Mehdi Razzaghi

Fourier series direct method for variational problems

A direct method for solving variational problems using Fourier series is presented. An operational matrix of integration is first introduced and is utilized to reduce a variational problem to the solution of algebraic equations. Illustrative examples are also given.

Solutions of convolution integral and Fredholm integral equations via double Fourier series

Double Fourier series are developed to approximate the solutions of the convolution integral and Fredholm integral equations. Properties of Fourier series are first briefly presented, and the operational matrix of integration together with the product operational matrix is utilized to reduce the computation of integral equations to a set of simultaneous linear algebraic equations. The method is computationally very attractive, and applications are demonstrated through illustrative examples.

Taylor series analysis of time-varying multi-delay systems

A method for finding an approximate solution of a multi-delay system with time-varying coefficients is proposed. The method is based upon expanding various time functions in the system as their truncated Taylor series, using the operational matrices for integration and delay and hence reducing the system into a set of algebraic simultaneous equations. The properties of Taylor series are first briefly reviewed and associated operational matrices are utilized to solve multi-delay systems. The method is computationally very attractive and applications are demonstrated through illustrative examples.

Aerobic ethanol production by leaves: Evidence for air pollution stress in trees of the Ohio River Valley, USA.

We measured the frequency with which leaves of trees in the Ohio River Valley produced ethanol aerobically, to determine if aerobic ethanol production might provide a viable field assay for air pollution stress. Leaves were collected from trees during the summers of 1985 and 1986 and ethanol production was determined using headspace GC. Frequency of ethanol production was compared with environmental factors, including air pollution concentrations. We found frequent foliar ethanol production and elevated alcohol dehydrogenase activity in the leaves of several species of trees in the Ohio River Valley, USA. The ethanol concentrations measured were often equivalent to those produced by anaerobic leaves. Ethanol production was associated with hot, hazy weather and elevated NO(2) concentrations. Ethanol production was more frequent in urban and industrialized areas. Ethanol production was not associated with natural stresses such as flooding and herbivory. We propose that aerobic ethanol production is the result of cell acidification due to the accumulation of acidic gases in the cytoplasm. The use of ethanol production as a diagnostic tool for detecting stress imposed by acidic gases is discussed.

Instabilities in the solution of a heat conduction problem using taylor series and alternative approaches

Abstract A direct method for solving a heat conduction problem using the Taylor series is discussed. It is shown that the implementation of Taylor series for thisproblem involves the use of an ill-conditioned matrix. Alternative approaches using shifted Legendre and shifted Chebyshev polynomials with satisfactory results are given.

Taylor series direct method for variational problems

Abstract A direct method for solving variational problems using Taylor series is discussed. Properties of Taylor series are briefly presented and an operational matrix is utilized to solve the variational problems by means of a direct method. An illustrative example is given.

Beta-normal distribution in dose–response modeling and risk assessment for quantitative responses

To establish allowable daily intakes for humans from animal bioassay experiments, benchmark doses corresponding to low levels of risk have been proposed to replace the no-observed-adverse-effect level for non-cancer endpoints. When the experimental outcomes are quantal, each animal can be classified with or without the disease. The proportion of affected animals is observed as a function of dose and calculation of the benchmark dose is relatively simple. For quantitative responses, on the other hand, one method is to convert the continuous data to quantal data and proceed with benchmark dose estimation. Another method which has found more popularity (Crump, Risk Anal 15:79–89; 1995) is to fit an appropriate dose–response model to the continuous data, and directly estimate the risk and benchmark doses. The normal distribution has often been used in the past as a dose–response model. However, for non-symmetric data, the normal distribution can lead to erroneous results. Here, we propose the use of the class of beta-normal distribution and demonstrate its application in risk assessment for quantitative responses. The most important feature of this class of distributions is its generality, encompassing a wide range of distributional shapes including the normal distribution as a special case. The properties of the model are briefly discussed and risk estimates are derived based on the asymptotic properties of the maximum likelihood estimates. An example is used for illustration.

Shifted-Jacobi series direct method for variational problems

The operational matrix or integration of a Jacobi vector whose elements are Jacobi polynomials is introduced and applied to solve variational problems by a direct method. Illustrative examples are given to demonstrate that only a small number of shifted-Jacobi polynomials are needed to obtain an accurate solution.

On using Lehmann alternatives with nonresponders

The problem of testing for treatment effect when some subjects in the treatment group may be unaffected by the treatment is considered. A form of the Lehmann alternative suggested by Conover and Salsburg is used that assumes that each control score has the same distribution as the minimum of the known number of responses in the treatment group. It is shown that the locally most powerful test leads to a test statistic that, under the hypothesis of no treatment effect, is the sum of independent pareto random variables whereas under the alternative hypothesis it is the sum of independent random variables from a mixture of two pareto distributions. The limiting distribution of the test statistic under both hypotheses is in the domain of attraction of a stable distribution whose indices are derived. The power of the test is given, and its properties are discussed. A set of data from clinical research involving development of a new drug is used to show application of the procedure and demonstrate its usefulness.

FOURIER SERIES APPROACH FOR THE SOLUTION OF LINEAR TWO-POINT BOUNDARY VALUE PROBLEMS WITH TIME-VARYING COEFFICIENTS

The solution of a linear two-point boundary value problem with time-varying coefficients is obtained by Fourier series approximation. Properties of Fourier series are first briefly discussed and the operational matrix of integration is presented. A transformation matrix relating the back vector to the current time vector together with the operational matrix are utilized to reduce the system to a set of linear algebraic equations and thereby obtain the approximate solution. Illustrative examples are included to demonstrate the validity and applicability of the technique.