Repeating Use of Integral Transforms--A New Method for Evaluation of Some Infinite Integrals

A new method for evaluation of infinite integrals is proposed.The integrals are derived by applying the same or differentintegral transforms twice. The integral transforms of Laplace,Fourier, Mellin, Hankel, K, Y-Bessel, H-Struve, Stieltjes, generalizedStieltjes, Kantrovich and Lebedev were used. Using the proposedmethod, a number of new infinite integrals of elementary andspecial functions were derived.     

Truncations of multilinear Hankel operators

We extend to multilinear Hankel operators the fact that some truncations of bounded Hankel operators are still bounded. We prove and use a continuity property of bilinear Hilbert transforms on products of Lipschitz spaces and Hardy spaces.

     

Axial Couette flow of a generalized Oldroyd-B fluid due to a longitudinal time-dependent shear stress

Abstract

The velocity field and the adequate shear stress corresponding to the unsteady flow of a generalized Oldroyd-B fluid in an infinite circular cylinder are determined by means of Hankel and Laplace transforms. The solutions that have been obtained, written in terms of the generalized G-functions, satisfy all imposed initial and boundary conditions. The similar solutions for generalized Maxwell fluids as well as those for ordinary fluids are obtained as limiting cases of our general solutions.     

Exact solutions for the unsteady axial flow of Non-Newtonian fluids through a circular cylinder

The velocity field and the shear stress corresponding to the motion of a generalized Oldroyd-B fluid due to an infinite circular cylinder subject to a longitudinal time-dependent shear stress are established by means of the Laplace and finite Hankel transforms. The exact solutions, written under series form, can be easily specialized to give the similar solutions for generalized Maxwell and generalized second grade fluids as well as for ordinary Oldroyd-B, Maxwell, second grade and Newtonian fluids performing the same motion. Finally, some characteristics of the motion as well as the influence of the material parameters on the behavior of the fluid are shown by graphical illustrations.     

Unsteady helical flows of a generalized Oldroyd-B fluid with fractional derivative

This paper deals with the unsteady helical flows of a generalized Oldroyd-B fluid between two infinite coaxial cylinders and within an infinite cylinder. The fractional calculus approach is used in the constitutive relationship of fluid model. Exact analytical solutions are obtained with the help of integral transforms (Laplace transform, Weber transform and finite Hankel transform). The corresponding solutions for generalized second grade and Maxwell fluids as well as those for the Newtonian and ordinary Oldroyd-B fluids are also given in limiting cases. Finally, the influence of model parameters on the velocity field is also analyzed by graphical illustrations.     

Identities for the Hankel transform and their applications

In the present paper the authors show that iterations of the Hankel transform with Kν-transform is a constant multiple of the Widder transform. Using these iteration identities, several Parseval-Goldstein type theorems for these transforms are given. By making use of these results a number of new Goldstein type exchange identities are obtained for these and the Laplace transform. The identities proven in this paper are shown to give rise to useful corollaries for evaluating infinite integrals of special functions. Some examples are also given as illustration of the results presented here.     

Exact analytic solutions for the unsteady flow of a non-Newtonian fluid between two cylinders with fractional derivative model

The velocity field and the associated shear stress corresponding to the torsional oscillatory flow of a generalized Maxwell fluid, between two infinite coaxial circular cylinders, are determined by means of the Laplace and Hankel transforms. Initially, the fluid and cylinders are at rest and after some time both cylinders suddenly begin to oscillate around their common axis with different angular frequencies of their velocities. The solutions that have been obtained are presented under integral and series forms in terms of generalized G and R functions. Moreover, these solutions satisfy the governing differential equation and all imposed initial and boundary conditions. The respective solutions for the motion between the cylinders, when one of them is at rest, can be obtained from our general solutions. Furthermore, the corresponding solutions for the similar flow of ordinary Maxwell fluid are also obtained as limiting cases of our general solutions. At the end, flows corresponding to the ordinary Maxwell and generalized Maxwell fluids are shown and compared graphically by plotting velocity profiles at different values of time and some important results are remarked.     

Ratio Mercerian Theorems with Applications to Hankel and Fourier Transforms

We complete and complement our recent work on Drasin-Shea-Jordantheorems for Fourier and Hankel transforms. In improving onthe methods of our previous work, we were led to certain ratioMercerian theorems for general kernels; these yield definitiveversions of our earlier results for Fourier and Hankel transforms.1991 Mathematics Subject Classification: primary 44A15; secondary47B38.     

Unsteady helical flows of Oldroyd-B fluids

The unsteady helical flow of an Oldroyd-B fluid, in an infinite circular cylinder, is studied by using finite Hankel transforms. The motion is produced by the cylinder that, at time t = 0⁺, is subject to torsional and longitudinal time-dependent shear stresses. The solutions that have been obtained, presented under series form, satisfy all imposed initial and boundary conditions. The corresponding solutions for Maxwell, second grade and Newtonian fluids are obtained as limiting cases of general solutions. Finally, the influence of the pertinent parameters on the fluid motion is underlined by graphical illustrations.     

Riordan arrays and the LDU decomposition of symmetric Toeplitz plus Hankel matrices

We examine a result of Basor and Ehrhardt concerning Hankel and Toeplitz plus Hankel matrices, within the context of the Riordan group of lower-triangular matrices. This allows us to determine the LDU decomposition of certain symmetric Toeplitz plus Hankel matrices. We also determine the generating functions and Hankel transforms of associated sequences.     

Dual integral equations related to the kontorovich-lebedev transform : PMM vol. 38, n 6, 1974, pp. 1090–1097

We study the dual integral equations related to the Kontorovich-Lebedev integral transforms arising in the course of solution of the problems of mathematical physics, in particular of the mixed boundary value problems for the wedge-shaped regions. We show that the solutions of these equations can be expressed in quadratures, using the auxilliary functions satisfying the integral Fredholm equation of second kind with a symmetric kernel.At present, the dual equations investigated in most detail are those connected with the Fourier and Hankel integral transforms. The results obtained and their applications are given in [1–3]. A large number of papers also deal with the theory and applications of the dual integral equations connected with the Mehler-Fock integral transform and its generalizations [4–11]., The dual integral transforms considered in the present paper belong to a more complex class than those listed above, and so far, no effective solution has been obtained for them. The only relevant results known to the authors are those in [12, 13]. In [12] a method of solving the equations (1.2) is given for a single particular value of the parameter γ = π/2, while in [13] the dual equations of the type under consideration are reduced to a solution of an infinite system of linear algebraic equations.     

Helical flow of an Oldroyd-B fluid due to a circular cylinder subject to time-dependent shear stresses

Exact solutions corresponding to the unsteady helical flow of an Oldroyd-B fluid due to an infinite circular cylinder subject to torsional and longitudinal time-dependent shear stresses are established using Hankel transforms. These solutions, presented under series form in terms of Bessel functions J ₀(·), J ₁(·) and J ₂(·), can be easily specialized to give the similar solutions for Maxwell, Second grade and Newtonian fluids performing the same motion. Some characteristics of the motion, as well as the influence of pertinent parameters on the velocity profiles, are underlined by graphical illustrations.     

Exact solutions for the flow of a viscoelastic fluid induced by a circular cylinder subject to a time dependent shear stress

The velocity field and the adequate shear stress corresponding to the flow of a Maxwell fluid with fractional derivative model, between two infinite coaxial cylinders, are determined by means of the Laplace and finite Hankel transforms. The motion is due to the inner cylinder that applies a longitudinal time dependent shear to the fluid. The solutions that have been obtained, presented under integral and series form in terms of the generalized G and R functions, satisfy all imposed initial and boundary conditions. They can be easy particularizes to give the similar solutions for ordinary Maxwell and Newtonian fluids. Finally, the influence of the relaxation time and the fractional parameter, as well as a comparison between models, is shown by graphical illustrations.     

Über die Wirbelstromdämpfung bei ferromagnetischen Schichten

Summary A ferromagnetic disc is placed between two conducting plates of infinite extent, so that the axis of the disc is perpendicular to the plates. The magnetization in the disc is homogeneous and varies with timet>0 in a known but arbitrary fashion. The eddy current distribution in the plates and the magnetic field which they generate in the disc are represented with the help of a Laplace transform with respect to time and a Hankel transform with respect to the distance from the center of the disc. In the special case where the distances perpendicular to the conducting plates are small compared to the radius of the disc, these two transforms can be inverted. The eddy currents and the generated magnetic fields are then easily visualized.     

Helical flow of an Oldroyd-B fluid due to a circular cylinder subject to time-dependent shear stresses

Exact solutions corresponding to the unsteady helical flow of an Oldroyd-B fluid due to an infinite circular cylinder subject to torsional and longitudinal time-dependent shear stresses are established using Hankel transforms. These solutions, presented under series form in terms of Bessel functions J ₀(·), J ₁(·) and J ₂(·), can be easily specialized to give the similar solutions for Maxwell, Second grade and Newtonian fluids performing the same motion. Some characteristics of the motion, as well as the influence of pertinent parameters on the velocity profiles, are underlined by graphical illustrations.     

Distributional Chébli-Trimèche transforms

In this paper we investigate the distributional Chébli-Trimèche transforms. We use the so-called kernel method and we are inspired by the papers of Dube and Pandey [L.S. Dube, J.N. Pandey, On the Hankel transform of distributions, Tôhoku Math. J. 27 (1975) 337-354] and Koh and Zemanian [E.L. Koh, A.H. Zemanian, The complex Hankel and I-transformations of generalized functions, SIAM J. Appl. Math. 16 (1968) 945-957] about distributional Hankel transforms. We note that our procedure, supported in a representation of the elements in the corresponding dual spaces, is simpler than the methods described in the above mentioned papers. Some applications of our distributional theory are presented.     

A criterion for a two-sided integral transform to be unitary

A criterion for a two-sided Watson transform to be unitary in the space L₂(R) is considered. It enables us to construct new examples of integral transforms with symmetric inversion formulas (only the Fourier and the Hartley transforms are known). We give some new examples of the indicated transforms, in particular, the symmetric Hankel transform with the sum of two Bessel functions in the kernal and the Hardy transform with the sum of a Neumann function and a Struve function in the kernel, and the Narain transform with a sum of two G-functions in the kernelDoctor of Physicomathematical sciences.Candidate of Physicomathematical sciences.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 5, pp. 697–699, May, 1992.     

In honor of Professor Dr. Wolfgang Haack on the occasion of his seventieth birthday on April 24, 1972

A discrete transform with a Bessel function kernel is defined, as a finite sum, over the zeros of the Bessel function. The approximate inverse of this transform is derived as another finite sum. This development is in parallel to that of the discrete Fourier transform (DFT) which lead to the fast Fourier transform (FFT) algorithm. The discrete Hankel transform with kernel Jo, the Bessel function of the first kind of order zero, will be used as an illustration for deriving the discrete Hankel transform, its inverse and a number of its basic properties. This includes the convolution product which is necessary for solving boundary problems. Other applications include evaluating Hankel transforms, Bessel series and replacing higher dimension Fourier transforms, with circular symmetry, by a single Hankel transform     

Fast integral equation solver for Maxwell’s equations in layered media with FMM for Bessel functions

The paper presents a new fast integral equation solver for Maxwell’s equations in 3-D layered media. First, the spectral domain dyadic Green’s function is derived, and the 0-th and the 1-st order Hankel transforms or Sommerfeld-type integrals are used to recover all components of the dyadic Green’s function in real space. The Hankel transforms are performed with the adaptive generalized Gaussian quadrature points and window functions to minimize the computational cost. Subsequently, a fast integral equation solver with O(N _z ² N _x N _y log(N _x N _y )) in layered media is developed by rewriting the layered media integral operator in terms of Hankel transforms and using the new fast multipole method for the n-th order Bessel function in 2-D. Computational cost and parallel efficiency of the new algorithm are presented.     

On a Certain Class of Integral Equations Associated with Hankel Transforms

This paper deals with a new class of Fredholm integral equations of the first kind associated with Hankel transforms of integer order. Analysis of the equations is based on operators transforming Bessel functions of the first kind into kernels of Weber–Orr integral transforms. Their inverse operators are established by means of new inversion theorems for the Hankel and Weber–Orr integral transforms of functions belonging to L ¹ and L ². These operators together with the proven Paley–Wiener’s theorem for the Weber–Orr transform enable to regularize the equations and, in special cases, to derive explicit solutions. The integral equations analyzed in this paper can be employed instead of dual integral equations usually treated with the Cooke–Lebedev method. An example manifests that it may be preferable because of the possibility to control norms of operators in the regularized equations.