An integral transform whose kernel is a stream function of a steady Euler flow with axisymmetry

A new integral transform whose kernel is a Stokes stream function of an exact solution to the axisymmetric stationary Euler equations is derived with its inverse by using a Coulomb wave function. In the process of its derivation, a set of integral transforms with respect to the radial variable is also found. Each of them is different from the Hankel (the Fourier–Bessel) transform.     

Wave packet analysis of Schrödinger equations in analytic function spaces

We consider a class of linear Schrödinger equations in R^d${\mathbb{R}}^d$, with analytic symbols. We prove a global-in-time integral representation for the corresponding propagator as a generalized Gabor multiplier with a window analytic and decaying exponentially at infinity, which is transported by the Hamiltonian flow. We then provide three applications of the above result: the exponential sparsity in phase space of the corresponding propagator with respect to Gabor wave packets, a wave packet characterization of Fourier integral operators with analytic phases and symbols, and the propagation of analytic singularities.     

Semi-integer derivatives of the Airy functions and related properties of the Korteweg-de Vries-type equations

Fractional integrals and derivatives of Airy functions (Riesz potentials) are considered. For half integrals D ^−1/2 Ai(x) and D ^−1/2 Gi(x) explicit representations are found in terms of the products of Airy functions. Here Ai(x) and Gi(x) are the Airy function of the first kind and the Scorer function, respectively. Based on that representations are obtained for all semi-integer derivatives of Ai(x) and Gi(x). Applications to Korteweg–de Vries type equations are provided.      

The obrechkoff integral transform: properties and relation to a generalized fractional calculus

This survey is devoted to one of the most general Laplace-type integral transforms, the so-called Obrechkoff integral transform, introduced and studied for the first time by Obrechkoff[25]. It has been modified by Dimovski [5],[6] and used as a basis of a Mikusinski-type operational calculus for the hyper-Bessel differential operators of arbitrary order. Later, in a series of papers Dimovski and Kiryakova [8],[9],[10] have found operational properties, complex and real inversion formulas, Abel-type theorems for the Obrechkoff transform. This theory has been further developed by Kiryakova [16],[17],[18] using the tools of the Meijer's G-functions and of the fractional calculus. Namely, a new definition as a G-transform has been given for the Obrechkoff transform. The hyper-Bessel operators themselves, have given rise to a new generalized fractional calculus and further extensive use of the G-functions. Many other generalized differentiation and integration operators happen to be special cases in this calculus, too. Special cases of the Obrechkoff transform have been "rediscovered" later by many authors. We give examples how their results could be derived from the general ones surveyed here.     

Stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow

We are concerned with the stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow. In this paper, we focus on the stability of steady four-wave configurations that are the solutions of the Riemann problem in the flow direction, consisting of two shocks, one vortex sheet, and one entropy wave, which is one of the core multi-wave configurations for the two-dimensional Euler equations. It is proved that such steady four-wave configurations in supersonic flow are stable in structure globally, even under the BV perturbation of the incoming flow in the flow direction. In order to achieve this, we first formulate the problem as the Cauchy problem (initial value problem) in the flow direction, and then develop a modified Glimm difference scheme and identify a Glimm-type functional to obtain the required BV estimates by tracing the interactions not only between the strong shocks and weak waves, but also between the strong vortex sheet/entropy wave and weak waves. The key feature of the Euler equations is that the reflection coefficient is always less than $1$, when a weak wave of different family interacts with the strong vortex sheet/entropy wave or the shock wave, which is crucial to guarantee that the Glimm functional is decreasing. Then these estimates are employed to establish the convergence of the approximate solutions to a global entropy solution, close to the background solution of steady four-wave configuration.     

On a transmission problem for two systems of elastostatics with polygonal interface

It is shown the invertibility of the boundary operator associated to a transmission problem for two systems of elastostatics with different Lamé moduli, and where the interface is a piecewiseC ¹ curve in the plane.Partially supported by MURST 60% and 40% grants     

The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on [−1,1]

We consider polynomials that are orthogonal on [−1,1] with respect to a modified Jacobi weight (1−x)^α(1+x)^βh(x), with α,β>−1 and h real analytic and strictly positive on [−1,1]. We obtain full asymptotic expansions for the monic and orthonormal polynomials outside the interval [−1,1], for the recurrence coefficients and for the leading coefficients of the orthonormal polynomials. We also deduce asymptotic behavior for the Hankel determinants and for the monic orthogonal polynomials on the interval [−1,1]. For the asymptotic analysis we use the steepest descent technique for Riemann-Hilbert problems developed by Deift and Zhou, and applied to orthogonal polynomials on the real line by Deift, Kriecherbauer, McLaughlin, Venakides, and Zhou. In the steepest descent method we will use the Szeg? function associated with the weight and for the local analysis around the endpoints ±1 we use Bessel functions of appropriate order, whereas Deift et al. use Airy functions.     

An equation whose Fujita critical exponent is not given by scaling

In this paper, we prove sharp blow up and global existence results for a heat equation with nonlinear memory. It turns out that the Fujita critical exponent is not the one which would be predicted from the scaling properties of the equation.     

Rigorous derivation of incompressible type Euler equations from non-isentropic Euler-Maxwell equations

In this paper, we investigate the convergence of the time-dependent and non-isentropic Euler-Maxwell equations to incompressible Euler equations in a torus via the combined quasi-neutral and non-relativistic limit. For well prepared initial data, the convergences of solutions of the former to the solutions of the latter are justified rigorously by an analysis of asymptotic expansions and energy method.     

On the Fourier transform of SO(d)-finite measures on the unit sphere

The paper presents a simple new approach to the problem of computing Fourier transforms of SO(d)-finite measures on the unit sphere in the euclidean space. Representing such measures as restrictions of homogeneous polynomials we use the canonical decomposition of homogeneous polynomials together with the plane wave expansion to derive a formula expressing such transforms under two forms, one of which was established previously by F. J. Gonzalez Vieli. We showthat equivalence of these two forms is related to a certain multi-step recurrence relation for Bessel functions, which encompasses several classical identities satisfied by Bessel functions. We show it leads further to a certain periodicity relation for the Hankel transform, related to the Bochner- Coifman periodicity relation for the Fourier transform. The purported novelty of this approach rests on the systematic use of the detailed form of the canonical decomposition of homogeneous polynomials, which replaces the more traditional approach based on integral identities related to the Funk-Hecke theorem. In fact, in the companion paper the present authors were able to deduce this way a fairly general expansion theorem for zonal functions, which includes the plane wave expansion used here as a special case.Received: 7 May 2004; revised: 11 October 2004     

Transonic shocks for steady Euler flows with cylindrical symmetry

In this paper, we prove the existence of transonic shocks adjacent to a uniform one for the full Euler system for steady compressible fluids with cylindrical symmetry in a cylinder, and consequently show the stability of such uniform transonic shocks. Mathematically we solve a free boundary problem for a quasi-linear elliptic–hyperbolic composite system. This reveals that the boundary conditions and equations interact in a subtle way. The key point is to “separate” in a suitable way the elliptic and hyperbolic parts of the system. The approach developed here can be applied to deal with certain multidimensional problems concerning stability of transonic shocks for the full Euler system.     

A fast transform for spherical harmonics

Spherical harmonics arise on the sphere S² in the same way that the (Fourier) exponential functions {e^ik}k arise on the circle. Spherical harmonic series have many of the same wonderful properties as Fourier series, but have lacked one important thing: a numerically stable fast transform analogous to the Fast Fourier Transform (FFT). Without a fast transform, evaluating (or expanding in) spherical harmonic series on the computer is slow—for large computations probibitively slow. This paper provides a fast transform.For a grid ofO(N²) points on the sphere, a direct calculation has computational complexityO(N⁴), but a simple separation of variables and FFT reduce it toO(N³) time. Here we present algorithms with timesO(N^5/2 log N) andO(N²(log N)²). The problem quickly reduces to the fast application of matrices of associated Legendre functions of certain orders. The essential insight is that although these matrices are dense and oscillatory, locally they can be represented efficiently in trigonometric series.     

Euler transformation formula for multiple basic hypergeometric series of type A and some applications

A multiple generalization of the Euler transformation formula for basic hypergeometric series ₂φ₁ is derived. It is obtained from the symmetry of the reproducing kernel for Macdonald polynomials by a method of multiple principal specialization. As applications, elementary proofs of the Pfaff-Saalschutz summation formula and the Gauss summation formula for basic hypergeometric series in U(n+1) due to S.C. Milne are given. Some other multiple transformation and summation formulas for very-well-poised ₁₀φ₉ and ₈φ₇ series, balanced ₄φ₃ series and ₃φ₂ series are also given.     

Reduction and quotient equations for differential equations with symmetries

The method of reduction previously known in the theory of Hamiltonian systems with symmetries is developed in order to obtain exact group-invariant solutions of systems of partial differential equations. This method leads to representations of quotient equations which are very convenient for the systematic analysis of invariant solutions of boundary value problems. In the case of partially invariant solutions, necessary and sufficient conditions of their invariance with respect to subalgebras of symmetry algebras are given. The concept of partial symmetries of differential equations is considered.     

An integral equation method for the electromagnetic scattering from a scatterer on an absorbing plane

Consider a time-harmonic electromagnetic plane wave incident on a scatterer on a grounded absorbing plane modelized as an infinite impedance plane. In this paper, a new integral representation formula is rigorously derived. Existence and uniqueness of weak solutions for the model problem are also established. The proof of existence is based on an extension of the Hodge decomposition technique to open boundaries. The results reported in this paper form a basis for numerical solutions of the electromagnetic scattering problem from a scatterer on an absorbing plane.     

Qualitative analysis of steady states to the Sel'kov model

In this work, we are concerned with a reaction-diffusion system well known as the Sel'kov model, which has been used for the study of morphogenesis, population dynamics and autocatalytic oxidation reactions. We derive some further analytic results for the steady states to this model. In particular, we show that no nonconstant positive steady state exists if 0<p?1 and θ is large, which provides a sharp contrast to the case of p>1 and large θ, where nonconstant positive steady states can occur. Thus, these conclusions indicate that the parameter p plays a crucial role in leading to spatially nonhomogeneous distribution of the two reactants. The a priori estimates are fundamental to our mathematical approaches.     

Explicit inverse of a tridiagonal k−Toeplitz matrix

Summary We obtain explicit formulas for the entries of the inverse of a nonsingular and irreducible tridiagonal k–Toeplitz matrix A. The proof is based on results from the theory of orthogonal polynomials and it is shown that the entries of the inverse of such a matrix are given in terms of Chebyshev polynomials of the second kind. We also compute the characteristic polynomial of A which enables us to state some conditions for the existence of A^–1. Our results also extend known results for the case when the residue mod k of the order of A is equal to 0 or k–1 (Numer. Math., 10 (1967), pp. 153–161.).The work was supported by CMUC (Centro de Matemática da Universidade de Coimbra) and by Acção Integrada Luso-Espanhola E-6/03     

An integral representation approach for valuing American-style installment options with continuous payment plan

In this paper, we present an integral equation approach for the valuation of American-style installment derivatives when the payment plan is assumed to be a continuous function of the asset price and time. The contribution of this study is threefold. First, we show that in the Black-Scholes model the option pricing problem can be formulated as a free boundary problem under very general conditions on payoff structure and payment schedule. Second, by applying a Fourier transform-based solution technique, we derive a system of coupled recursive integral equations for the pair of free boundaries along with an analytic representation of the option price. Third, based on these results, we propose a unified framework which generalizes the existing methods and is capable of dealing with a wide range of monotonic payoff functions and continuous payment plans. Finally, by using the illustrative example of American vanilla installment call options, an explicit pricing formula is obtained for time-varying payment schedules.