Search

• Using integration by parts and the characteristics of Chebyshev polynomials, four useful recursive relationships of the required modified moments are deduced. • We perform the strict error analysis on the presented method and acquire asymptotic error estimations in inverse powers of frequency ω. • Our method has the following advantages: when the interpolation node is fixed, the accuracy improves considerably as either the frequency or the interpolated multiplicities at endpoints increase. • Our proposed method is simple to construct, the algorithm is easy to implement and has less running times. Moreover, only a small number of interpolation nodes is needed to achieve high accuracy. Our proposed method is applicable to many kinds of highly oscillatory Fourier-type integrals. This fully shows the wide application of this method. In this article, we propose and analyze the Clenshaw–Curtis–Filon-type method for computing many highly oscillatory Fourier-type integrals with algebraic or logarithmic singularities at the endpoints. First, by using integration by parts and the characteristics of Chebyshev polynomials, four useful recursive relationships of the required modified moments are deduced. We perform the strict error analysis on the presented method and acquire asymptotic error estimations in inverse powers of frequency ω. Our method has the following advantages: when the interpolation node is fixed, the accuracy improves considerably as either the frequency or the interpolated multiplicities at endpoints increase. Numerical experiments verify the efficacy and correctness of the presented method. [ABSTRACT FROM AUTHOR]

Let d  ≥ 1. For any A ∈ Z d × d such that | det A | = 2 , we construct two families of Parseval wavelet frames with two generators. These generators have compact support, any desired number of vanishing moments, and any given degree of regularity. The first family is real valued while the second family is complex valued. To construct these families we use Daubechies low pass filters to obtain refinable functions, and adapt methods employed by Chui and He and Petukhov for dyadic dilations to this more general case. We also construct several families of Parseval wavelet frames with three generators having various symmetry properties. Our constructions are based on the same refinable functions and on techniques developed by Han and Mo and by Dong and Shen for the univariate case with dyadic dilations. [ABSTRACT FROM AUTHOR]

The spline theory can be derived from two sources: the algebraic one (where splines are understood as piecewise smooth functions satisfying some continuity conditions) and the variational one (where splines are obtained via minimization of some quadratic functionals with constraints). We show that the general variational approach called smooth interpolation introduced by Talmi and Gilat covers not only the cubic spline but also the tension spline (called also spline in tension or spline with tension) in one or more dimensions. We show the results of a 1D numerical example that present the advantages and drawbacks of the tension spline. [ABSTRACT FROM AUTHOR]

In this paper, we study one class of generalized convolution-type singular integral equations in class {0}. Such equations are turned into complete singular integral equations with nodal points and further turned into boundary value problems for analytic function with discontinuous coefficients by Fourier transforms. For such equations, we will propose one method different from classical one and obtain the general solutions and their conditions of solvability in class {0}. Thus, this paper generalizes the theory of classical equations of convolution type. [ABSTRACT FROM AUTHOR]

The numerical solution of the initial value problem for the two-dimensional Burgers equation on the whole plane is considered. Usual techniques, like finite difference methods and finite element methods cannot be directly applied for the solution of this problem, because the corresponding domain is unbounded. We propose a new method to overcome this difficulty. The efficiency of the proposed method is tested by several numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we study four classes of discrete type equations with harmonic singular operator and convolution. Such equations are turned into boundary value problems for analytic function with discontinuous coefficients by discrete Fourier transform. The general solutions and the conditions of solvability are obtained in class h by our method. Thus, this paper generalizes the theory of classical equations of convolution type. [ABSTRACT FROM AUTHOR]

The paper is primarily devoted to the problem of smooth interpolation of data in 1D. In addition to the exact interpolation of data at nodes, we are also concerned with the smoothness of the interpolating curve and its derivatives. The interpolating curve is defined as the solution of a variational problem with constraints. The system of functions exp ( - i kx ) , k being an integer, is taken for the basis of the space where we measure the smoothness of the result. It also generates the functions used for the interpolation itself. Choosing different norms when measuring the smoothness, we arrive at different interpolating functions. We also mention the problem of smooth curve fitting (data smoothing). We discuss the proper choice of different norms for this way of approximation and present the results of several 1D numerical examples that show the advantages and drawbacks of smooth interpolation. [ABSTRACT FROM AUTHOR]

We present a Fourier transform representation of the generalized hypergeometric functions. We then use this representation to evaluate integrals of products of two generalized hypergeometric functions. A number of integral identities for confluent and Gauss hypergeometric functions are presented. The results for Euler’s gamma function are deduced as special cases. [ABSTRACT FROM AUTHOR]

• A cracked orthotropic solid under symmetric heat flow and symmetric mechanical loading is studied. • The modified partially impermeable crack model is employed to simulate thermal load transfer. • With application of Fourier transform technique and superposition theory, the related physical quantities and fracture parameters are obtained in explicit forms. • It reveals that for the problem of combined thermal and mechanical loads the fracture parameter would be physically significant unless certain conditions are met. This paper addresses problem of two collinear cracks in an infinity orthotropic solid subject to combined thermal and mechanical loads. Based on a model called ' improved partially conductive crack model', the Fourier transform and superposition theorem, the analytical solutions to some physical quantities and fracture parameters are given. It is revealed that the dimensionless thermal resistance (R c) between the upper and below crack regions and the proposed coefficient (ε) exert a great influence on some physical quantities and fracture parameters, i.e., the mode-II stress intensity factors (K II) by numerical results. [ABSTRACT FROM AUTHOR]

The objective of this paper is to construct wave packet frames on local fields of positive characteristic. A necessary and sufficient condition for the wave packet system D p j T u ( n ) a E u ( m ) b ψ j ∈ Z , m , n ∈ N 0 to be a frame for L 2 ( K ) is given by means of the Fourier transform. [ABSTRACT FROM AUTHOR]

In this paper, the ill-posedness of computing the two dimensional Fourier transform is discussed. A regularized algorithm for computing the two dimensional Fourier transform of band-limited signals is presented. The convergence of the regularized Fourier series is studied and compared with the Fourier series by some examples. [ABSTRACT FROM AUTHOR]

We present four numerical methods to compute the Value-at-Risk and Expected Shortfall risk measure values of portfolios with financial options. The numerical methods are based on either wavelets or Fourier cosine approximations and belong to the class of Fourier inversion methods. We show that the risk measures can be efficiently calculated in terms of accuracy and CPU time. Besides, we provide a theoretical result about the shape of the resulting probability density. This a priori knowledge, allows us to enhance the efficiency and effectiveness of the proposed methods. Finally, we assess the accuracy of the approach in the presence of convexity or concavity properties of the financial portfolios. [ABSTRACT FROM AUTHOR]

We introduce a natural convolution of two suitable quaternion valued functions on R and list down its properties. Using this convolution, first we get the convolution theorem for Fourier transform on quaternion valued functions. Next, we modify the existing definition of wavelet transform on square integrable quaternion valued functions in a natural manner so that Parseval's identity is obtained without any additional conditions. Applying the Parseval's identity, we derive the inversion formula for the wavelet transform and we also prove the other properties like linearity, continuity and injectivity. Finally, we construct two Boehmian space of quaternion valued functions and extend the wavelet transform as a continuous linear injection from one Boehmian space into the other space. [ABSTRACT FROM AUTHOR]

Abstract: The Fourier Transform of a quartic Gaussian, , is important in the theory of radial basis functions with . We show that the transform is the exact sum of two generalized hypergeometric functions . This implies that is an entire function and also gives an analytical form for the power series coefficients of arbitrary degree. Using the method of steepest descent for integrals, we derive the first five terms in an asymptotic series in powers of . Through a simple hyperasymptotic analysis, we show that the magnitude of the error of the optimally-truncated series [“superasymptotic” error] is . The lowest order approximation, shows that the transform is an exponentially-decaying oscillation. We show that the steepest descent method [2,27] (Bender and Orszag, 1978; Miller, 2006) [26,33] (Lauwerier, 1966; Ursell, 1965) can be applied to the transform of for general integer n and give the first two terms. Another theme is that modern computer technology has simplified the steepest descent method. The steepest descent paths can be plotted by merely displaying the contours of the imaginary part of the phase function where is the logarithm of the integrand, and z is the large parameter (here proportional to ). The task of inverting to obtain a series in powers of w for the metric factor after the usual change of integration variable can now be done to high order by a few lines of an algebraic manipulation language as illustrated by a short table with the complete Maple code. [Copyright &y& Elsevier]

Abstract: In this paper we consider the pricing of an American call option whose underlying asset dynamics evolve under the influence of two independent stochastic volatility processes as proposed in Christoffersen, Heston and Jacobs (2009) [13]. We consider the associated partial differential equation (PDE) for the option price and its solution. An integral expression for the general solution of the PDE is presented by using Duhamel’s principle and this is expressed in terms of the joint transition density function for the driving stochastic processes. For the particular form of the underlying dynamics we are able to solve the Kolmogorov PDE for the joint transition density function by first transforming it to a corresponding system of characteristic PDEs using a combination of Fourier and Laplace transforms. The characteristic PDE system is solved by using the method of characteristics. With the full price representation in place, numerical results are presented by first approximating the early exercise surface with a bivariate log linear function. We perform numerical comparisons with results generated by the method of lines algorithm and note that our approach provides quite good accuracy. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a homogeneous backward heat conduction problem which appears in some applied subjects. This problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the final data. A new regularization method is applied to formulate regularized solutions which are stably convergent to the exact ones with Holder estimates. A numerical example shows that the computational effect of the method is all satisfactory. [Copyright &y& Elsevier]

Abstract: Connections between -solutions of the refinement equation of the form and distributional fixed points of a special random affine map are investigated. Using results on convergence of perpetuities the direct form of the Fourier transform of -solutions of the above refinement equation is obtained. [Copyright &y& Elsevier]

Abstract: Commonly used finite-difference numerical schemes show some deficiencies in the integration of certain types of stochastic partial differential equations with additive white noise. In this paper efficient predictor–corrector spectral schemes to integrate these equations are discussed. They are all based on the discretization of the system in Fourier space. The nonlinear terms are treated using a pseudospectral approach so as to speed up the computations without a significant loss of accuracy. The proposed schemes are applied to solve, both in one and two spatial dimensions, two paradigmatic continuum models arising in the context of nonequilibrium dynamics of growing interfaces: the Kardar–Parisi–Zhang and Lai–Das Sarma–Villain equations. Numerical results about the Lai–Das Sarma–Villain equation in two spatial dimensions have not been previously reported in the literature. [Copyright &y& Elsevier]

Abstract: We present a Fourier transform representation of the generalized gamma functions, which leads to a distributional representation for them as a series of Dirac-delta functions. Applications of these representations are shown in evaluation of the integrals of products of the generalized gamma function with other functions. The results for Euler’s gamma function are deduced as special cases. The relation of the generalized gamma function with the Macdonald function is exploited to deduce new identities for it. [Copyright &y& Elsevier]

Abstract: In this paper, we consider an inverse problem for a time-fractional diffusion equation with one-dimensional semi-infinite domain. The temperature and heat flux are sought from a measured temperature history at a fixed location inside the body. We show that such problem is severely ill-posed and further apply a spectral regularization method to solve it based on the solution given by the Fourier method. Convergence estimates are presented under a priori bound assumptions for the exact solution. Finally, numerical examples are given to show that the proposed numerical method is effective. [Copyright &y& Elsevier]

Abstract: This paper investigates the space fractional diffusion equation with fractional Feller’s operator. The Green’s function is obtained by using Fourier transform, and the analytical solutions of some space fractional diffusion equations with initial (or initial and boundary) condition are obtained in terms of Green’s function. In addition, numerical simulations are discussed. The results indicate that the effect range of skewness parameter has more effect on probability density than that of parameter . The results also explain the property of the skewness and long tail in the asymmetry diffusion process. [Copyright &y& Elsevier]

Abstract: Selim and Ahmed used the eigenvalue approach by assuming distinct eigenvalues to calculate the elastic deformation due to an inclined load at any point as a result of an inclined line load of initially stressed orthotropic elastic medium. They studied the plane strain problem and obtained the corresponding results for an unstressed orthotropic medium as a particular case. In the present paper, it is shown that all the eigenvalues do not remain distinct, but become repeated when the elastic medium is free from the initial compressive stresses. Further, the displacements and stresses for an unstressed elastic medium have been independently obtained. The variation of the displacements and stresses due to normal and tangential line load are also shown graphically. [Copyright &y& Elsevier]

Abstract: In this study we derive the general expression for the entries of the qth power for odd order complex circulant matrices of the type circ n (0, a,0,…, b). [ABSTRACT FROM AUTHOR]

Abstract: An initial value problem for a system of the second order partial differential equations, describing the electric wave propagation in vertically inhomogeneous electrically and magnetically anisotropic uniaxial media, is the main object of the study. The present paper suggests and justifies a new algorithm for solving this problem. This algorithm has several steps. On the first step the original initial value problem is written in terms of the Fourier images with respect to lateral variables. After that the obtained problem is transformed into an equivalent second kind vector integral equation of the Volterra type. A solution of this integral equation is constructed by successive approximations. At last, using the real Paley–Wiener theorem, a solution of the original initial value problem is found. [Copyright &y& Elsevier]

Abstract: In this survey we consider optical computers that encode data using images and compute by transforming such images. We give an overview of a number of such optical computing architectures, including descriptions of the type of hardware commonly used in optical computing, as well as some of the computational efficiencies of optical devices. We go on to discuss optical computing from the point of view of computational complexity theory, with the aim of putting some old, and some very recent, results in context. Finally, we focus on a particular optical model of computation called the continuous space machine. We describe some results for this model including characterisations in terms of well-known complexity classes. [Copyright &y& Elsevier]

Abstract: In this study we present the Fourier transform (FT) of the absolute amplitude spectrum for direct application to wood anatomy. In spite of its simplicity, to our knowledge there is no reference in the literature regarding the use of repeated FT of an arbitrary vibration. [Copyright &y& Elsevier]

Abstract: Laplace transform is a powerful method for enabling solving differential equation in engineering and science. Using the Laplace transform for solving differential equations, however, sometimes leads to solutions in the Laplace domain that are not readily invertible to the real domain by analytical means. Numerical inversion methods are then used to convert the obtained solution from the Laplace domain into the real domain. Four inversion methods are evaluated in this paper. Several test functions, which arise in engineering applications, are used to evaluate the inversion methods. We also show that each of the inversion methods is accurate for a particular case. This study shows that among all these methods, the Fourier transform inversion technique is the most powerful but also the most computationally expensive. Stehfest’s method, which is used in many engineering applications is easy to implement and leads to accurate results for many problems including diffusion-dominated ones and solutions that behave like e −t type functions. However, this method fails to predict e t type functions or those with an oscillatory response, such as sine and wave functions. [Copyright &y& Elsevier]

Abstract: The eigenvalue approach, using the Laplace and Fourier transforms, has been employed to find the analytical expressions for displacements and stresses at any point, as a result of an inclined line load, of an initially stressed orthotropic elastic medium. A plain strain problem has been studied. The results in the form of displacement and stress components have been obtained and discussed graphically for a particular model. In this paper it is shown that the displacement and stress components are affected by initial stresses in the medium. [Copyright &y& Elsevier]

Abstract: The discrete Fourier transform is used to determine the coefficients of a transfer function for n-order linear systems: . The algorithm is fast, straight forward and easily can be implemented. Two step-by-step examples, illustrating the application of the algorithm, are presented. [Copyright &y& Elsevier]

• We consider the pricing problem for GMDB products under general regime - switching jump diffusion models. • The complex Fourier series expansion method is applied to approximate the pricing formulas. • A lot of numerical examples are provided to show effectiveness of our pricing method. In this paper, we consider the valuation problem of equity-linked annuity with guaranteed minimum death benefit (GMDB) by complex Fourier series method under regime-switching jump diffusion models. We show that the price formulas can be expressed by some discounted density functions associated with the residual lifetime random variable. Fourier transforms for discounted density functions are derived, and complex Fourier series expansion method is applied to recover the discounted density functions. Some explicit formulas for computing the GMDB price are given. Finally, we give some numerical results to show effectiveness of our method. [ABSTRACT FROM AUTHOR]

In this paper, we establish some properties of the Hilbert transform in Clifford analysis setting, and mainly show the weak type (1,1) inequality for the Clifford Hilbert transform in Euclidean space R m (m ≥ 3) using the Clifford algebra and analysis generalizations of the one dimensional proofs. Furthermore, we prove Kolmogorov's inequality for Clifford Hilbert transform. With the help of the properties for the Hilbert transform and Clifford analytic techniques, we establish the Riemann–Hilbert problems in Clifford value Lebesgue p -integrable spaces. Based on the Newton embedding method, we prove an existence and uniqueness for the nonlinear Riemann–Hilbert problem and also give an error estimation for the approximate solutions in the Newton embedding procedure in Lebesgue spaces. [ABSTRACT FROM AUTHOR]