Double Exponential Formula Research Articles

Accurate numerical calculation of shielding current density in HTS thin film: application to noncontact measurement method of critical current density

The numerical method has been developed for calculating the shielding current density in a high temperature superconducting thin film. After formulated by use of the current-vector-potential method, the behavior of the shielding current density is expressed as an integral–differential equation. When the initial-boundary-value problem of the equation is discretized with respect to time and space, improper integrals appear as coefficients of the nonlinear system. In order to evaluate the coefficients accurately, the double exponential formula is applied to the improper integrals. Consequently, the shielding current density can be determined accurately. As an application of the method, two types of the noncontact methods for measuring the critical current density jC have been simulated numerically. The results of computations show that the maximum repulsive force is roughly proportional to jC in the permanent magnet method. In addition, it is found that an error of the inductive method is not more than 3% if jC exceeds a certain lower limit.

Numerical Simulation of Shielding Current Density in High-Temperature Superconducting Thin Film

The numerical code has been developed for simulating the time evolution of the shielding current density in the high-temperature superconducting (HTS) thin film. When the governing equation of the shielding current density is discretized by the use of the finite element method (FEM) and the completely implicit method, improper integrals appear as coefficients of the nonlinear system. In order to evaluate the coefficients accurately, the double exponential formula is applied to the integrals. As a result, the shielding current density can be calculated with a high accuracy. By using the code, the noncontact measurement method of the critical current density has been investigated numerically

On the evaluation of H and X functions

We indicate a Double Exponential Formula based numerical integration method for the evaluation of the Ambarzumian–Chandrasekhar H function and the X function of neutron transport for the single speed and isotropic case. This method is significantly more economical than our earlier scheme, which was based on IMT quadrature. For c < 5 , the present method converges faster than our earlier IMT scheme. This will be adequate for all radiative transport and transport theory applications. These findings are supported by appropriate error analysis. Unlike the IMT method, the DE quadrature nodes are generated by a simple algebraic expression which is a great advantage.

Numerical Calculation of Fractional Derivatives using Automatic Differentiation and Double Exponential Formula

Numerical Calculation of Fractional Derivatives using Automatic Differentiation and Double Exponential Formula

Discovery of the Double Exponential Transformation and Its Developments

This article is mainly concerned how the double exponential formula for numerical integration was discovered and how it has been developed thereafter. For evaluation of I = \int_{-1}^1 f(x)dx , H. Takahasi and M. Mori took advantage of the optimality of the trapezoidal formula over (−∞, ∞) with an equal mesh size and proposed in 1974 x = \phi(t) = \tanh(\frac{π}{2} \sinh t) as an optimal choice of variable transformation which transforms the original integral to I = \int^∞_{−∞} f(\phi(t))\phi' (t) dt . If the trapezoidal formula is applied to the transformed integral and the resulting infinite summation is properly truncated a quadrature formula I ≈ h ∑^n_{k=−n}f_(\phi(kh))\phi'(kh) is obtained. Its error is expressed as O(\exp(−CN/ \log\, N_)) as a function of the number N ( = 2n + 1 ) of function evaluations. Since the integrand decays double exponentially after the transformation it is called the double exponential (DE) formula. It is also shown that the formula is optimal in the sense that there is no quadrature formula obtained by variable transformation whose error decays faster than O(\exp(−CN/ \log\, N)) as N becomes large. Since the paper by Takahasi and Mori was published the DE formula has gradually come to be used widely in various fields of science and engineering. In fact we can find papers in which the DE formula is successfully used in the fields of molecular physics, fluid dynamics, statistics, civil engineering, financial engineering, in particular in the field of the boundary element method, and so on. The DE transformation has turned out to be also useful for evaluation of indefinite integrals, for solution of integral equations and for solution of ordinary differential equations, so that the scope of its applications is expected to spread also in the future.

A Double Exponential Formula for the Fourier Transforms

In this paper, we propose a new and efficient method that is applicable for the computation of the Fourier transform of a function which may possess a singular point or slowly converge at infinity. The proposed method is based on a generalization of the method of the double exponential (DE) formula; the DE formula is a powerful numerical quadrature proposed by H. Takahasi and M. Mori in 1974 [1]. Although it is a widely applicable formula, it is not effective in computing the Fourier transform of a slowly decreasing function. Actually it is not very efficient even if one wants to compute the value of a Fourier transform at a particular point, i.e., a Fourier-type integral. To conquer this weakness at least for a Fourier-type integral, M. Mori and the author proposed a new DE formula in 1991 [2]. See also [3] for a further improvement. The method proposed there is effective for Fourier-type integrals, but it is still weak in the computation of the Fourier transform. Here we propose another DE formula which is applicable to the computation of the Fourier transform. One point in the new method proposed here is that it makes use of fixed sampling points even if we change the point where the Fourier integral is evaluated. In this paper we propose the new method and illustrate the efficiency of the new method in several concrete examples through the comparison with the older methods.

Double exponential formulas for numerical indefinite integration

In this paper we derive a formula for indefinite integration of analytic functions over (−1, s) where −1< s<1, by means of the double exponential transformation and the Sinc method. The integrand must be analytic on −1< x<1 but may have a singularity at the end points x=±1. The error of the formula behaves approximately as exp(−c 1N/ log c 2N) where N is the number of function evaluations of the integrand. This error term shows a much faster convergence to zero when N becomes large than that of the known formula by Haber. Also we derive efficient double exponential formulas for numerical evaluation of indefinite integrals over (0, s), 0< s<∞ and over (−∞, s), −∞< s<+∞. Several numerical examples indicate high efficiency of the formulas.

An improved convergence test for the double exponential formula

An improved convergence test for the double exponential formula

A solution method for two-dimensional potential flow about bodies with smooth surfaces by direct use of the boundary integral equation

This work presents a novel boundary integral method to treat the two-dimensional potential flow due to a moving body with the Lyapunov surface. The singular integral equations are derived in singularity-free form by applying the Gauss flux theorem and the property of the equipotential body. The modified formulations are amenable to computation by directly using quadrature formulae. Instead of the conventional boundary element approach, the boundary surface is initially expressed in a parametric form. Further, applying the double exponential formula and the cubic spline method yield the corresponding geometrical coefficients which are involved in parametric form. For illustration, the flow field induced by a translating ellipse is examined. Numerical calculations indicate that the proposed method is more efficient than the flat-element constant-source boundary element method. Copyright © 1999 John Wiley & Sons, Ltd.

A solution method for two‐dimensional potential flow about bodies with smooth surfaces by direct use of the boundary integral equation

This work presents a novel boundary integral method to treat the two-dimensional potential flow due to a moving body with the Lyapunov surface. The singular integral equations are derived in singularity-free form by applying the Gauss flux theorem and the property of the equipotential body. The modified formulations are amenable to computation by directly using quadrature formulae. Instead of the conventional boundary element approach, the boundary surface is initially expressed in a parametric form. Further, applying the double exponential formula and the cubic spline method yield the corresponding geometrical coefficients which are involved in parametric form. For illustration, the flow field induced by a translating ellipse is examined. Numerical calculations indicate that the proposed method is more efficient than the flat-element constant-source boundary element method. Copyright © 1999 John Wiley & Sons, Ltd.

Theoretical Investigation of the Reaction of O(3P) with CH2Cl

The reaction of oxygen atom with chlorinated methyl radical has been studied using the G2MP2 level of theory. The computation reveals an association−elimination mechanism. The addition reaction of O(3P) with CH2Cl proceeds to the formation of an energy-rich intermediate OCH2Cl*. Four product channels of OCH2Cl* are found: H + CHClO, Cl + CH2O, CHClOH, and H2 + ClCO. The isomer CHClOH also has four product channels: H + CHClO, HCl + HCO, H2 + ClCO, and OH + CHCl. On the basis of this ab initio potential energy surface and the experimental rate constant at 297 K, the kinetics of the title reaction have been calculated by a “loose transition-state” model and RRKM theory. The overall rate constants can be fitted to a double-exponential formula k(T) = (1.51 ± 0.01) × 10-10 exp[(1.36 ± 0.01) kJ mol-1/(RT)] + (2.23 ± 0.09) × 10-10 exp[(−27.48 ± 0.73) kJ mol-1/(RT)] cm3 molecule-1 s-1 over the temperature range 250−2000 K. The energy-specific rate constants of the unimolecular decomposition of the activated add...

A new method for the numerical calculation of Cauchy principal value integrals in BEM applied to electromagnetics

A new method for the evaluation of singular boundary element integrals over three-dimensional isoparametric boundary elements of higher order is presented. This new procedure represents a Gaussian quadrature technique using polar coordinates for the calculation of the Gaussian points and the weighting coefficients. This method permits an efficient integration of singular kernels of order O(1/r) on curved surfaces. For a numerical example the proposed integration scheme is compared with other methods (subdivision technique, double exponential formula method, modified Gauss-quadrature) showing high efficiency and accuracy. The actual computation can be easily included in any existing computer code.

Optimality of the double exponential formula - functional analysis approach -

In the light of the functional analysis theory we establish the optimality of the double exponential formula. The argument consists of the following three ingredients: (1) introduction of a number of spaces of functions analytic in a strip region about the real axis, each space being characterized by the decay rate of their elements (functions) in the neighborhood of the infinity; (2) proof of the (near-) optimality of the trapezoidal formula in each space introduced in (1) by showing the (near-) equality between an upper estimate for the error norm of the trapezoidal formula and a lower estimate for the minimum error norm of quadratures; (3) nonexistence theorem for the spaces, the characterizing decay rate of which is more rapid than the double exponential.

Accuracy improvement using a modified Gauss-quadrature for integral methods in electromagnetics

A Gaussian quadrature technique for evaluating shape-function-boundary-element kernel produce integrals over three-dimensional isoparametric boundary elements is presented. The procedure allows the integration of singular kernels of O(1/r) on curved surfaces. The integration of the normal derivative of Green's function is also possible. Integrals which exist in the sense of Cauchy principal values are dealt with using the addition-subtraction technique. The accuracy of the numerical integration scheme is compared with that of the double exponential formula and the subdivision technique. Some examples show the effectiveness of the procedure. >

The double exponential formula for oscillatory functions over the half infinite interval

The double exponential formula is known to be very powerful for evaluation of various kinds of integrals, in particular integrals with end point singularities or integrals over the half infinite interval. It is also known that a weak point of this formula is the inefficiency when applied to a slowly decaying oscillatory integral over the half infinite interval such as I = ∫ 0 ∞f 1(x) sin x d x, f 1(x) is an algebraic function. In this paper we propose a new type of the double exponential formula which is quite efficient for evaluation of the integral mentioned above. It is based on such a transformation that makes the points of the formula after the transformation approach to the zeros of sin x double exponentially for large x.

Piezomagneti field associated with the Mogi model revisited: Analytic solution for finite spherical source.

The piezomagnetic field associated with the Mogi model was reexamined for the point source and the finite spherical source problems. In the point source case the question was how to deal with divergent stresses near the pressure source. In SASAI's (1979) method, the magnetized crust was divided into two layers, i.e. the upper half shallower than the source (0<z<D) and the lower half deeper than the source (D<z<H). The solution (type I) was simply the sum of the two contributions as calculated by the Fourier transform method. An alternative calculation (type II) integrates the whole magnetized area except for an infinitesimally small sphere involving the pressure source. This was achieved with the aid of a theorem for spherical harmonics. These two procedures gave different results. In order to identify the cause of the discrepancy, the finite source problem was further investigated. The piezomagnetic field was represented in the form of a one-dimensional integral containing complete elliptic integrals. It can be evaluated numerically with the aid of the double exponential formula (DEF). We called it the type III solution. The present analytic approach verifies the results of numerical 3-D integrations by DAVIS (1976), and SUZUKI and OSHIMAN (1990). The solution as the spherical radius diminishes to zero coincides with the type II solution. The type II solution had a gap across H=D. The potential value at H=D was just the average of the two limiting values for H=D-e and H=D+e. In the type I solution, the magnetization along a thin layer at H=D was ignored, which produced the pressure source term in the type II solution. Accordingly the type II solution was appropriate for the point source problem: the previous result by SASAI (1979) should be rejected.

Comment on double exponential formula.

A new method and its computer program for the double exponential formula for numerical integration is presented. The present method avoids underflow encountered in the computation of weights of the formula and sums the integral up to a sufficiently large number of terms, so that it does not deteriorate the accuracy of the double exponential formula even for integrands with strong end-point singularity.

The interaction between excited β-methylnaphthalene carboxylate and sulphuric acid

Abstract The interaction between excited state β-methylnaphthalene carboxylate (β-MN) and polymethylene-bis-naphthoates with sulphuric acid were studied. Absorption spectra were not observed to change detectably in dilute acid solution but the fluorescence spectra were dramatically changed. When the contribution of monomeric fluorescence was substracted the remaining spectra consisted of two bands. It was demonstrated that one band contributed by exciplex formed between β-MN and sulphuric acid and another one by protonated compound. Exciplex was an intermediate step in protonation of β-MN. By means of a photocounting method, fluorescence decay of β-MN was measured. It obeyed a double exponential formula. From the experimental data, the rate formation constant of exciplex was calculated to be equal to 5.4 × 10+8 M−1 s−1 which was much lower than the rate constant of diffusion. The fluorescence decay of protonated compound was also measured. It obeyed a single exponential formula. Its lifetime depended on the concentration of sulphuric acid. On the basis of experimental data, a protonation mechanism is proposed.

Boundary element analysis of axisymmetric resonant cavities

Two numerical techniques are applied to boundary element analysis of axisymmetric modes of a cylindrical resonant cavity. One of the techniques is based on a novel expression for the fundamental solution of the axisymmetric Helmholtz equation with double exponential formulas. The other is an integral equation method in which the fundamental solution of the Laplace equation is used by considering the axisymmetric Helmholtz equation as the axisymmetric Laplace equation with an inhomogeneous term. Axisymmetric modes of a cyclindrical resonant cavity are analyzed by these techniques, and the resonant wave numbers are obtained with high accuracy. These results show that the axisymmetric resonant cavities can be analyzed by the boundary element method. >

Efficient formulas for molecular integrals over the Hiller–Sucher–Feinberg identity using Cartesian Gaussian functions: Towards the improvement of spin density calculations

Recurrence relations are derived for the molecular integrals arising in the use of the Hiller–Sucher–Feinberg identity. In comparison with the formulas obtained by other methods, the present result gives simpler formulas, and efficient computations are feasible. To perform the numerical integration the double exponential formula developed by Takahashi and Mori is found suitable for accurate and expeditious calculations.