Abel Operators Research Articles

Generalized Abel equations and applications to translation invariant Radon transforms

Abstract Generalized Abel equations have been employed in the recent literature to invert Radon transforms which arise in a number of important imaging applications, including Compton Scatter Tomography (CST), Ultrasound Reflection Tomography (URT), and X-ray CT. In this paper, we present novel injectivity results and inversion methods for generalized Abel operators. We apply our theory to a new Radon transform, R j \mathcal{R}_{j} , of interest in URT, which integrates a square integrable function of compact support, 𝑓, over ellipsoid and hyperboloid surfaces with centers on a plane. Using our newly established theory on generalized Abel equations, we show that R j \mathcal{R}_{j} is injective and provide an inversion method based on Neumann series. In addition, using algebraic methods, we present image phantom reconstructions from R j ⁢ f \mathcal{R}_{j}f data with added pseudo-random noise.

An efficient algorithm based on the Pseudospectral method for solving Abel's integral equation using Hermite cubic spline scaling bases

This research is dedicated to the analysis and construction of an efficient method based on the Pseudospectral method for the second type of Abel's integral equation using Hermite cubic spline scaling bases (HCSSb). After expressing the Abel operator based on the HCSSb, the equation will be reduced to a system of linear or nonlinear algebraic equations. To solve this system, we apply Newton's method and the generalized minimal residual (GMRES) method for the nonlinear and linear types, respectively. The results demonstrate that the speed of convergence for this method is O(2−J) where J∈N0. One of the advantages of the method is that it is not too complicated in form, and it can be used to solve a variety of problems. Furthermore, the method is computationally desirable and gives very accurate results.

Raising the regularity of generalized Abel equations in fractional Sobolev spaces with homogeneous boundary conditions

The generalized (or coupled) Abel equations on the bounded interval have been well investigated in H$\ddot{\text{o}}$lderian spaces that admit integrable singularities at the endpoints and relatively inadequate in other functional spaces. In recent years, such operators have appeared in BVPs of fractional-order differential equations such as fractional diffusion equations that are usually studied in the frame of fractional Sobolev spaces for weak solution and numerical approximation; and their analysis plays the key role during the process of converting weak solutions to the true solutions. This article develops the mapping properties of generalized Abel operators $\alpha {_aD_x^{-s}}+\beta {_xD_b^{-s}}$ in fractional Sobolev spaces, where $0<\alpha,\beta$, $\alpha+\beta=1$, $ 0<s<1$ and $ {_aD_x^{-s}}$, $ {_xD_b^{-s}}$ are fractional Riemann-Liouville integrals. It is mainly concerned with the regularity property of $(\alpha {_aD_x^{-s}}+\beta {_xD_b^{-s}})u=f$ by taking into account homogeneous boundary conditions. Namely, we investigate the regularity behavior of $u(x)$ while letting $f(x)$ become smoother and imposing homogeneous boundary restrictions $u(a)=u(b)=0$.

Generalization of Abel's mechanical problem: The extended isochronicity condition and the superposition principle

U(X) being an unknown potential. Second, the isochronicity condition is extended for the possible Abel-transform approach to designing the isochronous trajectories of charged particles in spectrometers and/or accelerators for time-resolving experiments. Our approach is based on the integral formula for the oscillatory motion by Landau and Lifshitz [Mechanics (Pergamon, Oxford, 1976), pp. 27–29]. The same formula is used to treat the non-periodic motion that is driven by U(X). Specifically, this unknown potential is determined by the (linear) Abel transform X(U) ∝A[T(E)], where X(U) is the inverse function of U(X), A = (1/ √ π ) E 0 dU/ √ E − U is the so-called Abel operator, and T(E) is the prescribed transit-time for a particle with energy E to spend in the region of interest. Based on this Abel-transform approach, we have introduced the extended isochronicity condition: typically, τ = TA(E) + TN(E) where τ is a constant period, TA(E) is the transit-time in the Abel type [A-type] region spanning X > 0 and TN(E) is that in the Non-Abel type [N-type] region covering X 0, the unknown inverse function XA(U) is determined from TA(E) via the Abel-transform relation XA(U) ∝A[TA(E)]. In contrast, the N-type region in X < 0 does not ensure this linear relation: the region is covered with a predetermined potential UN(X) of some arbitrary choice, not necessarily obeying the Abel-transform relation. In discussing the isochronicity problem, there has been no attempt of N-type regions that are practically of full use for the charged-particle spectrometers and/or accelerators. In this Abel-transform approach, the superposition principle simplifies the derivation of XA(U) satisfying the extended isochronicity condition. Although the extended isochronicity condition inevitably discards the low-energy particles, there is no problem for handling accelerated particles because they do not involve the small-amplitude oscillations around the potential minimum. We present analytic examples of XA(U) that are instructive. In Appendix B, Urabe’s criterion is interpreted in the time domain, using the Abel-transform approach. C 2014 AIP Publishing LLC .[ http://dx.doi.org/10.1063/1.4865996]

Approximation of functions on the Sobolev space with a Gaussian measure

We discuss the best approximation of periodic functions by trigonometric polynomials and the approximation by Fourier partial summation operators, Vallee-Poussin operators, Ces`aro operators, Abel operators, and Jackson operators, respectively, on the Sobolev space with a Gaussian measure and obtain the average error estimations. We show that, in the average case setting, the trigonometric polynomial subspaces are the asymptotically optimal subspaces in the Lq space for 1 ⩽ q < ∞, and the Fourier partial summation operators and the Vallee-Poussin operators are the asymptotically optimal linear operators and are as good as optimal nonlinear operators in the Lq space for 1 ⩽ q < ∞.

Approximation of functions on the Sobolev space on the sphere in the average case setting

In this paper, we discuss the best approximation of functions by spherical polynomials and the approximation by Fourier partial summation operators, Vallée–Poussin operators, Cesàro operators, and Abel operators, on the Sobolev space on the sphere with a Gaussian measure, and obtain the average error estimates. We also get the asymptotic values for the average Kolmogorov and linear widths of the Sobolev space on the sphere and show that, in the average case setting, the spherical polynomial subspaces are the asymptotically optimal subspaces in the L q ( 1 ≤ q < ∞ ) metric, and Fourier partial summation operators and Vallée–Poussin operators are the asymptotically optimal linear operators and are (modulo a constant) as good as optimal nonlinear operators in the L q ( 1 ≤ q < ∞ ) metric.

Toward an equation of state for solid materials with memory by use of the half-order derivative

The time derivative operator does not depend upon the difference between the current time and the past times; however, the fractional time derivative operator does. Thus, it is reasonable to expect that the fractional derivative would be useful in describing the mathematical theory of the behavior of materials with memory. An equation of state is proposed for solid materials with memory by introducing the half-order fractional calculus derivative in order to relate to the empirical expression used in the fundamental work of Tobolsky and Catsiff. This theory replaces the three empirical functions used by Tobolsky and Catsiff in reducing their experimental data for the low temperature glassy region, the transition region and the quasi-static rubbery plateau region. The square root differential operator with respect to time, D 1/2, has built in memory since the kernel of this operator depends upon the difference between the current time and the past time. D 1/2 is a special case of the Abel operator, which is used in the theory of integral equations. The present theory introduces integrals into the standard linear solid resulting in an integral differential equation governing the stress, strain and temperature. It is shown that this proposed linear equation of state for a solid material, which undergoes a second order transition, requires only four phenomenological constants to completely determine the behavior of the solid material. These four phenomenological constants are two relaxation times and two creep times, both of which are functions of the temperature.

Regularization of Integral Equations in Spaces of Distribution

In this article the notion of multiplicative regularizator, a smooth function that by multiplication allows the extension of operators in spaces of distributions, is introduced, and several of the properties are obtained. Applications to Hilbert transforms, Carleman operators, fractional integration operators and generalized Abel operators are given.

Audible-frequency stiffness of a primary suspension isolator on a high-speed tilting bogie

The preload-dependent dynamic stiffness of a primary suspension isolator on a high-speed tilting bogie is examined via measurements and modelling within an audible frequency range. The stiffness is found to depend strongly on both frequency and preload. The former displays some resonance phenomena, such as stiffness peaks and troughs, while the latter exhibits a steep low-frequency stiffness increase in addition to an anti-resonance peak shifting to a higher frequency with increased preload. The problems of simultaneously modelling the preload and frequency dependence are removed by adopting a frequency-dependent waveguide approach, assuming incompressible rubber with an Abel operator kernel as its shear relaxation function. The preload dependence is modelled by a non-linear shape factor based approach, using a globally equivalent preload configuration. All the translational stiffnesses are modelled, including the vertical, longitudinal and lateral directions, and the vertical stiffness results are compared to those of measurements in a specially designed test rig. Good agreement is obtained for a wide frequency domain-covering 100-600 Hz-using a minimum number of parameters and for a wide preload domain-from vanishing to the maximum in service, 90 kN.

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.

Effects of crack surface convection for rapid crack growth in a thermoelastic solid

As a Green’s function for rapid steady-state crack growth with crack surface convection, semi-infinite Mode I crack growth at subcritical speeds in an unbounded solid under the action of compressive line loads moving on the crack surfaces is considered. A standard convection law that relates heat flux to change in temperature is employed, and the solid obeys the fully-coupled (dynamic) equations of thermoelasticity. The use of robust asymptotic forms reduces the problem to the solution of coupled integral equations. These exhibit both Cauchy and Abel operators, but an exact solution is possible. The solution indicates that convection can give rise to temperature changes in the crack plane that are both more prominent and extensive than those that occur for an insulated crack surface. Exact expressions for the thermoelastic Rayleigh speed, which is the critical crack speed, and for speeds that arise for a particular value of an important characteristic parameter are also presented.

Effects of Mixed-Mode and Crack Surface Convection in Rapid Crack Growth in Coupled Thermoelastic Solids

Two Green’s function problems for rapid two-dimensional steady-state crack growth governed by fully coupled (dynamic) linear thermoelasticity are analyzed. In Problem A, normal and in-plane shear line loads move on the insulated surfaces of a semi-infinite crack growing at a subcritical speed. Problem B involves only normal line loads, but crack surface convection is allowed. Problem A involves, therefore, mixed traction/displacement boundary conditions, while Problem B also exhibits mixed thermal boundary conditions. Robust asymptotic forms based on exact solutions for related problems reduce Problems A and B to coupled sets of integral equations. Both sets exhibit both Cauchy and Abel operators, but are solved exactly. The solutions show that Mode II loading couples the tangential crack face separation and discontinuity in crack-face temperature changes, while crack surface convection enhances thermal response, especially at large distances. [S0021-8936(00)03101-9]

Derivation of the solution of a simple hypersingular integral equation

Three straightforward methods of solution of a simple hypersingular integral equation are presented. While two of the methods, presented here, rely on the inversion of Cauchy‐type singular integral operators, the third method uses the inversion of a simple Abel's operator with a weak singularity. All the three methods lead to the known form of the solution of the hypersingular integral equation considered.

Hard-core Scattering for N -body Systems

We prove propagation properties (maximal and minimal velocity bounds) forpseudo-resolvents associated to N-body Hamiltonians with short-rangepotentials that are infinite on a star-shaped domain centred at the origin.Motivated by the fact that the invariance principle holds for usual N-bodysystems, we define the cluster wave operators in terms of pseudo-resolventsand prove that they exist and are asymptotically complete. For any clusterdecomposition a, these operators intertwine the hard-core pseudo-selfadjointHamiltonians corresponding to the pair of pseudo-resolvents R,Ra, and equal the Abel operators constructed in terms ofHamiltonians.

AXIALLY SYMMETRIC THERMAL STRESS OF A PENNY-SHAPED CRACK UNDER GENERAL HEAT FLUX

This paper presents a solution of the steady-state thermal stress and displacement in an infinite isotropic elastic solid containing a penny-shaped crack whose surfaces are exposed to the general mechanical loading and nonsymmetric heat flux. That is, the heat flux applied to both surfaces of the crack is not the same. The problem is solved using Hankel transforms and the Abel operator of the second kind. Using limiting values of the stress, displacement, and temperature fields at the crack plane in terms of the stress, displacement, temperature, and heat flux discontinuities and boundary conditions of the crack, the problem is reduced to that of solving Abel integral equations that admit closed-form solutions. For a special case of heat flux conditions, stresses at a general point of the medium have been computed and presented graphically.

Axially symmetric thermal stress of an external circular crack under general thermal conditions

The paper presents a solution for the linear thermoelastic problem of determining axisymmetric stress and displacement fields in an isotropic elastic solid of infinite extent weakened by an external circular crack under general mechanical loadings and general thermal conditions. The mechanical loadings and thermal conditions applied on the crack faces are axisymmetric, being non-symmetric about the crack plane. In similar lines of [7], equations of equilibrium of an elastic solid conducting heat have been solved using Hankel transforms and Abel operators of the first kind. Expressions for stress, displacement, temperature and heat flux functions are obtained in terms of Abel transforms of the first kind of the jumps of stress, displacement, temperature and heat flux at the crack plane. Two types of thermal conditions, that is, general surface temperatures and general heat flux on faces of the crack are considered. In both the cases, closed form solutions have been obtained for the unknown functions solving Abel type of integral equations. Explicit expressions for stresses, displacements, temperature fields, stress intensity factors have been obtained. Two special cases of thermal conditions in which: (i) crack faces are subjected to constant non-symmetric temperatures over a circular ring area, (ii) crack faces are subjected to constant non-symmetric heat flux over a circular ring area, have been considered. In some special cases, results have been compared with those from the literature.

Stress state in multilayer cylindrical tubes made of elastically compressible linear viscoelastic material

Through the use of Abel's operators and э operators, the authors obtained an implicit solution for N-layered cylindrical tubes with alternating elastic and viscoelastic layers under conditions of elastic compressibility of the viscoelastic medium. Explicit expressions were written for the reactive pressures in 3- and 2-layered cylindrical tubes with alternating elastic and viscoelastic layers; these are analyzed in detail. Based on the solution obtained earlier for a condition where the viscoelastic operator corresponding to Poisson's coefficient is taken as a constant, it has been shown that the hypotheses discussed will lead in time to qualitatively different stress states. The overall results obtained are illustrated on an example.

States of stress in multilayer spherical vessels, cylindrical tubes, and circular disks of linear viscoelastic material

By using the properties of Abel operators and эα-operators [3], unique implicit solutions for N-layer structures with alternating elastic and viscoelastic layers in the case where the viscoelastic operator corresponding to the Poisson ratio reduces to a constant. For an N-layer spherical vessel an analogous implicit solution is also obtained on the assumption that the viscoelastic medium is elastically compressible. Explicit expressions for the reaction pressures are written down for the case of all possible two- and three-layer structures with alternating elastic and viscoelastic layers and analyzed in detail. It is shown that the hypotheses in question lead to qualitatively different states of stress. The general results obtained are illustrated by an example.

The effect of internal pressure on a penny-shaped crack at the interface of two bonded dissimilar elastic half-spaces

The problem considered is that of determining the displacement field in the vicinity of a penny-shaped crack situated at the interface of two half-spaces of different elastic materials bonded together along their common plane boundary. The deformation in the two half-spaces is a result of the application of a symmetrically-distributed pressure to the faces of the crack. The representation of the displacement field in the form of Hankel transforms leads to a set of simultaneous dual integral equations for two unknown functions. These equations are transformed by the use of Abel operators to a similar set involving both the Fourier cosine and the Fourier sine transforms of each of these two unknown functions.These equations are transformed in turn to a singular integral equation for a complex-valued function in terms of whose real and imaginary parts it is possible to completely specify the stress and displacement fields. An explicit formula is obtained for the crack energy in the case in which the applied pressure is constant; it is also indicated how simple integral expressions may he obtained for the components of stress along the interface in this case.