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ferential equations in physics, chemistry, and engineering. Its two power series solutions are the Kummer function, M(a,b,z) often referred to as the confluent hypergeometric function of the first kind, and where a and b are parameters that appear in the differential equation. A third function, the Tricomi function, M(a, b, z), sometimes referred to as the confluent hypergeometric function of the second kind, is also a solution of the confluent hypergeometric equation that is routinely used. Contrary to common procedure, all three of these functions (and more) must be considered in a search for the two linearly independentsolutionsoftheconfluenthypergeometricequation. Therearesituations,when a and a-b are integers,where one of these functions is not defined, or two of the functions are not linearly independent, or one of the linearly independents ol utions of the differentia lequationis different from these three functions. Many of these special cases correspond precisely to cases needed to solve problems in physics. This leads to significant confusion about how to work with confluent hyper geometric equations, inspite of authoritative references such as the NIST Digital Library of Mathematical Functions. Here, we carefully describe all of the different cases one has to consider and what the explicit formulas are for the two linearly independent solutions of the confluent hypergeometric equation. The procedure to properly solve the confluent hypergeometric equation is summarized in a convenient table. As an example, we use these solutions to study the bound states of the hydrogenic atom, correcting the standard treatment in textbooks. We also briefly consider the cutoff Coulomb potential. We hope that this guide will aid physicists to properly solve problems that involve the confluent hypergeometric differential equation. [ABSTRACT FROM AUTHOR]

In this paper we present lower and upper bounds for Kummer's function ratios of the form \frac {{M(a, b, z)}'}{M(a, b, z)} when 0

A steady-state, two-dimensional flow of Rivlin-Ericksen magnetohydrodynamics (MHD) fluid flow induced by stretching of the sheet of porous medium considering heat and mass transfer is investigated in the present analysis. The fluid flow is influenced by a uniform magnetic field. The inverse Darcy model, as well as thermohydrodynamic characteristics, is taken into account. Within thermal analysis effects of temperature-dependent heat source/sink, viscous dissipation, heat generation due to the elastic deformation, and thermal radiation are considered. Mass transfer is concentrated to chemically reactive diffusive species by means of first-order chemical conversion rate. The similarity transformations are employed to convert highly non-linear governing partial differential equations into a set of ordinary differential equations. Then the analytical results of the temperature and mass transfer equations are expressed in the form of Kummer's function for two different cases namely prescribed surface temperature and prescribed heat flux cases. The presented closed-form analytical solution of this research can be used as a benchmark solution for the results of numerical methods and can find possible industrial and technological applications in fluid-based systems involving shrinkable/stretchable materials. A steady-state 2D flow of Rivlin-Ericksen MHD fluid flow induced by stretching of the sheet of porous medium considering heat and mass transfer is investigated in the present analysis. The fluid flow is influenced by a uniform magnetic field. The inverse Darcy model, as well as thermo-hydrodynamic characteristics, are taken into account. Within thermal analysis effects of temperature-dependent heat source/sink, viscous dissipation, heat generation due to the elastic deformation, and thermal radiation are considered. Mass transfer is concentrated to chemically reactive diffusive species by means of first-order chemical conversion rate. The similarity transformations are employed to convert highly non-linear governing partial differential equations into a set of ordinary differential equations. Then the analytical results of the temperature and mass transfer equations are expressed in the form of Kummer's function for two different cases namely prescribed surface temperature and prescribed heat flux cases. The presented closed-form analytical solution of this research can be used as a benchmark solution for the results of numerical methods and can find possible industrial and technological applications in fluid-based systems involving shrinkable/stretchable materials. [ABSTRACT FROM AUTHOR]

In this paper it is intended to analyse entropy generation by applying second law of thermodynamics to magnetohydrodynamic flow, heat and mass transfer of an electrically conducting viscoelastic liquid (Walters B′) past on a stretching surface, taking into account the effects of Joule dissipation, viscous dissipation and Darcy dissipation, and internal heat generation. The boundary layer equations are solved analytically by using Kummer’s function. The entropy generation has been computed considering Darcy dissipation besides viscous and Joule dissipation. Results for some special cases of the present analysis are in good agreement with the existing literature. Increase in viscoelastic and magnetic parameter reduces the velocity. Increase in elastic parameter causes a greater retardation in the velocity. Presence of porous matrix enhances temperature whereas increase in Prandtl number decreases the temperature. One striking result of the present study is that Darcy dissipation favours higher level entropy generation in all the cases except the flow of liquid with low thermal diffusivity assuming the process to be irreversible.

A confluent hypergeometric function (Kummer's function) is a generalized hypergeometric series introduced by Kummer in 1837 [De integralibus quibusdam definitis et seriebus infinitis. J Reine Angew Math (in Latin). 1837;17:228–242], given by M (α ; γ ; z) = 1 F 1 (α ; γ ; z) := ∑ n = 0 ∞ ((α) n / n ! (γ) n ) z n (γ ≠ 0 , − 1 , − 2 , ...) , which are of great applications in statistics, mathematical physics, engineering and so on. In this paper, we investigate some properties of Kummer's function from viewpoint of value distribution theory. Specifically, two different growth orders are obtained for α ∈ Z ≤ 0 and α ∉ Z ≤ 0 , which are corresponding to the degenerated and non-degenerated cases respectively. Moreover, we obtain an asymptotic estimate of characteristic function T (r , M (α ; γ ; z)) and calculate the logarithmic derivative m (r , M ′ (α ; γ ; z) / M (α ; γ ; z)) , the distribution of zeros of Kummer's function is also discussed. Finally, we show Kummer's function and an entire function are uniquely determined by their c-values. [ABSTRACT FROM AUTHOR]

The boundary layer flow, heat and mass transfer of an electrically conducting viscoelastic fluid over a stretching sheet embedded in a porous medium has been studied. The effect of transverse magnetic field, non-uniform heat source and chemical reaction on the flow has been analyzed. The Darcy linear model has been applied to account for the permeability of the porous medium. The method of solution involves similarity transformation. The confluent hypergeometric function (Kummer’s function) has been applied to solve the governing equations. Two aspects of heat equation namely, (1) prescribed surface ure and (2) prescribed wall heat flux are considered. The study reveals that the loss of momentum transfer in the main direction of flow is compensated by increasing in transverse direction vis-à-vis the corresponding velocity components due to magnetic force density. The application of magnetic field of higher density produces low solutal concentration and a hike in surface temperature. [ABSTRACT FROM AUTHOR]

In this paper it is intended to analyse entropy generation by applying second law of thermodynamics to magnetohydrodynamic flow, heat and mass transfer of an electrically conducting viscoelastic liquid (Walters B ′ ) past on a stretching surface, taking into account the effects of Joule dissipation, viscous dissipation and Darcy dissipation, and internal heat generation. The boundary layer equations are solved analytically by using Kummer’s function. The entropy generation has been computed considering Darcy dissipation besides viscous and Joule dissipation. Results for some special cases of the present analysis are in good agreement with the existing literature. Increase in viscoelastic and magnetic parameter reduces the velocity. Increase in elastic parameter causes a greater retardation in the velocity. Presence of porous matrix enhances temperature whereas increase in Prandtl number decreases the temperature. One striking result of the present study is that Darcy dissipation favours higher level entropy generation in all the cases except the flow of liquid with low thermal diffusivity assuming the process to be irreversible. [ABSTRACT FROM AUTHOR]

This article addresses the mass and heat transfer analysis over an electrically conducting viscoelastic (Walters B′) fluid over a stretching surface in presence of transverse magnetic field. The impact of chemical reaction, as well as non-uniform heat source, are also taken into account. Similarity transformations are employed to model the equations. The governing equations comprises of momentum, energy, and concentration which are modified to a set of non-linear differential equations and then solved by applying confluent hypergeometric function known as ' Kummer's function'. The exact solution for heat equation is obtained for two cases i.e. (1) Prescribed surface temperature, (2) Prescribed wall heat flux. Physical behavior of all the sundry parameters are against concentration, temperature, and velocity profile are presented through graphs. The inclusion of magnetic field is counterproductive in diminishing the velocity distribution whereas reverse effect is encountered for concentration and temperature profiles. [ABSTRACT FROM AUTHOR]

The study of flow and heat transfer in fluid as it passes over a porous surface has attracted considerable scientific attention, particularly in technologies where boundary-layer control is important. Therefore, this paper reports the effects of radiation and heat source over a stretching surface subjected to a power law heat flux, in the presence of transverse magnetic field on two-dimensional boundary layer steady flow and heat transfer of a viscous incompressible fluid. Heat transfer equation is reduced to a linear differential equation using non-dimensional parameters and the exact solution is obtained in the form of confluent hypergeometric function (Kummer’s Function) for prescribed power law wall temperature. The effects of the various parameters entering into the problem on the temperature distribution and wall temperature gradient are discussed.

The heat and mass transfer effects in a boundary layer flow through porous medium of an electrically conducting viscoelastic fluid subject to transverse magnetic field in the presence of heat source/sink and chemical reaction have been analyzed. It has been considered the effects of radiation, viscous and Joule dissipations and internal heat generation/absorption. Closed form solutions for the boundary layer equations of viscoelastic, second-grade and Walters' B′ fluid models are obtained. The method of solution involves similarity transformation. The transformed equations of thermal and mass transport are solved by applying Kummer's function. The solutions of temperature field for both prescribed surface temperature as well as prescribed surface heat flux are obtained. It is important to remark that the interaction of magnetic field is found to be counterproductive in enhancing velocity and concentration distribution whereas the presence of chemical reaction as well as porous matrix with moderate values of magnetic parameter reduces the temperature and concentration fields at all points of flow domain. [ABSTRACT FROM AUTHOR]

An analysis was carried out to study the flow and heat transfer characteristics in a second-grade fluid through a porous medium over a linear stretching surface with two general cases, namely PST and PHF, including the effect of viscous dissipation. The partial differential equations governing the flow and heat transfer are converted into nonlinear ordinary differential equations and boundary conditions by using suitable similarity transformation. The proposed problem was solved analytically by the power series method (using Kummer's function). The graphical results for velocity, skin-friction coefficient, and temperature (for PST and PHF cases) are presented and discussed. The numerical values of wall friction coefficient, wall temperature gradient θη(0), and wall temperature θ(0) are also calculated, tabulated, and discussed. [ABSTRACT FROM AUTHOR]

The study of flow and heat transfer in fluid as it passes over a porous surface has attracted considerable scientific attention, particularly in technologies where boundary-layer control is important. Therefore, this paper reports the effects of radiation and heat source over a stretching surface subjected to a power law heat flux, in the presence of transverse magnetic field on two-dimensional boundary layer steady flow and heat transfer of a viscous incompressible fluid. Heat transfer equation is reduced to a linear differential equation using non-dimensional parameters and the exact solution is obtained in the form of confluent hypergeometric function (Kummer's Function) for prescribed power law wall temperature. The effects of the various parameters entering into the problem on the temperature distribution and wall temperature gradient are discussed. [ABSTRACT FROM AUTHOR]

Two approaches to describe nonparaxial optical vortices were considered. One approach is to use a revised Kirchhoff integral, which does not neglect the relief of an optical element. Using this integral and the finite- difference time-domain method it is shown that an optical vortex generated by a refractive spiral plate with a relief step has an asymmetric profile. The annular diffraction pattern in the vortex beam cross-section is found to be disturbed not only for the near-field diffraction but also for the middle-field diffraction, at a distance of several Fresnel lengths. Another approach is to solve the Helmholtz equation without any approximations. An analytical solution to describe propagation of a light beam in the positive direction of the optical axis was found. The complex amplitude of such a beam is found to be in direct proportion to the product of two linearly independent solutions of Kummer's differential equation. Relationships for a particular case of such beams--namely, the Hankel-Bessel (HB) beams--are deduced. The auto-focusing of the HB beams is studied. [ABSTRACT FROM AUTHOR]

This article reports the effects of radiation and heat sink over a stretching surface in the presence of a transverse magnetic field on two-dimensional boundary layer steady flow and heat transfer of a viscous incompressible fluid. The heat transfer equation is reduced to a linear differential equation using nondimensional parameters, and the exact solution is obtained in the form of a confluent hypergeometric function (Kummer's function) for prescribed power-law wall temperature. The effects of the various parameters entering into the problem on the temperature distribution and wall temperature gradient are discussed. [ABSTRACT FROM AUTHOR]

An analysis has been carried out to investigate the analytical solution to the flow and heat transfer characteristics of a viscous flow over a stretching sheet in the presence of second-order slip in flow. The governing partial differential equations of flow and heat transfer are converted into non-linear ordinary differential equations by using suitable similarity transformations. The exact solution of momentum equation is assumed in exponential form and analytical solutions of heat transfer for both PST and PHF cases are obtained by the power series method in terms of Kummer's hypergeometric function. The temperature profiles are drawn for different governing parameters. The numerical values of wall temperature gradient and wall temperature are compared with earlier numerical results which have a good agreement. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21044 [ABSTRACT FROM AUTHOR]

In the present investigation the boundary layer steady flow and heat transfer of a viscous incompressible fluid due to a stretching porous sheet in presence of heat source are studied. The equations of motion and heat transfer are reduced to non-linear ordinary differential equations and the exact solutions are obtained in the form of confluent hypergeometric function (Kummer's function) for prescribed heat flux, when the wall is at prescribed second order power law heat flux or the prescribed heat flux at the stretching porous wall varies as the square of the distance from the origin. The effects of the various parameters entering into the problem on the temperature distribution and recovery temperature are discussed. [ABSTRACT FROM AUTHOR]

Abstract: By considering the recent developments in porous media, the correction to the fundamental error made by many researchers, while formulating the flow and heat transfer in porous medium over stretching surface has been incorporated in this present problem, which is still open in the literature. Here an analysis is carried out to study the flow and heat transfer characteristics in a viscoelastic fluid flow in porous medium over a stretching surface with two general cases namely PST and PHF cases, including the effects of viscous dissipation. The partial differential equations governing the flow and heat transfer are converted into ordinary differential equations and boundary conditions by suitable similarity transformation. The proposed problem has been solved analytically by power series method (using Kummer’s function). The graphical results for velocity, wall frictional coefficient and temperature are presented and discussed. Furthermore, it is shown that porous medium has same effect on the flow as viscoelasticity and it is also shown that the heat flow is always from the stretching sheet to the fluid. The numeric values of wall frictional coefficient , surface velocity gradient fηη (0) and wall temperature gradient θη (0) and wall temperature g(0) are also calculated, tabulated and discussed. [ABSTRACT FROM AUTHOR]

Abstract: An expression is obtained for the probability that a Weibull random variable falls after the truncation and within a finite interval. However small, the truncation in the Weibull distribution (when the value of the shape parameter is two, it is called the Rayleigh distribution) has an impact. An attempt is made to obtain generating functions for two fixed shape parameters. [Copyright &y& Elsevier]

Abstract: The effect of radiation on MHD steady asymmetric flow of an electrically conducting fluid past a stretching porous sheet in the presence of radiation has been analyzed. Exact solutions for the velocity and temperature fields have been derived and the effects of radiation, magnetic, Prandtl number, wall temperature and suction (or injection) parameters have been studied with the help of graphs. [Copyright &y& Elsevier]

In this short paper, we shall consider the arbitrage free Asian call option pricing under the standard Black-Scholes setting. Yor [11] studied this problem by using the bond as numéraire, whereas we use the stock as numéraire which enables us to construct a single variable Markov process for Asian option pricing. Then we show the results obtained by Yor easily through the backward equation treatment for this one dimensional Markov process. Furthermore we shall show the related results for Asian option pricing derived by German-Yor [4] and Eydeland-German [3] through our approach. [ABSTRACT FROM AUTHOR]