Exact Phase Retrieval for a Class of 2-D Parametric Signals

We address the problem of two-dimensional (2-D) phase retrieval from magnitude of the Fourier spectrum. We consider 2-D signals that are characterized by first-order difference equations, which have a parametric representation in the Fourier domain. We show that, under appropriate stability conditions, such signals can be reconstructed uniquely from the Fourier transform magnitude. We formulate the phase retrieval problem as one of computing the parameters that uniquely determine the signal. We show that the problem can be solved by employing the annihilating filter method, particularly for the case when the parameters are distinct. For the more general case of the repeating parameters, the annihilating filter method is not applicable. We circumvent the problem by employing the algebraically coupled matrix pencil (ACMP) method. In the noiseless measurement setup, exact phase retrieval is possible. We also establish a link between the proposed analysis and 2-D cepstrum. In the noisy case, we derive Cramer-Rao lower bounds (CRLBs) on the estimates of the parameters and present Monte Carlo performance analysis as a function of the noise level. Comparisons with state-of-the-art techniques in terms of signal reconstruction accuracy show that the proposed technique outperforms the Fienup and relaxed averaged alternating reflections (RAAR) algorithms in the presence of noise.

Adaptive Mesh Refinement for Fast Convergence of EFIE-Based 3-D Extraction

3-D full-wave method of moments (MoM) based electromagnetic analysis is a popular means toward accurate solution of Maxwell's equations. The time and memory bottlenecks associated with such a solution have been addressed over the last two decades by linear complexity fast solver algorithms. However, the accurate solution of 3-D full-wave MoM on an arbitrary mesh of a package-board structure does not guarantee accuracy, since the discretization may not be fine enough to capture spatial changes in the solution variable. At the same time, uniform over-meshing on the entire structure generates a large number of solution variables and therefore requires an unnecessarily large matrix solution. In this paper, different refinement criteria are studied in an adaptive mesh refinement platform. Consequently, the most suitable conductor mesh refinement criterion for MoM-based electromagnetic package-board extraction is identified and the advantages of this adaptive strategy are demonstrated from both accuracy and speed perspectives. The results are also compared with those of the recently reported integral equation-based h-refinement strategy. Finally, a new methodology to expedite each adaptive refinement pass is proposed.

Spectral solutions to the Korteweg-de-Vries and nonlinear Schrodinger equations

In this paper, we present the solutions of 1-D and 2-D non-linear partial differential equations with initial conditions. We approach the solutions in time domain using two methods. We first solve the equations using Fourier spectral approximation in the spatial domain and secondly we compare the results with the approximation in the spatial domain using orthogonal functions such as Legendre or Chebyshev polynomials as their basis functions. The advantages and the applicability of the two different methods for different types of problems are brought out by considering 1-D and 2-D nonlinear partial differential equations namely the Korteweg-de-Vries and nonlinear Schrodinger equation with different potential function. (C) 2015 Elsevier Ltd. All rights reserved.

Correlation equations for average deposition rate coefficients of nanoparticles in a cylindrical pore

Nanoparticle deposition behavior observed at the Darcy scale represents an average of the processes occurring at the pore scale. Hence, the effect of various pore-scale parameters on nanoparticle deposition can be understood by studying nanoparticle transport at pore scale and upscaling the results to the Darcy scale. In this work, correlation equations for the deposition rate coefficients of nanoparticles in a cylindrical pore are developed as a function of nine pore-scale parameters: the pore radius, nanoparticle radius, mean flow velocity, solution ionic strength, viscosity, temperature, solution dielectric constant, and nanoparticle and collector surface potentials. Based on dominant processes, the pore space is divided into three different regions, namely, bulk, diffusion, and potential regions. Advection-diffusion equations for nanoparticle transport are prescribed for the bulk and diffusion regions, while the interaction between the diffusion and potential regions is included as a boundary condition. This interaction is modeled as a first-order reversible kinetic adsorption. The expressions for the mass transfer rate coefficients between the diffusion and the potential regions are derived in terms of the interaction energy profile. Among other effects, we account for nanoparticle-collector interaction forces on nanoparticle deposition. The resulting equations are solved numerically for a range of values of pore-scale parameters. The nanoparticle concentration profile obtained for the cylindrical pore is averaged over a moving averaging volume within the pore in order to get the 1-D concentration field. The latter is fitted to the 1-D advection-dispersion equation with an equilibrium or kinetic adsorption model to determine the values of the average deposition rate coefficients. In this study, pore-scale simulations are performed for three values of Peclet number, Pe = 0.05, 5, and 50. We find that under unfavorable conditions, the nanoparticle deposition at pore scale is best described by an equilibrium model at low Peclet numbers (Pe = 0.05) and by a kinetic model at high Peclet numbers (Pe = 50). But, at an intermediate Pe (e.g., near Pe = 5), both equilibrium and kinetic models fit the 1-D concentration field. Correlation equations for the pore-averaged nanoparticle deposition rate coefficients under unfavorable conditions are derived by performing a multiple-linear regression analysis between the estimated deposition rate coefficients for a single pore and various pore-scale parameters. The correlation equations, which follow a power law relation with nine pore-scale parameters, are found to be consistent with the column-scale and pore-scale experimental results, and qualitatively agree with the colloid filtration theory. These equations can be incorporated into pore network models to study the effect of pore-scale parameters on nanoparticle deposition at larger length scales such as Darcy scale.

High Aspect Ratio (L/D) Riser Viv Prediction Using Wake Oscillator Model

A two-dimensional (2-D) vortex-induced vibration (VIV) prediction model for high aspect ratio (LID) riser subjected to uniform and sheared flow is studied in this paper. The nonlinear structure equations are considered. The near wake dynamics describing the fluctuating nature of vortex shedding is modeled using classical van der Pol equation. A new approach was applied to calibrate the empirical parameters in the wake oscillator model. Compared the predicted results with the experimental data and computational fluid dynamic (CFD) results. Good agreements are observed. It can be concluded that the present model can be used as simple computational tool in predicting some aspects of VIV of long flexible structures. (C) 2008 Elsevier Ltd. All rights reserved.

Basic equations for suspension flows with interphase mass-transfer

In the case of suspension flows, the rate of interphase momentum transfer M(k) and that of interphase energy transfer E(k), which were expressed as a sum of infinite discontinuities by Ishii, have been reduced to the sum of several terms which have concise physical significance. M(k) is composed of the following terms: (i) the momentum carried by the interphase mass transfer; (ii) the interphase drag force due to the relative motion between phases; (iii) the interphase force produced by the concentration gradient of the dispersed phase in a pressure field. And E(k) is composed of the following four terms, that is, the energy carried by the interphase mass transfer, the work produced by the interphase forces of the second and third parts above, and the heat transfer between phases. It is concluded from the results that (i) the term, (-alpha-k-nabla-p), which is related to the pressure gradient in the momentum equation, can be derived from the basic conservation laws without introducing the "shared-pressure presumption"; (ii) the mean velocity of the action point of the interphase drag is the mean velocity of the interface displacement, upsilonBAR-i. It is approximately equal to the mean velocity of the dispersed phase, upsilonBAR-d. Hence the work terms produced by the drag forces are f(dc) . upsilonBAR-d, and f(cd) . upsilonBAR-d, respectively, with upsilonBAR-i not being replaced by the mean velocity of the continuous phase, upsilonBAR-c; (iii) by analogy, the terms of the momentum transfer due to phase change are upsilonBAR-d-GAMMA-c, and upsilonBAR-d-GAMMA-d, respectively; (iv) since the transformation between explicit heat and latent heat occurs in the process of phase change, the algebraic sum of the heat transfer between phases is not equal to zero. Q(ic) and Q(id) are composed of the explicit heat and latent heat, so that the sum Q(ic) + Q(id)) is equal to zero.

Periodic solutions of integro-differential equations which arise in population dynamics

The problem of the existence and stability of periodic solutions of infinite-lag integra-differential equations is considered. Specifically, the integrals involved are of the convolution type with the dependent variable being integrated over the range (- ∞,t), as occur in models of population growth. It is shown that Hopf bifurcation of periodic solutions from a steady state can occur, when a pair of eigenvalues crosses the imaginary axis. Also considered is the existence of traveling wave solutions of a model population equation allowing spatial diffusion in addition to the usual temporal variation. Lastly, the stability of the periodic solutions resulting from Hopf bifurcation is determined with aid of a Floquet theory.

The first chapter is devoted to linear integro-differential equations with constant coefficients utilizing the method of semi-groups of operators. The second chapter analyzes the Hopf bifurcation providing an existence theorem. Also, the two-timing perturbation procedure is applied to construct the periodic solutions. The third chapter uses two-timing to obtain traveling wave solutions of the diffusive model, as well as providing an existence theorem. The fourth chapter develops a Floquet theory for linear integro-differential equations with periodic coefficients again using the semi-group approach. The fifth chapter gives sufficient conditions for the stability or instability of a periodic solution in terms of the linearization of the equations. These results are then applied to the Hopf bifurcation problem and to a certain population equation modeling periodically fluctuating environments to deduce the stability of the corresponding periodic solutions.

Exact solutions and transformation properties of nonlinear partial differential equations from general relativity

Various families of exact solutions to the Einstein and Einstein-Maxwell field equations of General Relativity are treated for situations of sufficient symmetry that only two independent variables arise. The mathematical problem then reduces to consideration of sets of two coupled nonlinear differential equations.

The physical situations in which such equations arise include: a) the external gravitational field of an axisymmetric, uncharged steadily rotating body, b) cylindrical gravitational waves with two degrees of freedom, c) colliding plane gravitational waves, d) the external gravitational and electromagnetic fields of a static, charged axisymmetric body, and e) colliding plane electromagnetic and gravitational waves. Through the introduction of suitable potentials and coordinate transformations, a formalism is presented which treats all these problems simultaneously. These transformations and potentials may be used to generate new solutions to the Einstein-Maxwell equations from solutions to the vacuum Einstein equations, and vice-versa.

The calculus of differential forms is used as a tool for generation of similarity solutions and generalized similarity solutions. It is further used to find the invariance group of the equations; this in turn leads to various finite transformations that give new, physically distinct solutions from old. Some of the above results are then generalized to the case of three independent variables.

The stability of traveling wave solutions of parabolic equations

A method for determining by inspection the stability or instability of any solution u(t,x) = ɸ(x-ct) of any smooth equation of the form u_t = f(u_(xx),u_x,u where ∂/∂a f(a,b,c) > 0 for all arguments a,b,c, is developed. The connection between the mean wavespeed of solutions u(t,x) and their initial conditions u(0,x) is also explored. The mean wavespeed results and some of the stability results are then extended to include equations which contain integrals and also to include some special systems of equations. The results are applied to several physical examples.

I. Exact solution of some nonlinear evolution equations. II. The similarity solution for the Korteweg-De Vries equation and the related Painleve Transcendent

In Part I, a method for finding solutions of certain diffusive dispersive nonlinear evolution equations is introduced. The method consists of a straightforward iteration procedure, applied to the equation as it stands (in most cases), which can be carried out to all terms, followed by a summation of the resulting infinite series, sometimes directly and other times in terms of traces of inverses of operators in an appropriate space.

We first illustrate our method with Burgers' and Thomas' equations, and show how it quickly leads to the Cole-Hopft transformation, which is known to linearize these equations.

We also apply this method to the Korteweg and de Vries, nonlinear (cubic) Schrödinger, Sine-Gordon, modified KdV and Boussinesq equations. In all these cases the multisoliton solutions are easily obtained and new expressions for some of them follow. More generally we show that the Marcenko integral equations, together with the inverse problem that originates them, follow naturally from our expressions.

Only solutions that are small in some sense (i.e., they tend to zero as the independent variable goes to ∞) are covered by our methods. However, by the study of the effect of writing the initial iterate u_1 = u_(1)(x,t) as a sum u_1 = ^∼/u_1 + ^≈/u_1 when we know the solution which results if u_1 = ^∼/u_1, we are led to expressions that describe the interaction of two arbitrary solutions, only one of which is small. This should not be confused with Backlund transformations and is more in the direction of performing the inverse scattering over an arbitrary “base” solution. Thus we are able to write expressions for the interaction of a cnoidal wave with a multisoliton in the case of the KdV equation; these expressions are somewhat different from the ones obtained by Wahlquist (1976). Similarly, we find multi-dark-pulse solutions and solutions describing the interaction of envelope-solitons with a uniform wave train in the case of the Schrodinger equation.

Other equations tractable by our method are presented. These include the following equations: Self-induced transparency, reduced Maxwell-Bloch, and a two-dimensional nonlinear Schrodinger. Higher order and matrix-valued equations with nonscalar dispersion functions are also presented.

In Part II, the second Painleve transcendent is treated in conjunction with the similarity solutions of the Korteweg-de Vries equat ion and the modified Korteweg-de Vries equation.

Existence, uniqueness, and stability of solutions of a class of nonlinear partial differential equations

In this study we investigate the existence, uniqueness and asymptotic stability of solutions of a class of nonlinear integral equations which are representations for some time dependent non- linear partial differential equations. Sufficient conditions are established which allow one to infer the stability of the nonlinear equations from the stability of the linearized equations. Improved estimates of the domain of stability are obtained using a Liapunov Functional approach. These results are applied to some nonlinear partial differential equations governing the behavior of nonlinear continuous dynamical systems.

New variational principles for systems of partial differential equations

The question of finding variational principles for coupled systems of first order partial differential equations is considered. Using a potential representation for solutions of the first order system a higher order system is obtained. Existence of a variational principle follows if the original system can be transformed to a self-adjoint higher order system. Existence of variational principles for all linear wave equations with constant coefficients having real dispersion relations is established. The method of adjoining some of the equations of the original system to a suitable Lagrangian function by the method of Lagrange multipliers is used to construct new variational principles for a class of linear systems. The equations used as side conditions must satisfy highly-restrictive integrability conditions. In the more difficult nonlinear case the system of two equations in two independent variables can be analyzed completely. For systems determined by two conservation laws the side condition must be a conservation law in addition to satisfying the integrability conditions.