Mohamed El-Gebeily

Existence and uniqueness of solutions of a class of two-point singular nonlinear boundary value problems

Abstract El-Gebeily, M.A., A. Boumenir and M.B.M. Elgindi, Existence and uniqueness of solutions of a class of two-point singular nonlinear boundary value problems, Journal of Computational and Applied Mathematics 46 (1993) 345–355. This paper is concerned with the existence and uniqueness of solution of a class of two-point singular nonlinear boundary value problems. It is shown that the problem has a unique solution only for certain boundary conditions under the assumption that the range of ∂ f /∂ y has empty intersection with the closure of the spectrum of the singular differential operator, where f denotes the nonlinearity.

Real Self-Adjoint Sturm–Liouville Problems

In this article we give three characterizations of real self-adjoint operators associated with the singular Sturm–Liouville expressions The first characterization is based on the construction of extensions of the domain of the minimal operator associated with ℓ, the second is based on the behavior of certain functions in their domain near the endpoints and the third is based on the boundary conditions satisfied by such extensions.

Modal frequencies of fiber-reinforced polymer pipes with wall-thinning using a wavelet-based finite element model

Wall-thinning due to chemical reactions, heat, erosion, or a combination of such influences is the most dominant type of internal surface damage in piping systems. In order to examine the effect of wall-thinning on the natural frequencies, the elastodynamic model of the fiber-reinforced polymer pipe is formulated using a wavelet-based finite element method. In this context, a new set of Hermite shape functions is developed. The generalized eigen value problem is solved and the natural frequencies are obtained for an fiber-reinforced polymer pipe with different depths and locations of the wall-thinning. Moreover, the effect of wall-thinning on the modal frequencies of the pipe was verified experimentally. Both the analytical and experimental results demonstrate the potential of using vibration signature to detect internal surface damage in fiber-reinforced polymer pipes.

Characterization of self-adjoint ordinary differential operators

Symmetric differential expressions @? of order n=2k with real valued coefficients give rise to self adjoint operators in the space of weighted square integrable functions. Characterization theorems exist in the literature that describe such self-adjoint operators. All such characterizations begin by constructing the maximal domain of definition of the expression @?. The Glazman-Krein-Naimark theorem constructs the maximal domain in terms of eigenfunctions corresponding to a nonreal parameter @l. Representations in terms of certain functions related to a real parameter @l can also be found in the literature. In this paper we construct the maximal domain from two complementary self-adjoint realizations of @?. One operator is assumed to be known and the other one is computed explicitly. From these two operators we explicitly give all other self-adjoint operators associated with @?. A special class of operators associated with @? is what we call Type I operators. They arise in connection with a certain bilinear form that results from the weak formulation of the expression @?. Depending on the deficiency index of @? and the properties of the bilinear form we can have two complementary self-adjoint operators (two Type I operators) and, as it turns out, one of them is the celebrated Friedrich Extension. The other operator appears to be new. As in the general case, using these two operators we give an explicit characterization of all other operators of the same Type I.

A CHARACTERIZATION OF SELF-ADJOINT OPERATORS DETERMINED BY THE WEAK FORMULATION OF SECOND-ORDER SINGULAR DIFFERENTIAL EXPRESSIONS

In this paper we describe a special class of self-adjoint operators associated with the singular self-adjoint second-order differential expression . This class is defined by the requirement that the sesquilinear form q (u, v) obtained from by integration by parts once agrees with the inner product 〈 u, v〉. We call this class Type I operators. The Friedrichs Extension is a special case of these operators. A complete characterization of these operators is given, for the various values of the deficiency index, in terms of their domains and the boundary conditions they satisfy (separated or coupled). AMS Subject Classification. 34B15, 34A34.

The interval versions of the Kalman filter and the EM algorithm

In this paper, we study state space models represented by interval parameters and noise. We introduce an interval version of the Expectation Maximization (EM) algorithm for the identification of the interval parameters of the system. We also introduce a suboptimal interval Kalman filter for the identification and estimation of the state vectors. The work requires the introduction of the concept of interval random variables which we also include in this work together with a study of their interval statistical properties such as expectation, conditional expectation and variance. Although the interval Kalman filter introduced here is suboptimal, it successfully recovers the state vectors to a high precision in the simulation examples we have run.

Filtering and identification of a state space model with linear and bilinear interactions between the states

In this paper, we introduce a new bilinear model in the state space form. The evolution of this model is linear-bilinear in the state of the system. The classical Kalman filter and smoother are not applicable to this model, and therefore, we derive a new Kalman filter and smoother for our model. The new algorithm depends on a special linearization of the second-order term by making use of the best available information about the state of the system. We also derive the expectation maximization (EM) algorithm for the parameter identification of the model. A Monte Carlo simulation is included to illustrate the efficiency of the proposed algorithm. An application in which we fit a bilinear model to wind speed data taken from actual measurements is included. We compare our model with a linear fit to illustrate the superiority of the bilinear model.

Continuous and discrete wavelet transforms based analysis of weather data of North Western Region of Saudi Arabia

The present study utilizes daily mean time series of meteorological parameters (air temperature, relative humidity, barometric pressure and wind speed) and daily totals of rainfall data to understand the changes in these parameters during 17 years period i.e. 1990 to 2006. The analysis of the above data is made using continuous and discrete wavelet transforms because it provides a time-frequency representation of an analyzed signal in the time domain. Moreover, in the recent years, wavelet methods have become useful and powerful tools for analysis of the variations, periodicities, trends in time series in general and meteorological parameters in particular. In present study, both continues and discrete wavelet transforms were used and found to be capable of showing the increasing or decreasing trends of the meterorological parameters with. The seasonal variability was also very well represented by the wavelet analysis used in this study. High levels of compressions were obtained retaining the originality of the signals.

Regular approximation of singular second-order differential expressions

Abstract In this paper we construct regular real self-adjoint approximations for real self-adjoint operators associated with the differential expression l(y)= 1 w −(py′)′+qy . If 0 is in the resolvent of the original operator, then the construction guarantees that 0 is a point of the resolvent set of the approximating operators. The notion of strong resolvent convergence is generalized and we prove the strong resolvent convergence of the approximations.

Necessary and sufficient conditions for the stability of linear parameter-dependent systems

In this paper we discuss the necessary and sufficient conditions for the Hurwitz stability of a linear system whose matrix depends continuously on several parameters. We also give an algorithm based on our derived condition to determine the stability of the system in a finite number of steps. Numerical examples are given to illustrate the results.