A hybrid fast Hankel transform algorithm for electromagnetic modeling

Geophysics ◽  
1989 ◽  
Vol 54 (2) ◽  
pp. 263-266 ◽  
Author(s):  
Walter L. Anderson
Keyword(s):  
Continued Fraction ◽  
Good Accuracy ◽  
Hybrid Approach ◽  
Hankel Transform ◽  
Kernel Functions ◽  
Electromagnetic Modeling ◽  
Electromagnetic Problems ◽  
Special Cases ◽  
Standard Fortran ◽  
Fraction Algorithm

A hybrid fast Hankel transform algorithm has been developed that uses several complementary features of two existing algorithms: Anderson’s digital filtering or fast Hankel transform (FHT) algorithm and Chave’s quadrature and continued fraction algorithm. A hybrid FHT subprogram (called HYBFHT) written in standard Fortran-77 provides a simple user interface to call either subalgorithm. The hybrid approach is an attempt to combine the best features of the two subalgorithms in order to minimize the user’s coding requirements and to provide fast execution and good accuracy for a large class of electromagnetic problems involving various related Hankel transform sets with multiple arguments. Special cases of Hankel transforms of double‐order and double‐argument are discussed, where use of HYBFHT is shown to be advantageous for oscillatory kernel functions.

Computation of Green’s tensor integrals for three‐dimensional electromagnetic problems using fast Hankel transforms

Geophysics ◽  
1984 ◽  
Vol 49 (10) ◽  
pp. 1754-1759 ◽  
Author(s):  
Walter L. Anderson
Keyword(s):  
Half Space ◽  
Three Dimensional ◽  
Hankel Transform ◽  
Electromagnetic Modeling ◽  
Electromagnetic Problems ◽  
Hankel Transforms ◽  
Homogeneous Half Space ◽  
Green's Tensor ◽  
Green’S Tensor ◽  
The Many

A new method is presented that rapidly evaluates the many Green’s tensor integrals encountered in three‐dimensional electromagnetic modeling using an integral equation. Application of a fast Hankel transform (FHT) algorithm (Anderson, 1982) is the basis for the new solution, where efficient and accurate computation of Hankel transforms are obtained by related and lagged convolutions (linear digital filtering). The FHT algorithm is briefly reviewed and compared to earlier convolution algorithms written by the author. The homogeneous and layered half‐space cases for the Green’s tensor integrals are presented in a form so that the FHT can be easily applied in practice. Computer timing runs comparing the FHT to conventional direct convolution methods are discussed, where the FHT’s performance was about 6 times faster for a homogeneous half‐space, and about 108 times faster for a five‐layer half‐space. Subsequent interpolation after the FHT is called is required to compute specific values of the tensor integrals at selected transform arguments; however, due to the relatively small lagged convolution interval used (same as the digital filter’s), a simple and fast interpolation is sufficient (e.g., by cubic splines).

Nonlinear integral equations for electromagnetic inverse problems

Geophysics ◽  
1987 ◽  
Vol 52 (9) ◽  
pp. 1297-1302 ◽  
Author(s):  
E. Gómez‐Treviño
Keyword(s):  
Electrical Conductivity ◽  
Inverse Problem ◽  
Integral Equations ◽  
Kernel Functions ◽  
Nonlinear Dependence ◽  
Field Equations ◽  
Nonlinear Integral Equations ◽  
Scaling Properties ◽  
Electromagnetic Problems ◽  
Special Cases

The scaling properties of Maxwell’s equations allow the existence of simple yet general nonlinear integral equations for electrical conductivity. These equations were developed in an attempt to reduce the generality of linearization to the exclusive scope of electromagnetic problems. The reduction is achieved when the principle of similitude for quasi‐static fields is imposed on linearized forms of the field equations. The combination leads to exact integral relations which represent a unifying framework for the general electromagnetic inverse problem. The equations are of the same form in both time and frequency domains and hold for all observations that scale as electric and magnetic fields do; direct current resistivity and magnetometric resistivity methods are considered as special cases. The kernel functions of the integral equations are closely related, through a normalization factor, to the Frechét kernels of the conventional equations obtained by linearization. Accordingly, the sensitivity functions play the role of weighting functions for electrical conductivity despite the nonlinear dependence of the model and the data. In terms of the integral equations, the inverse problem consists of extracting information about a distribution of conductivity from a given set of its spatial averages. The form of the new equations leads to the consideration of their numerical solution through an approximate knowledge of their kernel functions. The integral equation for magnetotelluric soundings illustrates this approach in a simple fashion.

scholarly journals Continued Fractions Related to $(t,q)$-Tangents and Variants

10.37236/2014 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Helmut Prodinger
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
General Version ◽  
Continued Fraction Expansion ◽  
Special Cases ◽  
Tangent Function

For the $q$-tangent function introduced by Foata and Han (this volume) we provide the continued fraction expansion, by creative guessing and a routine verification. Then an even more recent $q$-tangent function due to Cieslinski is also expanded. Lastly, a general version is considered that contains both versions as special cases.

The second-order deformation of a finite compressible isotropic elastic annulus subjected to circular shearing

1993 ◽  
Vol 442 (1916) ◽  
pp. 621-639 ◽  
Keyword(s):  
Hankel Transform ◽  
Outer Edge ◽  
Kernel Functions ◽  
Second Order ◽  
Cylindrical Hole ◽  
Angular Deformation ◽  
Interior Boundary ◽  
Order Deformation ◽  
Finite Hankel Transform

This work investigates the second-order deformation of a uniformly thick compressible isotropic elastic annulus with an axial cylindrical hole. The annulus is clamped at its outer edge and is subjected to a constant angular deformation on the interior boundary of the hole. The implicit m athematical solution is formulated in term s of finite Hankel transform s with Weber-Orr kernel functions which are then numerically inverted.

scholarly journals A continued fraction algorithm for real algebraic numbers

1972 ◽  
Vol 26 (119) ◽  
pp. 785-785 ◽  
Author(s):  
David G. Cantor ◽  
Paul H. Galyean ◽  
Horst G. Zimmer
Keyword(s):  
Continued Fraction ◽  
Algebraic Numbers ◽  
Fraction Algorithm

Numerical integration of related Hankel transforms of orders 0 and 1 by adaptive digital filtering

Geophysics ◽  
1979 ◽  
Vol 44 (7) ◽  
pp. 1287-1305 ◽  
Author(s):  
Walter L. Anderson
Keyword(s):  
Large Range ◽  
Digital Filtering ◽  
Hankel Transform ◽  
Kernel Functions ◽  
General Nature ◽  
Earth Model ◽  
Hankel Transforms ◽  
Quadrature Methods ◽  
Test Driver ◽  
Different Order

A linear digital filtering algorithm is presented for rapid and accurate numerical evaluation of Hankel transform integrals of orders 0 and 1 containing related complex kernel functions. The kernel for Hankel transforms is defined as the non‐Bessel function factor of the integrand. Related transforms are defined as transforms, of either order 0 or 1, whose kernel functions are related to one another by simple algebraic relationships. Previously saved kernel evaluations are used in the algorithm to obtain rapidly either order transform following an initial convolution operation. Each order filter is designed with identical abscissas over a large range so that an adaptive convolution procedure can be applied to a large class of kernels. Different order Hankel transforms with related kernels are often found in electromagnetic (EM) applications. Because of the general nature of this algorithm, the need to design new filters should not be necessary for most applications. Accuracy of the filters is comparable to that of single‐precision numerical quadrature methods, provided well‐behaved kernels and moderate values of the transform argument are used. Filtering errors of less than 0.005 percent are demonstrated numerically using known analytical Hankel transform pairs. The digital filter accuracy is also illustrated by comparison with other published filters for computing the apparent resistivity for a Schlumberger array over a horizontally layered earth model. The algorithm is written in Fortran IV and is listed in the Appendix along with a test driver program. Detailed comments are included to define sufficiently all calling parameter requirements.

scholarly journals Axisymmetric Vibration for Micropolar Porous Thermoelastic Circular Plate

2017 ◽  
Vol 22 (3) ◽  
pp. 583-600 ◽  
Author(s):  
R. Kumar ◽  
P. Kaushal ◽  
R. Sharma
Keyword(s):  
Circular Plate ◽  
Axisymmetric Problem ◽  
Volume Fraction ◽  
Hankel Transform ◽  
Specific Model ◽  
Axisymmetric Vibration ◽  
Physical Domain ◽  
Special Cases ◽  
Eigen Value ◽  
Field Temperature

AbstractThe present investigation is concerned with a two dimensional axisymmetric problem in a homogeneous isotropic micropolar porous thermoelastic circular plate by using the eigen value approach. The Laplace and Hankel transform are used to solve the problem. The expression of displacements, microrotation, volume fraction field, temperature distribution and stresses are obtained in the transformed domain subjected to thermomechanical sources. A computer algorithm is developed for numerical computations. To obtain the resulting quantities in a physical domain, a numerical inversion technique is used. The resulting quantities are depicted graphically for a specific model. Some special cases are also deduced.

scholarly journals Laguerre semigroup and Dunkl operators

2012 ◽  
Vol 148 (4) ◽  
pp. 1265-1336 ◽  
Author(s):  
Salem Ben Saïd ◽  
Toshiyuki Kobayashi ◽  
Bent Ørsted
Keyword(s):  
Unitary Representation ◽  
Fourier Transforms ◽  
Universal Covering ◽  
Hankel Transform ◽  
Kernel Functions ◽  
Difference Operators ◽  
Dunkl Transform ◽  
Dunkl Operators ◽  
Hermite Semigroup ◽  
Heisenberg Uncertainty

AbstractWe construct a two-parameter family of actionsωk,aof the Lie algebra 𝔰𝔩(2,ℝ) by differential–difference operators on ℝN∖{0}. Herekis a multiplicity function for the Dunkl operators, anda>0 arises from the interpolation of the two 𝔰𝔩(2,ℝ) actions on the Weil representation ofMp(N,ℝ) and the minimal unitary representation of O(N+1,2). We prove that this actionωk,alifts to a unitary representation of the universal covering ofSL(2,ℝ) , and can even be extended to a holomorphic semigroup Ωk,a. In thek≡0 case, our semigroup generalizes the Hermite semigroup studied by R. Howe (a=2) and the Laguerre semigroup studied by the second author with G. Mano (a=1) . One boundary value of our semigroup Ωk,aprovides us with (k,a) -generalized Fourier transforms ℱk,a, which include the Dunkl transform 𝒟k(a=2) and a new unitary operator ℋk (a=1) , namely a Dunkl–Hankel transform. We establish the inversion formula, a generalization of the Plancherel theorem, the Hecke identity, the Bochner identity, and a Heisenberg uncertainty relation for ℱk,a. We also find kernel functions for Ωk,aand ℱk,afora=1,2 in terms of Bessel functions and the Dunkl intertwining operator.

scholarly journals Non-extreme-Points Approach to Extreme Points of Integral Families of Analytic Functions

10.53733/87 ◽  
2021 ◽  
Vol 51 ◽  
pp. 39-48
Author(s):  
Keiko Dow
Keyword(s):  
Unit Circle ◽  
Extreme Point ◽  
Analytic Functions ◽  
Compact Convex ◽  
Extreme Points ◽  
Starlike Functions ◽  
Extremal Problems ◽  
Kernel Functions ◽  
Closed Convex Hull ◽  
Special Cases

Non extreme points of compact, convex integral families of analytic functions are investigated. Knowledge about extreme points provides a valuable tool in the optimization of linear extremal problems. The functions studied are determined by a 2-parameter collection of kernel functions integrated against measures on the torus. Families from classical geometric function theory such as the closed convex hull of the derivatives of normalized close-to-convex functions, the ratio of starlike functions of different orders, as well as many others are included. However for these families of analytic functions, identifying “all” the extreme points remains a difficult challenge except in some special cases. Aharonov and Friedland [1] identified a band of points on the unit circle which corresponds to the set of extreme points for these 2-parameter collections of kernel functions. Later this band of extreme points was further extended by introducing a new technique by Dow and Wilken [3]. On the other hand, a technique to identify a non extreme point was not investigated much in the past probably because identifying non extreme points does not directly help solving the optimization of linear extremal problems. So far only one point on the unit circle has beenidentified which corresponds to a non extreme point for a 2-parameter collections of kernel functions. This leaves a big gap between the band of extreme points and one non extreme point. The author believes it is worth developing some techniques, and identifying non extreme points will shed a new light in the exact determination of the extreme points. The ultimate goal is to identify the point on the unit circle that separates the band of extreme points from non extreme points. The main result introduces a new class of non extreme points.

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