Bruce R. Fabijonas

On the Reversion of an Asymptotic Expansion and the Zeros of the Airy Functions

The general theories of the derivation of inverses of functions from their power series and asymptotic expansions are discussed and compared. The asymptotic theory is applied to obtain asymptotic expansions of the zeros of the Airy functions and their derivatives, and also of the associated values of the functions or derivatives. A Maple code is constructed to generate exactly the coefficients in these expansions. The only limits on the number of coefficients are those imposed by the capacity of the computer being used and the execution time that is available. The sign patterns of the coefficients suggest open problems pertaining to error bounds for the asymptotic expansions of the zeros and stationary values of the Airy functions.

Algorithm 838: Airy Functions

We present a Fortran 90 module, which computes the solutions and their derivatives of Airys differential equation, both on the real line and in the complex plane. The module also computes the zeros and associated values of the solutions and their derivatives, and the modulus and phase functions on the negative real axis. The computational methods are numerical integration of the differential equation and summation of asymptotic expansions for large argument. These methods were chosen because they are simple, adaptable to any precision, and amenable to rigorous error analysis. The module can be used to validate other codes or as a component in programs that require Airy functions.

Algorithms and codes for the Macdonald function: recent progress and comparisons

The modified Bessel function Kiv(x), also known as the Macdonald function, finds application in the Kontorovich-Lebedev integral transform when x and v are real and positive. In this paper, a comparison of three codes for computing this function is made. These codes differ in algorithmic approach, timing, and regions of validity. One of them can be tested independent of the other two through Wronskian checks, and therefore is used as a standard against which the others are compared.

Computation of complex Airy functions and their zeros using asymptotics and the differential equation

We describe a method by which one can compute the solutions of Airys differential equation, and their derivatives, both on the real line and in the complex plane. The computational methods are numerical integration of the differential equation and summation of asymptotic expansions for large argument. We give details involved in obtaining all of the parameter values, and we control the truncation errors rigorously. Using the same computational methods, we describe an algorithm that computes the zeros and associated values of the Airy functions and their derivatives, and the modulus and phase functions on the negative real axis.

Probabilistic Rate Compartment Cancer Model: Alternate versus Traditional Chemotherapy Scheduling

A four-compartment model for the evolution of cancer based on the characteristics of the cells is presented. The model is expanded to account for intrinsic and acquired drug resistance. This model can be explored to see the evolution of drug resistance starting from a single cell. Numerical studies are performed illustrating the palliative nature of chemotherapeutic treatments. Computational results are given for traditional treatment schedules. An alternate schedule for treatments is developed increasing the life expectancy and quality of life for the patient. A framework for the alternate scheduling is presented that addresses life expectancy, quality of life, and risk of metastasis. A key feature of the alternate schedule is that information for a particular patient can be used resulting in a personalized schedule of treatments. Alternate scheduling is compared to traditional treatment scheduling.

Laplace's method on a computer algebra system with an application to the real valued modified Bessel functions

We examine a Maple implementation of two distinct approaches to Laplaces method used to obtain asymptotic expansions of Laplace-type integrals. One algorithm uses power series reversion, whereas the other expands all quantities in Taylor or Puiseux series. These algorithms are used to derive asymptotic expansions for the real valued modified Bessel functions of pure imaginary order and real argument that mimic the well-known corresponding expansions for the unmodified Bessel functions.

Craik-Criminale solutions and elliptic instability in nonlinear-reactive closure models for turbulence

The Craik–Criminale class of exact solutions is examined for a nonlinear-reactive fluids theory that includes a family of turbulence closure models. These may be formally regarded as either large eddy simulation or Reynolds-averaged Navier–Stokes models of turbulence. All of the turbulence closure models in the class under investigation preserve the existence of elliptic instability, although they shift its angle of critical stability as a function of the rotation rate Ω of the coordinate system, the wave number β of the Kelvin wave, and the model parameter α, the turbulence correlation length. Elliptic instability allows a comparison among the properties of these models. It is emphasized that the physical mechanism for this instability is not wave–wave interaction, but rather wave, mean-flow interaction as governed by the choice of a model’s nonlinearity.

Transients in the decay of isotropic turbulence

Using a shell model, we examine how initial conditions affect the decay of isotropic 3D turbulence when no confining spatial boundaries exist, Re λ is high and the spectrum has a single peak initially. We base our investigation on large ensembles of realizations with the same initial energy spectrum. The initial shell phases, however, consist of a coherent part, common to all realizations within an ensemble, and a random part. The effects of the initial conditions are transient as a decay law ⟨ E ⟩ = C0(t − t0)α0 with a common exponent α0 emerges from all initial conditions. In addition, the same energy spectrum emerges, and the normalized helicity disappears from all ensembles whatever their initial helicity. Thus, we argue that asymptotically a single state of decay exists. By considering the compensated decay, we find the transient is itself a power law to leading order. What is more, our data suggest a universal transient exponent. Our analysis focuses on the decay signature F (⟨ E ⟩), which we define...

Self-similar enstrophy divergence in a shell model of isotropic turbulence

We focus on the early evolution of energy E and enstrophy Z when the dissipation grows in significance from negligible to important. By considering a sequence of viscous shell model solutions we find that both energy and dissipation are continuous functions of time in the inviscid limit. Inviscidly, Z takes only a finite time t* to diverge, where t* depends on initial conditions

Euler–Poincaré formulation and elliptic instability for nth-gradient fluids

The energy of an nth-gradient fluid depends on its Eulerian velocity gradients of order n. A variational principle is introduced for the dynamics of nth-gradient fluids and their properties are reviewed in the context of Noethers theorem. The stability properties of Craik–Criminale solutions for first and second gradient fluids are examined.