The fundamental solution (Green function) for the Cauchy problem of the space-time fractional diffusion equation is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation. Then, by using the Mellin transform, a general representation of the Green function in terms of Mellin-Barnes integrals in the complex plane is derived. This allow...

We prove existence and uniqueness of the global solution to the Cauchy problem on a universe fireworks model with finite total mass at the initial state when the ratio of the mass surviving the explosion, the probability of the explosion of fragments and the probability function of the velocity change of a surviving particle satisfy the corresponding physical conditions. Although the nonrelativ...

This paper deals with the Legendre wavelet (LW) collocation method for the numerical solution of the radial Schrodinger equation for hydrogen atom. Energy eigenvalues for the hydrogen bound system is derived -13.6 eV. Numerical results of the ground state modes of wave function for the hydrogen R(r) or the electron probability density function, has been presented. The numerical results ha...

Order statistics arising from exponentiated gamma (EG) distribution are considered. Closed from expressions for the single and double moments of order statistics are derived. Measures of skewness and kurtosis of the probability density function of the rth order statistic for different choices of r, n and /theta are presented. Recurrence relations between single and double moments of r...

We explore two generalizations of the Cauchy-Schwarz Bessel’s inequality and the Selberg inequality and their application to probability theory. We then give a tautological proof of the De Caen-Selberg Inequality and a proof of the second Borel-Cantelli Lemma with negative dependence. We finish with a suggestion of how linear operator theory can help us understand the tightness of many Selberg-...

We express the generalized Cauchy-Stieltjes transforms of some particular Beta distributions depending on a positive parameter λ as λ-powered Cauchy-Stieltjes transforms of some probability measures. The Cauchy-Stieltjes transforms of the latter measures are shown to be the geometric mean of the Cauchy-Stieltjes transform of the Wigner law together with another one. Moreover, they are absolutel...

In this article, we have introduced the new distribution named exponentiated inverse power Cauchy distribution, which presents more flexibility in modeling a real lifetime dataset. The proposed is analytically appealing and easy to work with can be used efficiently analyze data sets. Its probability density function include various shapes according value of parameters. Different explicit expres...

In this paper, we investigate the composite midpoint rule for the evaluation of Cauchy principal value integral in an interval and place the key point on its pointwise superconver-gence phenomenon. The error expansion of the rule is obtained, which shows that the superconvergence phenomenon occurs at the points of each subinterval whose local coordinate is the zeros of some function. Then, by a...

In this work, we consider the multidimensional time-fractional diffusion equation with $$\psi $$ -Hilfer derivative. This fractional derivative enables interpolation between Riemann–Liouville and Caputo derivatives its kernel depends on an arbitrary positive monotone increasing function ,$$ thus encompassing several in literature. allows us to obtain general results for different families of pr...

The packing of hard-core particles in contact with their neighbors is a statically determinate problem. probability functional method allows deriving the full system equations for stress tensor components analytically. For an isotropic and homogeneous two-dimensional packing, we derive classical Euler–Cauchy Navier equations; latter expressing terms Airy function.