Abel-type Equations Research Articles

A parallel algorithm for large systems of Volterra integral equations of Abel type

A significative number of recent applications require numerical solution of large systems of Abel–Volterra integral equations. Here we propose a parallel algorithm to numerically solve a class of these systems, designed for a distributed-memory MIMD architecture. In order to achieve a good efficiency we employ a fully parallel and fast convergent waveform relaxation (WR) method and evaluate the lag term by using FFT techniques. To accelerate the convergence of the WR method and to best exploit the parallel architecture we develop special strategies. The performances of the resulting code, NSWR4, are illustrated on some examples.

On monotonic solutions of an integral equation of Abel type

We present an existence theorem for monotonic solutions of a quadratic integral equation of Abel type in $C[0,1]$. The famous Chandrasekhar's integral equation is considered as a special case. The concept of measure of noncompactness and a fixed point theorem due to Darbo are the main tools in carrying out our proof.

Superconvergence of collocation methods for a class of weakly singular Volterra integral equations

We discuss the application of spline collocation methods to a certain class of weakly singular Volterra integral equations. It will be shown that, by a special choice of the collocation parameters, superconvergence properties can be obtained if the exact solution satisfies certain conditions. This is in contrast with the theory of collocation methods for Abel type equations. Several numerical examples are given which illustrate the theoretical results.

An efficient and fast parallel method for Volterra integral equations of Abel type

In this paper we present an efficient and fast parallel waveform relaxation method for Volterra integral equations of Abel type, obtained by reformulating a nonstationary waveform relaxation method for systems of equations with linear coefficient constant kernel. To this aim we consider the Laplace transform of the equation and here we apply the recurrence relation given by the Chebyshev polynomial acceleration for algebraic linear systems. Back in the time domain, we obtain a three term recursion which requires, at each iteration, the evaluation of convolution integrals, where only the Laplace transform of the kernel is known. For this calculation we can use a fast convolution algorithm. Numerical experiments have been done also on problems where it is not possible to use the original nonstationary method, obtaining good results in terms of improvement of the rate of convergence with respect the stationary method.

A connection between Abel and $_p{\rm F}_q$ hypergeometric differential equations

In a recent paper, a new 3-parameter class of Abel type equations, so-called AIR, all of whose members can be mapped into Riccati equations, is shown. Most of the Abel equations with solution presented in the literature belong to the AIR class. Three canonical forms were shown to generate this class, according to the roots of a cubic. In this paper, a connection between those canonical forms and the differential equations for the hypergeometric functions 2F1, 1F1 and 0F1 is unveiled. This connection provides a closed form pFq solution for all Abel equations of the AIR class.

Using wavelet methods to solve noisy Abel-type equations with discontinuous inputs

One way of estimating a function from indirect, noisy measurements is to regularise an inverse of its Fourier transformation, using properties of the adjoint of the transform that degraded the function in the first place. It is known that when the function is smooth, this approach can perform well and produce estimators that have optimal convergence rates. When the function is unsmooth, in particular when it suffers jump discontinuities, an analogue of this approach is to invert the wavelet transform and use thresholding to decide whether wavelet terms should be included or excluded in the final approximation. We evaluate the performance of this approach by applying it to a large class of Abel-type transforms, and show that the smoothness of the target function and the smoothness of the transform interact in a particularly subtle way to determine the overall convergence rate. The most serious difficulties arise when the target function has a jump discontinuity at the origin; this has a considerably greater, and deleterious, impact on performance than a discontinuity elsewhere. In the absence of a discontinuity at the origin, the rate of convergence is determined principally by an inequality between the smoothness of the function and the smoothness of the transform.

A New Method of Solving Noisy Abel-Type Equations

A new approach to solving noisy integral equations of the first kind is applied to the family of Abel equations. Such equations play a role in stereology (Wicksell's unfolding problem), medicine, engineering, and astronomy. The method is based on an expansion in an arbitrary orthonormal basis, coupled with exact inversion of the integral operator. The inverse appears in the Fourier coefficients of the expansion, where it can be carried over to the usually well-behaved basis elements in the form of the adjoint. This method is an alternative to Tikhonov regularization, regularization of the inverse of the operator itself, or a wavelet-vaguelette/singular-value decomposition. The method is particularly interesting in irregularity of the kernel, the input, or both. Because knowledge of the spectral properties of the operator is not required, the method is also of interest in regular cases where these spectral properties are not sufficiently known or are hard to deal with. For smooth input functions, the simple basis of trigonometric functions yields input estimators whose mean integrated squared error converges at the optimal rate for the entire family of Abel operators. This can be shown when smooth wavelets are used for Abel operators with index smaller than 1/2, and when the Haar wavelet is used for operators with index larger than 1/2.

The statistical mechanics of the self-gravitating gas: equation of state and fractal dimension

We provide a complete picture of the self-gravitating non-relativistic gas at thermal equilibrium using Monte Carlo simulations (MC), analytic mean field methods (MF) and low density expansions. The system is shown to possess an infinite volume limit, both in the canonical (CE) and in the microcanonical ensemble (MCE) when N, V→∞, keeping N/ V 1/3 fixed. We compute the equation of state (we do not assume it as is customary), the entropy, the free energy, the chemical potential, the specific heats, the compressibilities, the speed of sound and analyze their properties, signs and singularities. The MF equation of state obeys a first order non-linear differential equation of Abel type. The MF gives an accurate picture in agreement with the MC simulations both in the CE and MCE. The inhomogeneous particle distribution in the ground state suggest a fractal distribution with Haussdorf dimension D with D slowly decreasing with increasing density, 1≲ D<3.

The method of abel-type integral equations for identification of nonlinear systems of structural seismodynamics

The problem on determination of the nonlinear dissipative and elastic characteristics of some vibrating systems that are encountered in structural seismodynamics is considered. Systems of integral Volterra equations of the first kind (Abel-type equations) are constructed on the basis of approximate analytical solutions to problems on the forced vibration of quasilinear vibrating systems. Such equations relate the nonlinear stiffness and dissipation characteristics with the characteristics of motion, which can be obtained experimentally. The solutions of the integral equations derived are represented in the form of quadrature Stieltjes-integral formulas

A self-similar inhomogeneous dust cosmology

A detailed study of an inhomogeneous dust cosmology contained in a -law family of perfect-fluid metrics recently presented by Mars and Senovilla is performed. The metric is shown to be the most general orthogonally transitive, Abelian, on solution admitting an additional homothety such that the self-similar group is of Bianchi type VI and the fluid flow is tangent to its orbits. The analogous cases with Bianchi types I, II, III, V, VIII and IX are shown to be impossible thus making this metric privileged from a mathematical viewpoint. The differential equations determining the metric are partially integrated and the line-element is given up to a first-order differential equation of Abel type of first kind and two quadratures. The solutions are qualitatively analysed by investigating the corresponding autonomous dynamical system. The spacetime is regular everywhere except for the big bang and the metric is complete both into the future and in all spatial directions. The energy density is positive, bounded from above at any instant of time and with an spatial profile (in the direction of inhomogeneity) which is oscillating with a rapidly decreasing amplitude. The generic asymptotic behaviour at spatial infinity is a homogeneous plane wave. Well known dynamical system results indicate that this metric is very likely to describe the asymptotic behaviour in time of a much more general class of inhomogeneous dust cosmologies.

Closed Form Solutions of the Two-Dimensional Turning Bands Equation

The turning bands method generates realizations of isotropic Gaussian random fields by means of appropriately summed line processes. For two-dimensional simulations the relation between the isotropic correlation function of the random field and the correlation function to be simulated along the lines is given by an integral equation of Abel type. We present closed form solutions of this integral equation for almost all two-dimensional correlation models encountered in practice and discuss their numerical implementation. As an additional benefit, our tables and illustrations serve as a concise guide to correlation models useful in geostatistics.

Spatio-temporal variations of density of the particles generating the optical emission of meteor plasma

The data of absolute spectrophotometry of meteors permit to determine number densities of particles of different elements. It is possible to solve mathematically the problem on space-time structure of the meteor with the model of a spherically symmetrical flare. The space distribution of radiating matter can be estimated by the integral equation of Abel type. Variation of density of particles in time can be found from the differential equations of kinetics.

Chebyshev approximation for solving the radon integral equation

A Volterra integral equation, which relates the Fourier coefficients of the projection (in polar coordinates) with the corresponding coefficients of the unknown density function is deduced. The proposed method describes a new technique based on Chebyshev approximation of the integral equation of Abel type which is deduced by using the Radon inversion transformation, the recursive evaluation of certain integrals is calculated. We obtain also the same integral equation by other inversion formulae based on Hankel transformations. Numerical examples are treated and the method compares quite favourably with other known methods.

Prolate galaxy models with thin-tube orbits

view Abstract Citations (17) References (20) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Prolate Galaxy Models with Thin Tube Orbits Hunter, C. ; de Zeeuw, P. T. ; Park, C. ; Schwarzschild, M. Abstract We consider the construction of self-consistent distribution functions for prolate Stackel models. The orbits are confined by three independent integrals of motion and form two different families, the inner and outer long-axis tubes. The models constructed are built exclusively with infinitesimally thin orbits, and orbits of both families are needed. Once the density of the model has been split into two components, one for each family, two phase-space distribution functions can be found uniquely by solving one-dimensional integral equations of Abel type. But the models are not unique, because a variety of splits of the density is possible at all except extreme values of the axis ratio. Except at these extremes, physical models require that both families of orbits are populated. We have constructed a range of models for perfect prolate spheroids of all axis ratios, and have computed several of their other basic physical properties such as mean-streaming velocities, dispersions, angular momenta and the relative masses of their two orbit families. The mean- streaming motions exceed the circular velocity in the inner regions of these models. Publication: The Astrophysical Journal Pub Date: November 1990 DOI: 10.1086/169351 Bibcode: 1990ApJ...363..367H Keywords: Astronomical Models; Computational Astrophysics; Galactic Rotation; Galactic Structure; Orbital Mechanics; Distribution Functions; Integral Equations; Astrophysics; GALAXIES: INTERNAL MOTIONS; GALAXIES: STRUCTURE; STARS: STELLAR DYNAMICS full text sources ADS |

On two coupled Abel-type differential equations arising in a magnetostatic problem

The question of polar instability of spherical shells due to core magnetic polarization examined briefly in a preceding communication is treated in more detail regarding Dirac's theory of magnetic charges and physical solutions of differential equations of Abel type resulting from the demand of rotational invariance. Both localized approximate solutions and several singular solutions or quasi-solutions with physical appeal are proposed, for the deformed spherical shell.

Error analysis for the stereological estimation of sphere size distribution: Abel type integral equation

Computational errors are analyzed for existing schemes and their variants of estimating the sphere size distribution from the observed size distribution of cross sections. First, possible methods are classified according to their discrete approximation schemes of the basic integral equation, and the condition numbers of the coefficient matrices are examined. Then, asymptotic forms of error are derived. It is shown that the main source of error is the singularity of the basic integral equation of Abel type and that the magnitude of the error largely depends on the treatment of the singularity. Numerical examples are also given.

THERMAL PROBLEMS WITH RADIATION BOUNDARY CONDITIONS

Contact or crack problems in thermoelasticity are usually analysed with the idealised boundary conditions of perfect conduction or perfect insulation. These boundary conditions, while simplifying the mathematics, sometimes lead to unrealistic, singular thermoelastic fields. This paper formulates axisymmetric static thermal problems for a half-space when one boundary condition corresponds to partial insulation, either inside or outside the circle r = a, z = 0. Four important cases are considered and the problems are reduced to the solution of integro-differential equations of Abel type. In each case it is shown that the equation can be solved by using two simultaneous Fourier expansions of the unknown function.

A note on yawed slender wings

An integral equation of Abel type is derived for the wake strength in steady flow over a yawed slender wing. An analytic solution is obtained for a triangular plan form, and numerical results for other configurations.

Abel type integral equations in stereology. II. Computational methods of solution and the random spheres approximation

SUMMARYAs mentioned in Part I of this paper, Part II is mainly concerned with the construction of stable computational methods for the solution of integral equations of Abel type which occur in stereology. However, in order to develop a rationale for the types of methods which we shall propose, it is first necessary to review the different types of methods which have been suggested to date in the stereological literature. This is done in section 2. From the review, we conclude that the construction of stable methods can be based on the use of spectral differentiation and product integration to evaluate appropriate inversion formulae, when they are known. Using the Random Spheres Model as exemplification, we examine the nature of such methods in section 3. The potential of the proposed methods is illustrated in section 4 by examining the reliability of the random spheres approximation—an unsolved problem in stereology proposed by Moran (1972). We conclude from this examination that the random spheres approximation is quite reliable for particles which approximate spheres.

Abel type integral equations in stereology: I. General discussion

SUMMARYIn developing numerical methods for the solution of computational problems of an involved and pragmatic nature, as occur for example in stereology, the approach should be first to classify the distinct types of numerical problems which arise. If available, appropriate results from numerical analysis can then be used to develop stable numerical processes for their solution. In fact, many of the results relating to the construction of basic numerical methods, such as spectral methods for the stable differentiation of experimental data and product integration methods for the evaluation of integrals with oscillatory and singular integrands, have been resolved. Thus, we are now in a position to apply the above approach to the more complex computational problems which arise in stereology.Many of these problems belong to one of the following two classes: (i) the solution of integral equations of Abel type; and (ii) the solution of some numerical differentiation problem.In Part I of this paper, we illustrate the generality in stereology of the Abel type integral equation and numerical differentiation formulations. As well, we derive the specific properties required in Part II which is mainly concerned with the construction of stable computational methods for (i). Included among these properties are the explicit inversion formulae which are known for general Abel type integral equations. It is these inversion formulae which we use to construct stable computational methods.Often, when estimates for linear properties of the solutions of (i) and (ii) are required, the numerical solution of (i) and (ii) can be circumvented by estimating the linear properties directly from the given observational data. In deriving such estimates, use of the properties of the Abel type integral equation and differentiation formulations plays an essential role. Because of the close connection between such estimation problems and (i) and (ii), the estimation of linear properties from truncated observational data is also examined in Part I.