Integrand Function Research Articles

A modification for solving Fredholm‐Hammerstein integral equation by using wavelet basis

PurposeThe purpose of this paper is to use Alpert wavelet basis and modify the integrand function approximation coefficients to solve Fredholm‐Hammerstein integral equations.Design/methodology/approachL2[0, 1] was considered as solution space and the solution was projected to the subspaces of L2[0, 1] with finite dimension so that basis elements of these subspaces were orthonormal.FindingsIn this process, solution of Fredholm‐Hammerstein integral equation is found by solving the generated system of nonlinear equations.Originality/valueComparing the method with others shows that this system has less computation. In fact, decreasing of computations result from the modification.

A Very Fast and Accurate Boundary Element Method for Options with Moving Barrier and Time-Dependent Rebate

A numerical method to price options with moving barrier and time-dependent rebate is proposed. In particular, using the so-called Boundary Element Method, an integral representation of the barrier option price is derived in which one of the integrand function is not given explicitly but must be obtained solving a Volterra integral equation of the first kind. This equation is affected by several kinds of singularities, some of which are removed using a suitable change of variables. Then the transformed equation is solved using a low-order finite element method based on product integration. Several numerical experiments are carried out showing that the method proposed is extraordinarily fast and accurate. In particular a high level of accuracy is achieved also when the initial price of the underlying asset is close to the barrier, when the barrier and the rebate are not differentiable functions, or when the option's maturity is particularly long. Finally the numerical method proposed in this paper performssignificantly better than binomial and trinomial lattices that are commonly employed in barrier option pricing.

Elliptical cracks arbitrarily oriented in 3D-anisotropic elastic media

An elliptical crack in an infinite anisotropic elastic medium is considered. For a polynomial external stress field, an efficient numerical algorithm of the solution of the problem is developed. The discontinuity of the displacement field on the crack surface (the crack opening vector) is presented in the form of 2D-regular integrals that can be evaluated numerically for any crack orientation with respect to the principal axes of the anisotropy of the medium. The integrand functions in these integrals are expressed via the Fourier transform of the Green function of the medium. The cases of constant and linear external stress fields are considered in detail. Simplifications for particular crack orientations in the cases of orthotropic and transversely isotropic media are indicated. The stress intensity factors are obtained in the form of regular integrals that depend only on the Fourier transform of the Green function of the medium. An equation for the tensor of effective elastic constants of an anisotropic media containing a random set of elliptical cracks is presented.

Numerical integration using wavelets

In this paper, an algorithm for obtaining approximate value of a definite integral as well as double integral using wavelets will be illustrated. This approximation depends on the pure scaling functions expansion of the integrand function.

Evaluation of the run-length distribution for a combined Shewhart-EWMA control chart

A simple algorithm is introduced for computing the run length distribution of a monitoring scheme combining a Shewhart chart with an Exponentially Weighted Moving Average control chart. The algorithm is based on the numerical approximation of the integral equations and integral recurrence relations related to the run-length distribution. In particular, a Clenshaw-Curtis product-integration rule is applied for handling discontinuities in the integrand function due to the simultaneous use of the two control schemes. The proposed algorithm, implemented in R and publicy available, compares favourably with the Markov chain approach originally used to approximate the run length properties of the combined Shewhart-EWMA.

Analysis of the BCS equations for anisotropic superconductivity

Abstract Anisotropic superconducting states have been studied within a generalized Hubbard model and the BCS formalism, which leads to two coupled integral equations that determine the chemical potential and the superconducting gap. In this work, these equations are analyzed. Their integrand functions have sharp walls at the Fermi surface, which mainly determine the integrals. Furthermore, the chemical potential obtained from these equations is close to that from the mean-field density of states (DOS) and quite different to that from non-interacting DOS. Finally, the calculated condensation energy as a function of doping is compared with experimental data obtained from La 2− x Sr x CuO 4 and Y 0.8 Ca 0.2 Ba 2 Cu 3 O 7− δ .

On quadrature formulae based on derivative collocation

Abstract The recent article “A new type of weighted quadrature rules and its relation to orthogonal polynomials” by Masied-Jamei [M. Masjed-Jamei, A new type of weighted quadrature rules and its relation to orthogonal polynomials, Appl. Math. Comput. 188 (2007) 154–165] introduces quadrature rules based on the evaluation of the derivative(s) of the integrand function rather the function itself. The approach appears useful when a number of derivatives, including the integrand, vanish at a point λ , leading to increased order of accuracy compared to standard Gaussian rules. It is also shown by Masjed-Jamei (2007) how the nodes and weights of the resulting quadrature formula relate to nodes and weights of standard Gaussian quadratures applied to a weight function w to be determined by solving a specific system of integral equations. We give here an explicit expression for w and provide strategies for the practical computation of the quadrature nodes and weights. Additional comments on the examples used by Masjed-Jamei (2007) as well as a generalization involving multiple λ ’s, are also included.

Quadratic convergence algorithms for the problem of capture of a point by a family of convex bodies

We consider the nonlinear optimal control problem with an integral functional in which the integrand function is the characteristic function of a given closed set in the phase space. The approximation method is applied to prove the necessary conditions of optimality in the form of the Pontryagin maximum principle without any prior assumptions on the behavior of the optimal trajectory. Similarly to the case of phase-constrained problems, we derive conditions of nondegeneracy and pointwise nontriviality of the maximum principle.

On the Integral Identities Consisting of Two Spherical Bessel Functions

When deriving dyadic Green's functions for the spherical structures with gyrotropic or bianisotropic materials, an integral whose integrand function consists of two spherical Bessel functions and a power function needs to be evaluated. Therefore, this paper revisits thoroughly the evaluation of the integral of Il,l'(kappa,kappa'). Starting from pointing out an error, it provides the correct solution to the integral in spherical coordinates in terms of distribution, in particular, step functions and delta functions. The formulation is further extended to a more generalized integral Hl,l'lambda(kappa,kappa'); and it is newly found that the solution to the generalized integral varies differently in the cases of even and odd values of l-l'. The mistakes that we found in the previous literature can also be proved easily by some of our intermediate solutions

Numerical integration over smooth surfaces in [formula omitted] via class [formula omitted] variable transformations. Part II: Singular integrands

Numerical integration over smooth surfaces in [formula omitted] via class [formula omitted] variable transformations. Part II: Singular integrands

Stochastic Programming with Equilibrium Constraints

In this paper, we discuss here-and-now type stochastic programs with equilibrium constraints. We give a general formulation of such problems and study their basic properties such as measurability and continuity of the corresponding integrand functions. We discuss also the consistency and rate of convergence of sample average approximations of such stochastic problems

A Prototype Four-Dimensional Galerkin-Type Integral

An error functional expansion is constructed for a four-dimensional integral over \[0, 1]4 whose integrand function has a singularity structure of a type that occurs in integrals in the Galerkin boundary element method. We show with an example how extrapolation may be used to evaluate this integral.

Numerical integration over smooth surfaces in via class variable transformations. Part I: Smooth integrands

Class Sm variable transformations with integer m for finite-range integrals were introduced by the author about a decade ago. These transformations ''periodize'' the integrand functions in a way that enables the trapezoidal rule to achieve very high accuracy, especially with even m. In a recent work by the author, these transformations were extended to arbitrary real m, and their role in improving the convergence of the trapezoidal rule for different classes of integrands was studied in detail. It was shown that, with m chosen appropriately, exceptionally high accuracy can be achieved by the trapezoidal rule. For example, if the integrand function is smooth on the interval of integration including the endpoints, and vanishes at the endpoints, then excellent results are obtained by taking 2m to be an odd integer. In the present work, we consider the use of these transformations in the computation of integrals on surfaces of simply connected bounded domains in R^3, in conjunction with the product trapezoidal rule. We assume these surfaces are smooth and homeomorphic to the surface of the unit sphere, and we treat the cases in which the integrands are smooth. We propose two approaches, one in which the product trapezoidal rule is applied with the integrand as is, and another, in which the integrand is preprocessed before the rule is applied. We give thorough analyses of the errors incurred in both approaches, which show that surprisingly high accuracies can be achieved with suitable values of m. We also illustrate the theoretical results with numerical examples.

Extension of a class of periodizing variable transformations for numerical Integration

ClassSm{\mathcal S}_mvariable transformations with integermm, for numerical computation of finite-range integrals, were introduced and studied by the author in the paper [A. Sidi, A new variable transformation for numerical integration,Numerical Integration IV,1993 (H. Brass and G. Hämmerlin, eds.), pp. 359–373.] A representative of this class is thesinm\sin ^m-transformation that has been used with lattice rules for multidimensional integration. These transformations “periodize” the integrand functions in a way that enables the trapezoidal rule to achieve very high accuracy, especially with evenmm. In the present work, we extend these transformations toarbitraryvalues ofmm, and give a detailed analysis of the resulting transformed trapezoidal rule approximations. We show that, with suitablemm, they can be very useful in different situations. We prove, for example, that if the integrand function is smooth on the interval of integration and vanishes at the endpoints, then results of especially high accuracy are obtained by taking2m2mto be an odd integer. Such a situation can be realized in general by subtracting from the integrand the linear interpolant at the endpoints of the interval of integration. We also illustrate some of the results with numerical examples via the extendedsinm\sin ^m-transformation.

Numerical evaluation of Theis and Hantush-Jacob well functions

This paper presents a new approach where a numerical quadrature scheme is applied to compute special functions described by integrals, appearing in many groundwater hydrology and pollution problems. The basic idea is to apply a smoothing procedure to the singular integrand and a non-linear coordinate transformation in the quadrature domain. Techniques for lowering the degree of singularity of integrand functions have been successfully applied in physics and computational mechanics, and the self-adaptive co-ordinate transformation used in this work is a well-established procedure in general boundary element integrals quadrature. The main advantage of this joint technique is that it can be directly used to other related functions without further approximations and the user can choose freely the degree of accuracy. Tables included in this paper show the improvement of several orders of magnitude when both techniques are applied simultaneously.

Application of class [formula omitted] variable transformations to numerical integration over surfaces of spheres

Class S m variable transformations with integer m for finite-range integrals were introduced by the author (Numerical Integration IV, International series of Numerical Mathematics, Basel, 1993, pp. 359–373) about a decade ago. These transformations “periodize” the integrand functions in a way that enables the trapezoidal rule to achieve very high accuracy, especially with even m . In a recent work by the author (Math. Comp. (2005)), these transformations were extended to arbitrary m , and their role in improving the convergence of the trapezoidal rule for different classes of integrands was studied in detail. It was shown that, with m chosen appropriately, exceptionally high accuracy can be achieved by the trapezoidal rule. In the present work, we make use of these transformations in the computation of integrals on surfaces of spheres in conjunction with the product trapezoidal rule. We treat integrands that have point singularities of the single-layer and double-layer types. We propose different approaches and provide full analyses of the errors incurred in each. We show that surprisingly high accuracies can be achieved with suitable values of m . We also illustrate the theoretical results with numerical examples. Finally, we also recall analogous procedures developed in another work by the author (Appl. Math. Comput. (2005)) for regular integrands.

Method for calculation of ionization profiles caused by cosmic rays in giant planet ionospheres from Jovian group

Abstract As a continuation of our studies of cosmic ray (CR) ionization in the atmospheres of the planets in the Solar system [Adv. Space Res. 27 (11) 1901, 2001a] we present a new method for the calculation of the electron production rate q ( h ) profiles due to particles of all energy intervals: galactic CR, anomalous CR component and other types of high energy particles. In the above mentioned paper, ionospheres of terrestrial planets are investigated, where the spherical model is used. For giant planets which have significant oblateness in spite of the isotropic penetration of the galactic CR in their atmospheres, the trivial integration on the azimuth angle is not applicable, because of the presentation of the planets as rotational ellipsoids and the azimuth dependence of the integrand function. The difference between profiles for spherical q S and ellipsoidal q E models of the terrestrial planets (Earth, Venus and Mars) is small. These differences in the q S and q E profiles increase significantly in the upper atmospheric layers of outer major planets. This requires the introduction of a modified Chapman function for oblate planet in the particle depth parameter (PDP), while considering the CR influence and ionization processes in the ionospheres of the giant planets. New calculations for relative profiles q E / q S in the atmosphere of Saturn are presented. For this purpose an improved primary CR spectrum with a new type of smoothing function f with tangens hyperbolicus is used.

A bilinear approximation of the surface relief in terrain correction computations

A new method is presented for the computation of the gravitational attraction of topographic masses when their height information is given on a regular grid. It is shown that the representation of the terrain relief by means of a bilinear surface not only offers a serious alternative to the polyhedra modeling, but also approaches even more smoothly the continuous reality. Inserting a bilinear approximation into the known scheme of deriving closed analytical expressions for the potential and its first-order derivatives for an arbitrarily shaped polyhedron leads to a one-dimensional integration with - apparently - no analytical solution. However, due to the high degree of smoothness of the integrand function, the numerical computation of this integral is very efficient. Numerical tests using synthetic data and a densely sampled digital terrain model in the Bavarian Alps prove that the new method is comparable to or even faster than a terrain modeling using polyhedra.

Variational problems with several volume constraints on the level sets

We consider functionals of the kind \(I(u)= \int F(x,u,\nabla u)\) on \(W^{1,p}\), and we study the problem \(\min_{\mathcal{K}} I\), where \(\mathcal{K}\subset W^{1,p}\) consists of those functions u whose level sets satisfy certain volume constraints \(\Big \vert{\{u=l_i\}\Big \vert=\alpha_i>0\), where \(\Big \vert \cdot\Big \vert\) denotes Lebesgue measure, and \(\{l_i\}\), \(\{\alpha_i\}\) are given numbers. Examples show that this problem may have no solution, even for simple smooth F. As a consequence, we relax the constraint \(u\in\mathcal{K}\) to \(u\in\mathcal{K}_+\), i.e. \(\Big \vert \{u=l_i\}\Big \vert\geq \alpha_i\), and we show that the minimizers over \(\mathcal{K}_+\) exist and are Holder continuous. Then we prove several existence theorems for the original problem, showing that, under suitable assumptions on the integrand function F, every minimizer over \(\mathcal{K}_+\) actually belongs to \(\mathcal{K}\).

Error bounds for interpolatory quadrature rules on the unit circle

For the construction of an interpolatory integration rule on the unit circle T T with n n nodes by means of the Laurent polynomials as basis functions for the approximation, we have at our disposal two nonnegative integers p n p_n and q n , q_n, p n + q n = n − 1 , p_n+q_n=n-1, which determine the subspace of basis functions. The quadrature rule will integrate correctly any function from this subspace. In this paper upper bounds for the remainder term of interpolatory integration rules on T T are obtained. These bounds apply to analytic functions up to a finite number of isolated poles outside T . T. In addition, if the integrand function has no poles in the closed unit disc or is a rational function with poles outside T T , we propose a simple rule to determine the value of p n p_n and hence q n q_n in order to minimize the quadrature error term. Several numerical examples are given to illustrate the theoretical results.