Mellin Convolution Research Articles

An optimal order collocation method for first kind boundary integral equations on polygons

This paper discusses the convergence of the collocation method using splines of any order $k$ for first kind integral equations with logarithmic kernels on closed polygonal boundaries in ${\Bbb R}^2$ . Before discretization the equation is transformed to an equivalent equation over $[-\pi,\pi]$ using a nonlinear parametrization of the polygon which varies more slowly than arc--length near each corner. This has the effect of producing a transformed equation with a solution which is smooth on $[-\pi,\pi]$ . This latter integral equation is shown to be well--posed in appropriate Sobolev spaces. The structure of the integral operator is described in detail, and can be written in terms of certain non--standard Mellin convolution operators. Using this information we are able to show that the collocation method using splines of order $k$ (degree $k-1$ ) converges with optimal order $O (h^k)$ . (The collocation points are the mid--points of subintervals when $k$ is odd and the break--points when $k$ is even, and stability is shown under the assumption that the method may be modified slightly.) Using the numerical solutions to the transformed equation we construct numerical solutions of the original equation which converge optimally in a certain weighted norm. Finally the method is shown to produce superconvergent approximations to interior potentials such as those used to solve harmonic boundary value problems by the boundary integral method. The convergence results are illustrated with some numerical examples.

On the boundary transmission problem of thermoelastostatics in a plane domain with interface corners

AbstractThis paper deals with bimetal problems of thermoelastostatics. By means of an explicit particular solution a reduction to problems of elastostatics is given. An indirect boundary integral method is applied for solving the traction boundary value problem. The solution is represented by a potential of single layer type having Green's contact tensor as the kernel. Thus, from the first the transmission conditions are satisfied. The Fredholm property of the boundary integral operator as well as the asymptotics of the potential density at an interface corner depend on the symbol of a Mellin convolution operator. The singular functions at corners can be obtained by calculating the potential for terms in the asymptotic expansion of the density.

Convolution integral equations with Gegenbauer function kernel

In this paper we develop a concise and transparent approach for solving Mellin convolution equations where the convolutor is the product of an algebraic function and a Gegenbauer function. Our method is primarily based on 1. the use of fractional integral/differential operators; 2. a formula for Gegenbauer functions which is a fractional extension of the Rodrigues formula for Gegenbauer polynomials (see Theorem 3); 3. an intertwining relation concerning fractional integral/differential operators (see Theorem 1), which in the integer case reads ( d/ dx) 2 n+1 = ( x −1 d/ dx) n x 2 n+1 ( x −1 d/ dx) n+1 . Thus we cover most of the known results on this type of integral equations and obtain considerable extensions. As a special illustration we present the Gegenbauer transform pair associated to the Radon transformation.

Complete SAXS data analysis and synthesis of lamellar two-phase systems. Deduction of a simple model for the layer statistics

A structural model for the analysis of SAXS data from lamellar two-phase systems is proposed and applied on data sets from three injection moulded poly(ethylene terephthalate) (PET) samples. The concept of data analysis is based on Ruland's interface distribution function (IDF). The suggested model is defined by few parameters of physical meaning. It unifies the well known concepts of an ensemble of non-uniform stacks, finite stack height and one-dimensional paracrystalline disorder in an analytical expression. In order to deduce this expression, the notion of an inhomogeneous structure within the sample is mathematically treated in terms of compansion, a general superposition principle. Its mathematical equivalent in one dimension is the Mellin convolution. The theory of the Mellin convolution may be used to find analytical functions even for the convoluted. An exampled is given, which in future work may be used to describe this thickness distributions of amorphous and crystalline layers. In the application part of this study gaussians are used to describe the thickness distributions in each local stack. The introduction of compansion adds one extra parameter, which describes the heterogeneity of the sample. Compansion makes the global thickness distributions become more asymmetrical

The convolution method for the development of new leibniz rules involving fractional derivatives and of their integral analogues

Leibniz rules for the operator of fractional derivatives and their integral analogues were considered, among other workers on the subject of fractional calculus, by T.J. Osler and Y. Watanabe. This operator of fractional derivatives is known to belong to a class of integral transforms associated with the Mellin convolution. Since Mejer' G-function transformation happens to be one of the most general integral transforms of this class, it would seem natural to obtain some new Leibniz rules and their integral analogues for operators involving the G-transforms. The object of the present paper is to summarize the development and applications of a general method for constructing such Leibniz rules and their integral analogues based upon the notion of the G-convolution and upon various representations of the kernels fo G-convolutions.

The $h$-$p$-Version of Spline Approximation Methods for Mellin Convolution Equations

The $h$-$p$-Version of Spline Approximation Methods for Mellin Convolution Equations

Fractional integral operators involving a general class of polynomials

In the present paper the authors derive a number of interesting expressions for the composition of certain multidimensional fractional integral operators involving a general class of polynomials with essentially arbitrary coefficients. It is shown how these fractional integral operators can be identified with elements of the algebra of functions having the multidimensional Mellin convolution as the product. Inversion formulas for the multidimensional fractional integrals are also established. The fractional integral operators studied here are fairly general in character, since (by suitably specializing the coefficients involved) the general class of polynomials can be reduced to each of the classical orthogonal polynomials, the Bessel polynomials, and numerous other classes of generalized hypergeometric polynomials studied in the literature.

Distribution of the quotient of noncentral normal random variables

The Mellin convolution is used to derive in analytical form an exact 3-parameterprobabilitydensity function of the quotient of two noncentral normal random variables. In contrast with the 5-parameter probability density function previously derivedby Fieller (1932) and Hinkley (1969), this 3-parameter probability density function is feasible for computer evaluation of the mean and cumulative distribution function, which are needed, for example, when dealing with estimation and distribution problems in regression analysis and sampling theory. When the normal variables are independent, the probability density function reduces to a 2-parameter function, for which a computer program is operational. An illustrative example is given for one set of parameters when the normal variables are independent, in which themean and functional form of the probability density function are presented, together with a brief tabulation of the probability density function.

On spline collocation for convolution equations

We consider the numerical solution of Wiener-Hopf integral equations and Mellin convolution equations by collocation methods and their iterated and discrete variants, using piecewise polynomials as basis functions. In the present paper we obtain results on stability and optimal convergence in the Lp norm generalizing those of [4]–[6] and [9].

On a method of solving integral equations and its application to the problem of the bending of a plate with a cruciform inclusion

A scheme for a method of solving a certain integral equation is proposed consisting of the construction of a system of solutions of the characteristic equation with a complete system of special right-hand sides (polynomials, say) and writing the approximate solution of the initial equation in the form of a linear combination of the functions constructed. The scheme is realized in the case when the characteristic equation is reduced to a Mellin convolution and is solved exactly by the factorization method using the Mellin transform. In particular, such an approach provides an efficient solution of an integral equation with a fixed singularity in the kernel. A system of two integral equations obtained in the problem of the bending of a rectangular plate with a cruciform inclusion is solved as an illustration. In the case of branches of the inclusion of identical length the system is solved by the ordinary factorization method, in the case of branches of the inclusion of different length the factorization method is modified substantially.

On algebras generated by convolutions and discontinuous functions

Banach algebras generated by Fourier and Mellin convolution operators with discontinuous presymbols and by discontinuous functions in Lp (IR+, xα) spaces with weight are investigated. The Fredholm properties are characterized by a symbol calculus and an index formula for such operators is presented. These results were obtained by H. O. Cordes in [3] for the case p=2, α=0 and presymbols, which are discontinuous only at infinity and generalized in [20] for 1<p<∞.

Generalized Mellin Convolutions and their Asymptotic Expansions

A large number of important integral transforms, such as Laplace, Fourier sine and cosine, Hankel, Stieltjes, and Riemann- Liouville fractional integral transforms, can be put in the form1.1where f(t) and the kernel, h(t), are locally integrable functions on (0,∞), and x is a positive parameter. Recently, two important techniques have been developed to give asymptotic expansions of I(x) as x → + ∞ or x → 0+. One method relies heavily on the theory of Mellin transforms [8] and the other is based on the use of distributions [24]. Here, of course, the integral I(x) is assumed to exist in some ordinary sense.If the above integral does not exist in any ordinary sense, then it may be regarded as an integral transform of a distribution (generalized function). There are mainly two approaches to extend the classical integral transforms to distributions.

Calculation ofO(1) interferences of double-scattering waves in 3-3 collision

Recent techniques on the asymptotic expansion of Mellin convolutions allow the evaluation of the interference of waves with discontinuous asymptotic behaviour. As an application, I consider the flux of representative points in theR6 co-ordinate space for 3-3 collision and address myself toO(1) self-interferences of the anomalous double scattering. It is shown how the contribution from the classically forbidden region can be summed up to the ridge with a quite simple (and mandatory) regularization procedure. In addition, the matching across the ridge, where the wave has no uniform asymptotic behaviour, contributes noO(1) flux.

An inverse for the Gohberg-Krupnik symbol map

SynopsisIt is shown that the elements of the closed operator algebra generated by one-dimensional singular integral operators with piecewise continuous coefficients with a fixed finite set of points of discontinuity can be written as the sum of a singular integral operator, a compact operator, and generalized Mellin convolutions. Their Gohberg-Krupnik symbol is given in terms of the Mellin transform. This gives an explicit construction of an operator with prescribed Gohberg—Krupnik symbol.

Performances of the method of linear systems identification by Mellin deconvolution

The authors proposed in a previous paper (Prost and Goutte 1976) a new method for linear systems identification. This identification method requires the resolving of a Mellin convolution equation. In this second paper the authors describe the algorithm used to resolve this equation and study its accuracy as well as the effect of noise. They give examples of identification of simulated systems.

Linear systems identification by Mellin deconvolution

This proposed method of identification is particularly well, adapted to the study of linear systems whose transfer function has real simple poles. It also permits the identification of systems with real multiple poles as well as complex conjugate poles. No prior knowledge of the number of poles and of zeros is necessary. The identification is accomplished by means of a digital process of the impulse response of the system. This process corresponds to the resolution of a Mellin convolution equation.

Simultaneous saturation in connection with dirichlet's problem for a wedge

This paper is concerned with an extension of the classical concept of staturation to systems of coupled approximations. Let {T j , ϑ}, j=1,2 be two families of bounded linear operators of the product space Y×Y into (the Banach space) y which constitute a strong simultaneous approximation process on Y×Y in the sense that for each and j=1,2 where . The simultaneous process {T j , ϑ} is then said to be saturated on Y×Y if there exist φ j (ϑ) with such that every for which is an invariant elements under {T j , ϑ} and if the set , the so-called Favard class, contains at least one noninvariant element. On the basis of this concept the boundary behavior of the solution of Dirichlet's problem for a wedge is studied in detail as an application of general results on simultaneous approximation processes of Mellin convolution;matrix-methods are employed.

The application of the Mellin convolution to the solution of integral equations of the first kind with a product-type kernel

The application of the Mellin convolution to the solution of integral equations of the first kind with a product-type kernel