Time-dependentX- andY-functions by integral equation method

The time-dependent equation of radiative transfer for isotropic scattering has been solved by integral equation technique in terms ofX- andY-functions appropriate for the problem. It is seen thatX- andY-functions are reducible to the corresponding function for steady-state problems by simply changing the Laplace transform parameters-i.e., byS0.     

Solution of Riemann–Hilbert problems for determination of new decoupled expressions of Chandrasekhar’s <Emphasis Type="Italic">X</Emphasis>- and <Emphasis Type="Italic">Y</Emphasis>-functions for slab geometry in radiative transfer

In radiative transfer, the intensities of radiation from the bounding faces of a scattering atmosphere of finite optical thickness can be expressed in terms of Chandrasekhar’s X- and Y-functions. The nonlinear nonhomogeneous coupled integral equations which the X- and Y-functions satisfy in the real plane are meromorphically extended to the complex plane to frame linear nonhomogeneous coupled singular integral equations. These singular integral equations are then transformed into nonhomogeneous Riemann–Hilbert problems using Plemelj’s formulae. Solutions of those Riemann–Hilbert problems are obtained using the theory of linear singular integral equations. New forms of linear nonhomogeneous decoupled expressions are derived for X- and Y-functions in the complex plane and real plane. Solutions of these two expressions are obtained in terms of one known N-function and two new unknown functions N ₁- and N ₂- in the complex plane for both nonconservative and conservative cases. The N ₁- and N ₂-functions are expressed in terms of the known N-function using the theory of contour integration. The unknown constants are derived from the solutions of Fredholm integral equations of the second kind uniquely using the new linear decoupled constraints. The expressions for the H-function for a semi-infinite atmosphere are obtained as a limiting case.     

A direct numerical approach to the Chandrasekhar'sH-functions for arbitrary characteristic functions

The purpose of this paper is to present a numerical technique to directly compute the Chandrasekhar'sH ()-function for anisotropic scattering in terms of the roots of the characteristic equations as well as the quadrature points of a certain degreen employed to approximate the definite integral involved in the basic equation. The principal feature of the algorithm proposed here is a compact computer code to enumerate _n C _m combinations ofn distinct integers {1,...,n} takenm at a time. With these quantities available, the coefficients of the polynomial equation of the characteristics equation can be readily computed for any given characteristic function, so that a standard technique such as the Laguerre method can be applied to find all the roots.It is shown that the results obtained for some representativeH()-functions using the present technique with relatively low-order formula (e.g.,n=7) are sufficiently accurate for all practical purposes.     

On the new computable solution of the generalized fractional kinetic equations involving the generalized function for the fractional calculus and related functions

In view of the usefulness and a great importance of the kinetic equation in certain astrophysical problems the authors develop a new and further generalized form of the fractional kinetic equation involving the G-function, a generalized function for the fractional calculus. This new generalization can be used for the computation of the change of chemical composition in stars like the Sun. The Mellin-Barnes contour integral representation of the G-function is also established. The manifold generality of the G-function is discussed in terms of the solution of the above fractional kinetic equation. A compact and easily computable solution is established. Special cases, involving the generalized Mittag-leffler function and the R-function, are considered. The obtained results imply more precisely the known results.     

An exact solution of transfer equations for interlocked multiplets

The transfer equations for non-coherent scattering arising from interlocking of principal lines without redistribution is exactly solved in a very simple way by Laplace tranform and Wiener-Hopf technique which are easily applied by the use of the new representation ofH-functions obtained recently by the author (1977). The emergent intensity in therth line is expressed in terms of anH-function and a Cauchy type integral admitting of closed form evaluation.     

The solution of resonance radiative transfer equation for diffusion by the method of laplace transform and linear singular operators

The equation for radiative transfer in the case of resonance radiation for isotropic scattering has been solved by the method of the Laplace transformation and linear singular operators. The solution for emergent intensities have come out in terms ofX- andY-functions.     

Polytropic spheres with isothermal cores

The paper deals with massive fluid spheres with an isothermal core (having finite central density) and a polytropic envelope in terms of the General Relativity. The matching of the solutions in the core and envelope has been done using Bondi's condition,H=0 and also without it. The study reveals that since this condition does not correspond to any particular physical situation the maximum values of fractional core size, fractional core mass and the redshift do not occur atH=0, but that they occur at some other point. Within the permissible physical conditions (dP/dρ≤1) the maximum values ofM _core/M,R _core/R and the surface redshift, for an isothermal coreP=ρ/3, are respectively 0.473, 0.554, and 0.565. Using the conditionH=0, it has been shown that for isothermal cores corresponding to the equation of the stateP≥0.6ρ, the configurations are pulsationally unstable.     

Fourier analysis of the light curves of eclipsing variables,XXII

The theoretical values of the momentsA _2m for any type of eclipses, expressed in terms of the elementsL ₁,a andc ₀, have been derived in the simple forms of rapidly convergent expansions to the series of Chebyshev polynomials, Jacobi polynomials and KopalJ-integrals (Kopal, 1977c) and hold good for any real (not necessarily integral) value ofm0.The aim of the present paper has been to establish explicit expressions for the Jacobian and its fast enough computation in the light changes of close eclipsing systems, arising from the partial derivative of different pairs ofg-functions (Kopal and Demircan, 1978, Paper XIV) with respect toa andc ₀ ² , for any type of eclipses (be these occultations or transit, partial, total or annular) and for any arbitrary degreel of the adopted law of limb-darkening. The functional behaviour of this Jacobian would determine the reasonable light curve in connection with geometrical determinacy of the parametersa andc ₀. In the expansion of Jacobian, the terms consist of two polynomials which satisfy certain three-term recursion relations having the eclipse parametersa andc ₀, as their arguments.Closed form expressions forf-functions, as well as of the Jacobian (e.g.,m=1, 2, 3), obtaining in the case of total eclipses, are given for a comparative discussion with the theoretical values of Jacobian derived from partial derivative of different pairs ofg-functions.The numerical magnitude of Jacobian would determine the best combination of the momentsA _2m in the different pairs ofg-functions and definite results would follow in the subsequent paper of this series (Edalati, 1978c, Paper XXIV).     

Fourier analysis of the light curves of eclipsing variables,XXIV

The aim of the present paper will be to evaluate numerically Jacobian and other functions which have been discussed in more detail in a previous paper of this series (Edalati, 1978b, Paper XXII), and also choose the most convenient moments to obtain a good determination for the unknown eclipse parametersa andc ₀. More than 12 different pairs ofg-functions for real values ofm have been investigated numerically and diagrammatically. The behaviour ofg-functions depends but very little on different combination of the moments, and related diagrams are approximately the same asg ₂ andg ₄ (Kopal and Demircan, 1978, Paper XIV).The behaviour of the vanishing Jacobian, arising from different pairs ofg-functions for real values ofm1 has been shown diagrammatically in terms ofa andc ₀. Accordingly, we obtain the optimum combination of the moments (i.e.,A ₆,A ₇,A ₈ andA ₉) ing-functionsg ₇ andg ₈. It has been noted that the behaviour of theg-functions which depend on the combinations of the higher order moments (i.e.,m5) have been ruled out, because the proportional error of the momentsA _2m increases with increasing values of realm. The automated method has been tested successfully on the light curve of RT Per (Mancusoet al., 1977; Edalati, 1978a). Finally, a comparison is given of the elements of RT Per arising from two different pairs ofg-functions, i.e.,g ₂,g ₄ (Edalati, 1978a) andg ₇,g ₈ for the light curves analysis.     

Fractional Reaction-Diffusion Equations

In a series of papers, Saxena et al. (2002, 2004a, 2004b) derived solutions of a number of fractional kinetic equations in terms of generalized Mittag-Leffler functions which provide the extension of the work of Haubold and Mathai (1995, 2000). The subject of the present paper is to investigate the solution of a fractional reaction-diffusion equation. The results derived are of general nature and include the results reported earlier by many authors, notably by Jespersen et al. (1999) for anomalous diffusion and del-Castillo-Negrete et al. (2003) for reaction-diffusion systems with Lévy flights. The solution has been developed in terms of the H-function in a compact form with the help of Laplace and Fourier transforms. Most of the results obtained are in a form suitable for numerical computation.     

Time-dependentH-,X- andY-functions for an anisotropically-scattering atmosphere

We have considered a homogeneous atmosphere scattering anisotropically with Dirac -function type time-dependent incidence. We used the method of integral operator developed by Ambartsumian and the theory ofN-solutions developed by Busbridge to find the correspondingH-function (in semi-infinite atmosphere) andX- andY-functions (in finite atmosphere).     

Time-dependentX- andY-functions for anisotropically scattering inhomogeneous atmosphere by principle of invariance

The nonlinear integral equations forX- andY-functions have been developed for an inhomogeneous atmosphere scattering anisotropically using the principle of invariance. The anisotropy is represented by means of a phase function expressed in terms of finite-order Legendre polynomials.     

Equilibrium of self-gravitating polytropic cylinders with a magnetic field

The effect of a prevalent magnetic field on static and uniformly rotating self-gravitating cylinders of infinite length is examined. The magnetic field is assumed to consist ofH andH _z components, which are taken to be functions of the radial coordinate alone. A variety of magnetic-field configurations are shown to be admissible solutions of equations of motion, from which some feasible cases are presented. A particular magnetic-field configuration having bothH andH _z components is studied in detail. The configuration is such that the assumption of a polytropic equation of state reduces the equation governing the density function to a non-homogeneous cylindrical analogue of the Lane-Emden equation for spherical polytropes. The homogeneous case is also studied and shows interesting magnetic-field patterns.     

On a Fortran procedure for rotating spherical-harmonic coefficients

The authors describe a Fortran subroutine that rotates the coefficients of a given spherical-harmonic model (in particular the geopotential). It is based on a paper by T. Risbo, working with the d-functions fundamental to axis rotations in Quantum Mechanics, his approach being equally applicable to the inclination functions of satellite geodesy (which we obtain as an option in the procedure). The subroutine applies Risbo’s approach to evaluate, for a given inclination, stably and accurately, the necessary d-functions up to a specified degree, whilst at the same time ‘rotating’ (for each degree in turn) the values of the harmonic coefficients. We follow Risbo’s helpful example by including a listing of the new subroutine.     

Heat and mass transfer of an oscillatory flow with Hall current

In the first part of these notes new expressions—simpler than any previously obtained—are presented in integral form for the derivatives of the α _n ⁰ -functions (required for an interpretation of the observed light changes of eclipsing variables) with respect to the fractional radiir _{1, 2} and projected separation δ of their centres in terms of the modified Bessel functionsK _{0, 1} (x) of the second kind; and utilized for establishing new asymptotic formulae for the computation of ‘boundary integrals’ of the formJ _?1 ⁰ ,n(μ). In the second part of this paper, by a resort to bi-polar coordinates, we shall establish a new type of expansions for the α _n ⁰ -functions valid for any type of eclipses, and converging faster than the expansions of the cross-correlation integral of the form (1) for α _n ⁰ that have so far been established.     

Das Korrespondenzprinzip in der Kosmologie. II. Klassische und relativistische Darstellungen der Weltmodelle

According to the equivalence between the FRIEDMANN equation of relativistic cosmology and the condition for the time-independence H = o of the HAMILTON ian H of an isotropic particle-system in the NEWTON ian mechanics (which equivalence is proved in the part I of our paper) we construct the corresponding classical HAMILTON ians to the relativistic world-models. Each cosmological model which is resulting from a physically meaningful gravitation theory must give a FRIEDMANN equation as the cosmological formulation of the time-independence condition of the energy H for the corresponding NEWTON ian N-particle system. In general relativity, EINSTEIN's field equations are including EINSTEIN's strong principle of equivalence and are giving the constance f = o and M = o of the gravitation-number f and of the mass M of the universe additional to FRIEDMANN's equation. – In special relativity, we have fM = o and this MILNE -universe is possessing a NEWTON ian and a general relativistic interpretation, too. – However, if the postulate together with the “cosmological principle” other principles about the world structure, too (p. e. MACH'S or DIRAC'S principle or the “perfect cosmological principle” by the steady-state cosmology), then EINSTEIN'S weak principle of equivalence can be fulfilled, only. In these world models the gravity-mass fM becomes a function of the cosmic time t [d/dt(fM) ± o] and this variability of fM is compatible with the constance H = o of the energy H of the NEWTON ian particle-system. For flat three-dimensional cosmological spaces (with H = Ḣ = o) a creation of rest-mass (M > o) is possible. This creation is the pecularity of the steady-state cosmos (with M > o, f = o) and of JORDAN'S cosmos (with M > o, f < o). The MACH -EINSTEIN -doctrine about the perfect determination of the inertia and of the space-time-metric by the cosmic gravitation is founded on the substitution of the NEWTON ian HAMILTON ian by a GAUSS -RIEMANN ian gravitation potential U^*(r_AB' v_AB) (TREDER 1972). Therefore, the FRIEDMANN equation for a universe with MACH'S principle is resulting from the analytical expression of the time-independence of this RIEMANNian potential U* = 0. In the case of such MACH-EINSTEIN's-Universes EINSTEIN'S condition 3fM = c8r between the mass A4 and the radius Y of the universe is valid additional to FRIEDMANN'S equation. For these universes, the EINSTEIN condition determinates the instantaneous value of the gravitation-number f. - The explicite form of the conditions H = o or h' = o gives the equation of motion for the cosmic fundamental particles with attraction and repulsion forces, generally.     

Derivation of frequency-dependentX- andY-functions for a non-coherent scattering problem

The equation of transfer for the case of non-coherent scattering (Hummer, 1968; Ivanov, 1973; McCormick and Siewert, 1970) has been considered. The correspondingX- andY-functions have been derived by a combination of eigenfunction method developed by Case, and from the principle of invariance as developed by Chandrasekhar (1960).     

Generalizations of reaction rate integrals with applications to nuclear synthesis in astrophysics and some corrections to published literature

Under the Maxwell-Boltzmann approach, the study of nuclear reactions in dense astrophysical plasmas under various cases (such as resonant, non-resonant, modified non-resonant, non-resonant under electron screening, and so on) leads to a class of complicated reaction rate integrals. It is shown that this general class of integrals can be identified with an integral involving the product of twoH-functions. This latter integral is evaluated in this article, and following Buschman (1979), several similar results in the published literature are shown to be incorrect.