numerical procedure adaptable to desk calculators and digital computers. We have studied the efficacy of this method with the help of an illustrative example with known exact solution. We have also compared it with Runge-Kutta method. We have then applied this method to a physical problem, namely, the study of the temperature distribution in a semi-infinite solid homogeneous medium for temperature-dependent conductivity coefficient.

Coherent quantum Boltzmann equations from cQPA

We reformulate and extend our recently introduced quantum kinetic theory for interacting fermion and scalar fields. Our formalism is based on the coherent quasiparticle approximation (cQPA) where nonlocal coherence information is encoded in new spectral solutions at off-shell momenta. We derive explicit forms for the cQPA propagators in the homogeneous background and show that the collision integrals involving the new coherence propagators need to be resummed to all orders in gradient expansion. We perform this resummation and derive generalized momentum space Feynman rules including coherent propagators and modified vertex rules for a Yukawa interaction. As a result we are able to set up self-consistent quantum Boltzmann equations for both fermion and scalar fields. We present several examples of diagrammatic calculations and numerical applications including a simple toy model for coherent baryogenesis.

Use of Abel integral equations in water wave scattering by two surface-piercing barriers

In this paper the classical problem of water wave scattering by two partially immersed plane vertical barriers submerged in deep water up to the same depth is investigated. This problem has an exact but complicated solution and an approximate solution in the literature of linearised theory of water waves. Using the Havelock expansion for the water wave potential, the problem is reduced here to solving Abel integral equations having exact solutions. Utilising these solutions,two sets of expressions for the reflection and transmission coefficients are obtained in closed forms in terms of computable integrals in contrast to the results given in the literature which,involved six complicated integrals in terms of elliptic functions. The two different expressions for each coefficient produce almost the same numerical results although it has not been possible to prove their equivalence analytically. The reflection coefficient is depicted against the wave number in a number of figures which almost coincide with the figures available in the literature wherein the problem was solved approximately by employing complementary approximations. (C) 2009 Elsevier B.V. All rights reserved.

An approximate analysis of non-linear non-conservative systems subjected to step function excitation

This paper deals with the approximate analysis of the step response of non-linear nonconservative systems by the application of ultraspherical polynomials. From the differential equations for amplitude and phase, set up by the method of variation of parameters, the approximate solutions are obtained by a generalized averaging technique based on ultraspherical polynomial expansions. The Krylov-Bogoliubov results are given by a particular set of these polynomials. The method has been applied to study the step response of a cubic spring mass system in presence of viscous, material, quadratic, and mixed types of damping. The approximate results are compared with the digital and analogue computer solutions and a close agreement has been found between the analytical and the exact results.

Numerical simulations of crack tip fields in polycrystalline plastic solids

In this paper, modes I and II crack tip fields in polycrystalline plastic solids are studied under plane strain, small scale yielding conditions. Two different initial textures of an Al–Mg alloy, viz., continuous cast AA5754 sheets in the recrystallized and cold rolled conditions, are considered. The former is nearly-isotropic, while the latter displays distinct anisotropy. Finite element simulations are performed by employing crystal plasticity constitutive equations along with a Taylor-type homogenization as well as by using the Hill quadratic yield theory. It is found that significant texture evolution occurs close to the notch tip which profoundly influences the stress and plastic strain distributions. Also, the cold rolling texture gives rise to higher magnitude of plastic strain near the tip.

Numerical simulations of crack tip fields in polycrystalline plastic solids

In this paper, modes I and II crack tip fields in polycrystalline plastic solids are studied under plane strain, small scale yielding conditions. Two different initial textures of an Al-Mg alloy, viz.,continuous cast AA5754 sheets in the recrystallized and cold rolled conditions, are considered. The former is nearly-isotropic, while the latter displays distinct anisotropy. Finite element simulations are performed by employing crystal plasticity constitutive equations along with a Taylor-type homogenization as well as by using the Hill quadratic yield theory. It is found that significant texture evolution occurs close to the notch tip which profoundly influences the stress and plastic strain distributions. Also, the cold rolling texture gives rise to higher magnitude of plastic strain near the tip. (C) 2010 Elsevier Ltd. All rights reserved.

Computationally efficient optical tomographic reconstructions through waveletizing the normalized quadratic perturbation equation - art. no.61640R

In this paper, we present a wavelet - based approach to solve the non-linear perturbation equation encountered in optical tomography. A particularly suitable data gathering geometry is used to gather a data set consisting of differential changes in intensity owing to the presence of the inhomogeneous regions. With this scheme, the unknown image, the data, as well as the weight matrix are all represented by wavelet expansions, thus yielding the representation of the original non - linear perturbation equation in the wavelet domain. The advantage in use of the non-linear perturbation equation is that there is no need to recompute the derivatives during the entire reconstruction process. Once the derivatives are computed, they are transformed into the wavelet domain. The purpose of going to the wavelet domain, is that, it has an inherent localization and de-noising property. The use of approximation coefficients, without the detail coefficients, is ideally suited for diffuse optical tomographic reconstructions, as the diffusion equation removes most of the high frequency information and the reconstruction appears low-pass filtered. We demonstrate through numerical simulations, that through solving merely the approximation coefficients one can reconstruct an image which has the same information content as the reconstruction from a non-waveletized procedure. In addition we demonstrate a better noise tolerance and much reduced computation time for reconstructions from this approach.

Optimal Blood Glucose Regulation using Single Network Adaptive Critics

Diabetes is a serious disease during which the body's production and use of insulin is impaired, causing glucose concentration level toincrease in the bloodstream. Regulating blood glucose levels as close to normal as possible, leads to a substantial decrease in long term complications of diabetes. In this paper, an intelligent neural network on-line optimal feedback treatment strategy based on nonlinear optimal control theory is presented for the disease using subcutaneous treatment strategy. A simple mathematical model of the nonlinear dynamics of glucose and insulin interaction in the blood system is considered based on the Bergman's minimal model. A glucose infusion term representing the effect of glucose intake resulting from a meal is introduced into the model equations. The efficiency of the proposed controllers is shown taking random parameters and random initial conditions in presence of physical disturbances like food intake. A comparison study with linear quadratic regulator theory brings Out the advantages of the nonlinear control synthesis approach. Simulation results show that unlike linear optimal control, the proposed on-line continuous infusion strategy never leads to severe hypoglycemia problems.

A note on the equivalence of shock manifold equations

The shock manifold equation is a first order nonlinear partial differential equation, which describes the kinematics of a shockfront in an ideal gas with constant specific heats. However, it was found that there was more than one of these shock manifold equations, and the shock surface could be embedded in a one parameter family of surfaces, obtained as a solution of any of these shock manifold equations. Associated with each shock manifold equation is a set of characteristic curves called lsquoshock raysrsquo. This paper investigates the nature of various associated shock ray equations.

A systolic evaluator for linear, quadratic, and cubic expressions

High-speed evaluation of a large number of linear, quadratic, and cubic expressions is very important for the modeling and real-time display of objects in computer graphics. Using VLSI techniques, chips called pixel planes have actually been built by H. Fuchs and his group to evaluate linear expressions. In this paper, we describe a topological variant of Fuchs' pixel planes which can evaluate linear, quadratic, cubic, and higher-order polynomials. In our design, we make use of local interconnections only, i.e., interconnections between neighboring processing cells. This leads to the concept of tiling the processing cells for VLSI implementation.

Sliding Modes for the Simulation of Mechanical and Electrical Systems Defined by Differential-Algebraic Equations

We offer a technique, motivated by feedback control and specifically sliding mode control, for the simulation of differential-algebraic equations (DAEs) that describe common engineering systems such as constrained multibody mechanical structures and electric networks. Our algorithm exploits the basic results from sliding mode control theory to establish a simulation environment that then requires only the most primitive of numerical solvers. We circumvent the most important requisite for the conventionalsimulation of DAEs: the calculation of a set of consistent initial conditions. Our algorithm, which relies on the enforcement and occurrence of sliding mode, will ensure that the algebraic equation is satisfied by the dynamic system even for inconsistent initial conditions and for all time thereafter. [DOI:10.1115/1.4001904]