Iterative method for solving a nonlinear fourth order boundary value problem

AbstractIn the study of transverse vibrations of a hinged beam there arises a boundary value problem for fourth order ordinary differential equation, where a significant difficulty lies in a nonlinear term under integral sign. In recent years several authors considered finite approximation of the problem and proposed an iterative method for solving the system of nonlinear equations obtained. The essence of the iteration is the simple iteration method for a nonlinear equation, although this is not shown in the papers of the authors.In this paper we propose a new approach to the solution of the problem, which is based on the reduction of it to finding a root of a nonlinear equation. In both cases, when the explicit form of this equation is found or not, the use of the Newton or Newton-type methods generate fast convergent iterative process for the original problem. The results of many numerical experiments confirm the efficiency of the proposed approach

Current Based Automated Design of Realizable Metasurface Antennas With Arbitrary Pattern Constraints

We present a 3-D method to numerically design a realizable metasurface, which transforms a given incident field into a radiated field that satisfies mask-type (inequality) constraints. The method is based on an integral equation formulation, with local impedance boundary condition (IBC) approximation. The procedure yields the spatial distribution of the impedance, yet the process involves the synthesis of the equivalent current only. This current is constrained to correspond to a realizable surface impedance, i.e., passive, lossless, and with reactance values bounded by practical realizability limits. The current-based design avoids any solution to the forward problem, and the impedance is obtained from the synthesized current only at the end of the process. The procedure is gradient-based, with the gradient expressed in closed form. This allows handling large metasurfaces, with full spatial variability of the impedance in two dimensions. The method requires no a priori information, and all relevant operations in the iterative process can be evaluated with O(N log N) complexity. Application examples concentrate on the case of on-surface excitation and far-field (FF) pattern specifications; they show designs of circular and rectangular metasurface antennas of 20 wavelengths in size, with pencil- and shaped-beam patterns, and for both circular and linear polarizations

On the approximation of L\'evy driven Volterra processes and their integrals

Volterra processes appear in several applications ranging from turbulence to energy finance where they are used in the modelling of e.g. temperatures and wind and the related financial derivatives. Volterra processes are in general non-semimartingales and a theory of integration with respect to such processes is in fact not standard. In this work we suggest to construct an approximating sequence of L\'evy driven Volterra processes, by perturbation of the kernel function. In this way, one can obtain an approximating sequence of semimartingales. Then we consider fractional integration with respect to Volterra processes as integrators and we study the corresponding approximations of the fractional integrals. We illustrate the approach presenting the specific study of the Gamma-Volterra processes. Examples and illustrations via simulation are given.Comment: 39 pages, 3 figure

Approximate pricing of barrier options in Lévy models

This thesis deals with pricing of a certain type of derivatives, namely European barrier options. We consider the question of pricing this option in geometric Lévy models, which in contrast to the famous Black-Scholes model allow jumps in the stock price. This increases the difficulty of computing an option price enormously due to the fact that the stock price does not necessarily cross the barrier continuously, but is able to jump over it from different space points. The main idea of our approach will be the interpretian of the jump model as a perturbed Black-Scholes model, where we compute a first order correction term. In the process of evaluating the approximation it will be necessary to split up our option into two, one of them paying a polynomial of the overshoot of the underlying Lévy process. The approximation for the first option will consist of moments of the stock price as well as sensitivities (so called 'greeks') of the Black-Scholes derivative price. The approximation for the overshoot option will consist of an independent result on the approximation of overshoot moments using Lévy process fluctuation theory. The correction term consists of a 2-dimensional complex integral formula depending only on the characteristic exponent of the underlying Lévy process, which may be efficiently evaluated numerically. We show in a numerical illustration for several parametric models used in practice, that our approximation yields good results if the Lévy model is reasonably close to a Black-Scholes model with same volatility in the sense that the fourth order cumulant of the Lévy process should not be too large, yet arguably being robust and simple to evaluate

Retarded versus time-nonlocal quantum kinetic equations

The finite duration of the collisions in Fermionic systems as expressed by the retardation time in non-Markovian Levinson-type kinetic equations is discussed in the quasiclassical limit. We separate individual contributions included in the memory effect resulting in (i) off-shell tails of the Wigner distribution, (ii) renormalization of scattering rates and (iii) of the single-particle energy, (iv) collision delay and (v) related non-local corrections to the scattering integral. In this way we transform the Levinson equation into the Landau-Silin equation extended by the non-local corrections known from the theory of dense gases. The derived nonlocal kinetic equation unifies the Landau theory of quasiparticle transport with the classical kinetic theory of dense gases. The space-time symmetry is discussed versus particle-hole symmetry and a solution is proposed which transforms these two exclusive pictures into each other.Comment: slightly revised, 19 page