Related papers: On Singular Points and Oscillatory Integrals

Finite dimensional representations of extended Weyl-Heisenberg algebra are studied both from mathematical and applied viewpoints. They are used to define unitary phase operator and the corresponding eigenstates (phase states). It is also…

We propose the generalization of the Fourier modal method aimed at calculating localized eigenmodes of integrated optical resonators. The method is based on constructing the analytic continuation of the structure's scattering matrix and…

In this paper, we resort to the Laplace transform method in order to show its efficiency when approaching some types of fractional differential equations. In particular, we present some applications of such methods when applied to possible…

Using the method of the Laplace transform, we consider fractional oscillations. They are obtained by the time-clock randomization of ordinary harmonic vibrations. In contrast to sine and cosine, the functions describing the fractional…

Equations arising in General Relativity are usually too complicated to be solved analytically and one has to rely on numerical methods to solve sets of coupled partial differential equations. Among the possible choices, this paper focuses…

We calculate the partition function of a harmonic oscillator with quasi-periodic boundary conditions using the zeta-function method. This work generalizes a previous one by Gibbons and contains the usual bosonic and fermionic oscillators as…

In this article, we consider some generalizations of polynomial and exponential B-splines. Firstly, the extension from integral to complex orders is reviewed and presented. The second generalization involves the construction of uncountable…

We derive an integral expression for the plane-wave expansion of the time-varying (nonstationary) random field inside a mode-stirred reverberation chamber. It is shown that this expansion is a so-called oscillatory process, whose kernel can…

In this paper, we study time-asymptotic propagation phenomena for a class of dispersive equations on the line by exploiting precise estimates of oscillatory integrals. We propose first an extension of the van der Corput Lemma to the case of…

We report on the possibilities of using the method of normal fundamental systems for solving some problems of oscillation theory. Large elastic dynamical systems with continuous and discrete parameters are considered, which have many…

In this brief note the operatorial methods are applied to the study of the Airy function and its generalizations.

In this note we provide an algorithm for computing the fractional integrals of orthogonal polynomials, which is more stable than that using the expression of the polynomials w.r.t. the canonical basis. This algorithm is aimed at solving…

We propose an asymptotic expansion formula for matrix integrals, including oscillatory terms (derivatives of theta-functions) to all orders. This formula is heuristically derived from the analogy between matrix integrals, and formal matrix…

In this paper we generalize notions of iterated integral with regard to an unpredictable process. We establish a formula of integration by parts, the existence of a continuous modification and give an expression of the increasing process.

We describe a generalized formalism, addressing the fundamental problem of reflection and transmission of complex optical waves at a plane dielectric interface. Our formalism involves the application of generalized operator matrices to the…

Local iterated function systems are an important generalisation of the standard (global) iterated function systems (IFSs). For a particular class of mappings, their fixed points are the graphs of local fractal functions and these functions…

Based on the local fractional calculus, we establish some new generalizations of H\"{o}lder's inequality. By using it, some results on the generalized integral inequality in fractal space are investigated in detail.

We consider a general schema involving measure spaces, contractions and linear and continuous operators. Within the framework of this schema we use our sesquilinear uniform integral and introduce some integral operators on continuous vector…

It is well known that phase function methods allow for the numerical solution of a large class of oscillatory second order linear ordinary differential equations in time independent of frequency. Unfortunately, these methods break down in…

The paper presents a new formula for the fractional integration, which generalizes the Riemann-Liouville and Hadamard fractional integrals into a single form, which when a parameter fixed at different values, produces the above integrals as…