Terms Of H-function Research Articles

General approach to stochastic resetting.

We address the effect of stochastic resetting on diffusion and subdiffusion process. For diffusion we find that mean square displacement relaxes to a constant only when the distribution of reset times possess finite mean and variance. In this case, the leading order contribution to the probability density function (PDF) of a Gaussian propagator under resetting exhibits a cusp independent of the specific details of the reset time distribution. For subdiffusion we derive the PDF in Laplace space for arbitrary resetting protocol. Resetting at constant rate allows evaluation of the PDF in terms of H function. We analyze the steady state and derive the rate function governing the relaxation behavior. For a subdiffusive process the steady state could exist even if the distribution of reset times possesses only finite mean.

A fast algorithm for fractional Helmholtz equation with application to electromagnetic waves propagation

A fractional Helmholtz equation with the fractional Laplacian is investigated. Fundamental solutions of this equation and their factorized representations in terms of H-functions are constructed using Fourier and Mellin integral transforms. Multipole expansion for integral representation of the fractional Helmholtz equation’s solution is derived. A technique for evaluating H-functions from the multipole expansion is proposed. A modification of the multipole method for solving considered equation is developed. Numerical results demonstrating high efficiency of the proposed approach are presented. A fractional generalization of the mathematical model for a plane polarized electromagnetic wave propagation in the inhomogeneous medium, leading to a fractional Helmholtz equation with the fractional Laplacian, is derived and investigated using the proposed algorithm.

Analytic Representation of Maxwell—Boltzmann and Tsallis Thermonuclear Functions with Depleted Tail

The closed forms of the non-resonant thermonuclear function in the Maxwell–Boltzmann and Tsallis case with depleted tail are obtained in generalized special functions. The results are written in terms of H-function of two variables. The importance of the results in this paper lies in the fact that the reaction rate probability integrals in Maxwell-Boltzmann and Tsallis cases are not obtained by the conventional method of approximation or by means of a single variable transform technique but by means of a two variable transform method. The behaviour of the depleted non-resonant thermonuclear functions are examined using graphs. The results in the paper are of much interest to astrophysicists and statisticians in their future work in this area.

Effects of slip on oscillating fractionalized Maxwell fluid

AbstractThe flow of an incompressible fractionalized Maxwell fluid induced by an oscillating plate has been studied, where the no-slip assumption between the wall and the fluid is no longer valid. The solutions obtained for the velocity field and the associated shear stress, written in terms of H-functions, using discrete Laplace transform, satisfy all imposed initial and boundary conditions. The no-slip contributions, that appeared in the general solutions, as expected, tend to zero when slip parameter

Generalized Fractional Integral Operators andM-Series

Two fractional integral operators associated with FoxH-function due to Saxena and Kumbhat are applied toM-series, which is an extension of both Mittag-Leffler function and generalized hypergeometric functionpFq. The Mellin and Whittaker transforms are obtained for these compositional operators withM-series. Further some interesting properties have been established including power function and Riemann-Liouville fractional integral operators. The results are expressed in terms ofH-function, which are in compact form suitable for numerical computation. Special cases of the results are also pointed out in the form of lemmas and corollaries.

On the Ratios of Pathway Random Variables

A study of the densities of ratios of independently distributed random variables following the pathway model, Mathai (2005), is carried out. The density functions of these random variables are obtained in terms of H-function. The particular cases of the integral forms are shown to be associated with Tsallis statistics and Beck–Cohen superstatistics. Many other special functions coming under the general density are also included. We plotted the density function of the ratio of these random variables for the different values of the pathway parameters. Real-life application of the results in communication theory related to the signal–noise ratio is illustrated.

Effects of Environmental Pollution on the Growth and Existence of Biological Populations Involving H-Function

In this paper we discus about 'Effect of environmental pollution on the growth and existence biological populations' in terms of H-function

Exact solutions of fractional Schrödinger-like equation with a nonlocal term

We study the time-space fractional Schrödinger equation with a nonlocal potential. By the method of Fourier transform and Laplace transform, the Green function, and hence the wave function, is expressed in terms of H-functions. Graphical analysis demonstrates that the influence of both the space-fractal parameter α and the nonlocal parameter ν on the fractional quantum system is strong. Indeed, the nonlocal potential may act similar to a fractional spatial derivative as well as fractional time derivative.

Solutions of certain fractional kinetic equations and a fractional diffusion equation

In view of the usefulness and importance of kinetic equations in certain physical problems, the authors derive an explicit solution of a fractional kinetic equation of general character that unifies and extends earlier results. Further, an alternative shorter method based on a result developed by the authors is given to derive the solution of a fractional diffusion equation. Solutions are represented in terms of H-functions and generalized Mittag–Leffler functions.

Extension of thermonuclear functions through the pathway model including Maxwell–Boltzmann and Tsallis distributions

The Maxwell–Boltzmannian approach to nuclear reaction rate theory is extended to cover Tsallis statistics [C. Tsallis, Possible generalization of Boltzmann–Gibbs statistics, Journal of Statistical Physics 52 (1988) 479–487] and more general cases of distribution functions. An analytical study of respective thermonuclear functions is being conducted with the help of statistical techniques. The pathway model, recently introduced by Mathai [A.M. Mathai, A Pathway to matrix-variate gamma and normal densities, Linear Algebra and its Applications 396 (2005) 317–328], is utilized for thermonuclear functions and closed-form representations are obtained in terms of H-functions and G-functions. Maxwell–Boltzmannian thermonuclear functions become particular cases of the extended thermonuclear functions. A brief review on the development of the theory of analytic representations of nuclear reaction rates is given.

H-function representations for stretched exponential relaxation and non-Debye susceptibilities in glassy systems.

Analytical expressions in the time and frequency domains are derived for non-Debye relaxation processes. The complex frequency-dependent susceptibility function for the stretched exponential relaxation function is given for general values of the stretching exponent in terms of H-functions. The relaxation functions corresponding to the complex frequency-dependent Cole-Cole, Cole-Davidson, and Havriliak-Negami susceptibilities are given in the time domain in terms of H-functions. It is found that a commonly used correspondence between the stretching exponent of Kohlrausch functions and the stretching parameters of Havriliak-Negami susceptibilities are not generally valid.

Analytical representations for relaxation functions of glasses

Analytical representations in the time and frequency domains are derived for the most frequently used phenomenological fit functions for non-Debye relaxation processes. In the time domain the relaxation functions corresponding to the complex frequency dependent Cole–Cole, Cole–Davidson and Havriliak–Negami susceptibilities are also represented in terms of H-functions. In the frequency domain the complex frequency dependent susceptibility function corresponding to the time dependent stretched exponential relaxation function is given in terms of H-functions. The new representations are useful for fitting to experiment.

THE FRACTIONAL KINETIC EQUATION AND THERMONUCLEAR FUNCTIONS

The paper discusses the solution of a simple kinetic equation of thetype used for the computation of the change of the chemical compositionin stars like the Sun. Starting from the standard form of the kineticequation it is generalized to a fractional kinetic equation and itssolutions in terms of H-functions are obtained. The role of thermonuclearfunctions, which are also represented in terms of G- and H-functions,in such a fractional kinetic equation is emphasized. Results containedin this paper are related to recent investigations of possibleastrophysical solutions of the solar neutrino problem.

Stability against phase mixing of collisionless self-gravitating matter

view Abstract Citations (7) References (45) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Stability against Phase Mixing of Collisionless Self-gravitating Matter Hjorth, Jens Abstract It is suggested how to define a macroscopic steady state of a collisionless self-gravitating system with Newtonian interactions in terms of H-functions. A new condition for stability is formulated as a consequence thereof: Any single-variable distribution function, f = f(Q), which is a stationary point of some entropy-like functional, must have df/dQ <= 0 to be stable against phase mixing. For the special class of Osipkov-Merritt models, Q = E + L^2^/2r_a_^2^, this is found to agree with results of published numerical experiments. The stability criterion may have important implications for the equilibria of galaxies and dark- matter halos. One consequence is that stable spherical galaxies apparently have an anisotropy radius, r_a_, greater than ~40% of the half-mass radius. This finding is consistent with dissipationless- collapse simulations. Publication: The Astrophysical Journal Pub Date: March 1994 DOI: 10.1086/173875 Bibcode: 1994ApJ...424..106H Keywords: Boltzmann Transport Equation; Dark Matter; Galactic Evolution; Galactic Halos; Many Body Problem; Mathematical Models; Gravitational Effects; Hamiltonian Functions; Steady State; Astrophysics; GALAXIES: KINEMATICS AND DYNAMICS; INSTABILITIES; METHODS: ANALYTICAL full text sources ADS |

Anisotropic Milne's problem solution in terms of H-functions: Numerical results

Abstract A solution of anisotropic Milne's problem in terms of the function H(μ) for non conservative polynomial phase functions has been subjected to numerical analysis and used to obtain numerical results. For the evaluation of H(μ) over the non negative half-line a constructive method is presented and discussed. Several checks, based mainly on the analiticity properties of H(μ), have been introduced to test the computational accuracy. Some sample cases with anisotropic phase functions of order up to N = 5 are reported and discussed

On the distributions of a class of statistics in multivariate analysis

The noncentral distributions of Y = Π i=1 p θ i a(1 − θ i) b are obtained, where a and b are known real numbers and θ i 's stand for latent roots of a matrix arising in each of three situations in multivariate normal theory, namely, test of equality of two covariance matrices, MANOVA, and canonical correlation. The study is extended to the complex case as well. The distributions are derived in terms of H-functions as a result of inverse Mellin transforms. Further, asymptotic expansions of the distribution of Y have been obtained in the case of two covariance matrices for selected values of ( a, b).