Pushpa N. Rathie

Exact Solutions for Normal Depth Problem

Normal depth is a key parameter occurring in the design of canals. It also occurs in the analysis of varied flow in canals and natural streams. The open channel resistance equation involves implicit form for all the practical canal sections. The solution of the implicit equation for normal depth involves tedious method of trial and error. Presented herein are the exact equations for normal depth for various open channel sections.

Exponentially damped Lévy flights

Since real processes seem to departure from standard Levy distributions, modifications to the latter have been suggested in literature. These include (abruptly) truncated (Phys. Rev. Lett. 73 (1994) 2946), smoothly truncated (Phys. Rev. E 52 (1995) 1197; Phys. Lett. A 266 (2000) 282) and gradually truncated Levy flights (Physica A 268 (1999) 231; Physica A 275 (2000) 531). We put forward what we call an exponentially damped Levy flight which encompasses the previous cases. In the presence of increasing and positive feedbacks, our distribution is assumed to deviate from the Levy in both a smooth and gradual fashion. We estimate the truncation parameters by nonlinear least squares to optimally fit the distribution tails. That is a novel approach for estimating parameters α and γ of the Levy. The method is illustrated with daily data on exchange rates for 15 countries against the US dollar. Our results show that the exponentially damped Levy flight fits the data well when increasing and positive deviations are present.

Exact equations for pipe-flow problems/ Des équations exactes pour les problèmes d'écoulement en conduite

The three basic and problems encountered in hydraulic engineering practice are the determination of the head loss, pipe diameter, and discharge. Out of these problems Swamee and Jain [J. Hydraul. Engng. ASCE 102 (1976) 657] gave exact solution for the discharge problem. Exact solutions of the remaining two pipe flow problems are not possible because of the implicit form of Colebrook equation that gives the friction factor for commercial pipes. For the friction factor problem a large number of approximate solutions have been proposed from time to time. Reported herein are exact equations for friction factor and pipe diameter. These exact equations can also be utilized with advantage in optimization studies of pipelines and water distribution systems.

ANALYTICAL SOLUTIONS FOR ALTERNATE DEPTHS

ABSTRACT Triangular, parabolic rectangular and trapezoidal sections are practical open channel sections. For rectangular channel section it is possible to express the alternate depths analytically. For other sections the alternate depths are presently obtained by trial and error procedure. In this paper exact analytical solutions of alternate depths for exponential and trapezoidal open channel sections have been obtained in the form of fast converging infinite series and algebraic equations.

Stable random variables: Convolution and reliability

In this work, first we obtain a representation of the convolution between two asymmetric stable random variables of different parameters in terms of the Fox H-function. In the symmetric case we obtain, in terms of the Meijer G-function, a computable expression of the convolution for rational stability indices. Physically, this density function can be interpreted as the probabilistic generalization of the Voigt profile function. Second, we obtain the reliability P(X>Y) for symmetric stable random variables of different stability indices, scale and shift parameters. Graphical illustrations and error analysis are given for the convolution, and the reliability is tested by Monte Carlo simulation.

Corrigendum to “Exact and approximate expressions for the reliability of stable Levy random variables with applications to stock market modelling” [J. Comput. Appl. Math. 321 (2017) 314–322]

Abstract The authors of the above titled paper (Rathie and Ozelim, 2017), have become aware that the reliability series equations in that paper were presented with errors. In the present document, we provide correct equations as well as the correct values to be considered in a tabulated example of application.

Normal Depth Equations for Parabolic Open Channel Sections

AbstractNormal depth is a key parameter occurring in open-channel hydraulics. For all practical canal sections, open-channel resistance equation involves implicit form for normal depth. Therefore, the determination of the normal depth involves the tedious method of trial and error. Presented in this paper is the good approximate explicit equation for normal depth in parabolic open-channel sections.