A Quadruple Integral Involving Product of the Struve Hv(βt) and Parabolic Cylinder Du(αx) Functions

The objective of the present paper is to obtain a quadruple infinite integral. This integral involves the product of the Struve and parabolic cylinder functions and expresses it in terms of the Hurwitz–Lerch Zeta function. Almost all Hurwitz-Lerch Zeta functions have an asymmetrical zero distributionSpecial cases in terms fundamental constants and other special functions are produced. All the results in the work are new.

Joint law of an Ornstein–Uhlenbeck process and its supremum

Let X be an Ornstein–Uhlenbeck process driven by a Brownian motion. We propose an expression for the joint density / distribution function of the process and its running supremum. This law is expressed as an expansion involving parabolic cylinder functions. Numerically, we obtain this law faster with our expression than with a Monte Carlo method. Numerical applications illustrate the interest of this result.

Distributions of the Ratio and Product of Two Independent Weibull and Lindley Random Variables

In this paper, we derive the cumulative distribution functions (CDF) and probability density functions (PDF) of the ratio and product of two independent Weibull and Lindley random variables. The moment generating functions (MGF) and the k-moment are driven from the ratio and product cases. In these derivations, we use some special functions, for instance, generalized hypergeometric functions, confluent hypergeometric functions, and the parabolic cylinder functions. Finally, we draw the PDF and CDF in many values of the parameters.

Stationary Schrödinger equation in the semi-classical limit: WKB-based scheme coupled to a turning point

This paper is concerned with the efficient numerical treatment of 1D stationary Schrödinger equations in the semi-classical limit when including a turning point of first order. As such it is an extension of the paper [3], where turning points still had to be excluded. For the considered scattering problems we show that the wave function asymptotically blows up at the turning point as the scaled Planck constant $$\varepsilon \rightarrow 0$$ε→0, which is a key challenge for the analysis. Assuming that the given potential is linear or quadratic in a small neighborhood of the turning point, the problem is analytically solvable on that subinterval in terms of Airy or parabolic cylinder functions, respectively. Away from the turning point, the analytical solution is coupled to a numerical solution that is based on a WKB-marching method—using a coarse grid even for highly oscillatory solutions. We provide an error analysis for the hybrid analytic-numerical problem up to the turning point (where the solution is asymptotically unbounded) and illustrate it in numerical experiments: if the phase of the problem is explicitly computable, the hybrid scheme is asymptotically correct w.r.t. $$\varepsilon $$ε. If the phase is obtained with a quadrature rule of, e.g., order 4, then the spatial grid size has the limitation $$h=\mathcal{O}(\varepsilon ^{7/12})$$h=O(ε7/12) which is slightly worse than the $$h=\mathcal{O}(\varepsilon ^{1/2})$$h=O(ε1/2) restriction in the case without a turning point.