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In this paper, we show that efficient separated sum-of-exponentials approximations can be constructed for the heat kernel in any dimension. In one space dimension, the heat kernel admits an approximation involving a number of terms that is of the order $O(\log (\frac {T}{\delta }) (\log (\frac {1}{\epsilon })+\log \log (\frac {T}{\delta })))$ for any $x\in \mathbb R$ and δ≤ t≤ T, where 휖 is the desired precision. In all higher dimensions, the corresponding heat kernel admits an approximation involving only $O(\log ^{2}(\frac {T}{\delta }))$ terms for fixed accuracy 휖. These approximations can be used to accelerate integral equation-based methods for boundary value problems governed by the heat equation in complex geometry. The resulting algorithms are nearly optimal. For N points in the spatial discretization and N time steps, the cost is $O(N_{S} N_{T} \log ^{2} \frac {T}{\delta })$ in terms of both memory and CPU time for fixed accuracy 휖. The algorithms can be parallelized in a straightforward manner. Several numerical examples are presented to illustrate the accuracy and stability of these approximations. [ABSTRACT FROM AUTHOR]

The X-ray computerized tomography is concerned to the reconstruction of a function f from line (ray) integrals of f. In general, we may have the parallel ray or divergent ray geometry. In this paper we develop an analytical singular value decomposition method for the cone-beam transform in divergent ray geometry. For the parallel ray transform a singular value decomposition has been derived by P. Maass in 1987. [ABSTRACT FROM AUTHOR]

The classical Talbot method for the computation of the inverse Laplace transform is improved for the case where the transform is analytic in the complex plane except for the negative real axis. First, by using a truncated Talbot contour rather than the classical contour that goes to infinity in the left half-plane, faster convergence is achieved. Second, a control mechanism for improving numerical stability is introduced. [ABSTRACT FROM AUTHOR]

We give an affirmative answer to the conjecture raised in Soltani and Roozegar [On distribution of randomly ordered uniform incremental weighted averages: divided difference approach. Statist Probab Lett. 2012;82(5):1012–1020] that a certain class of power semicircle distributions, parameterized byn, gives the distributions of the average ofnindependent and identically Arcsine random variables weighted by the cuts of (0,1) by the order statistics of a uniform (0, 1) sample of sizen−1, for eachn. Then we establish the central limit theorem for this class of distributions. We also use the Demni [On generalized Cauchy–Stieltjes transforms of some beta distributions. Comm Stoch Anal. 2009;3:197–210] results on the connection between the ordinary and generalized Cauchy or Stieltjes transforms, and introduce new classes of randomly weighted average distributions. [ABSTRACT FROM AUTHOR]

Analytic reconstruction of a function defined in an affine space from data of its integrals along lines or rays is in focus of the paper. Basic tools are the Fourier transform of homogeneous distributions and a self-duality equation in integral geometry. Three dimensional case is of special interest. [ABSTRACT FROM AUTHOR]

The Hermite functions are an orthonormalbasis of the space of square integrable functions with favourable approximation properties. Allowing for a flexible localization in position and momentum, the Hagedorn wavepackets generalize the Hermite functions also to several dimensions. Using Hagedorn's raising and lowering operators, we derive explicit formulas and recurrence relations for the Wigner and FBI transform of the wavepackets and show their relation to the Laguerre polyomials. [ABSTRACT FROM AUTHOR]

The data acquisition in computerized tomography takes a certain amount of time since the x-ray source has to be rotated around the specimen. An object that changes during the scanning causes inconsistent data sets. To avoid the motion artefacts in reconstructions, the algorithm has to take the dynamic behavior of the specimen into account. In this context, some a priori information about the movement is required. A reconstruction method is proposed that compensates for the motion with a special focus on affine deformations. It also permits the combination of reconstruction and image analysis tools to extract features of the object without motion artefacts. The algorithm is validated with a numerical example from medical imaging. [ABSTRACT FROM AUTHOR]

A novel method for solving boundary value problems for linear partial differential equations in convex polygons was developed by A.S. Fokas in the late 1990s. A spectrally accurate numerical implementation for the case of the Laplace equation led the way for the present implementation for both the Helmholtz equation and the modified Helmholtz equation. We obtain again spectrally accurate numerical solutions for Dirichlet to Neumann maps, without any discretization of the domain interior. The eigenvalue problem is also examined using the present Legendre expansion method. A key feature of the current approach is that all integrals that arise in the map problem can be evaluated in closed form, reducing it to the solution of a small overdetermined linear system of equations. [ABSTRACT FROM PUBLISHER]

In this work, we discuss the numerical computation of the eigenvalues and eigenfunctions of the finite (truncated) Hankel transform, important for numerous applications. Due to the very special behavior of the Hankel transform eigenfunctions, their direct numerical calculation often causes an essential loss of accuracy. Here, we present several simple, efficient and robust numerical techniques to compute Hankel transform eigenfunctions via the associated singular self-adjoint Sturm-Liouville operator. The properties of the proposed approaches are compared and illustrated by means of numerical experiments. [ABSTRACT FROM AUTHOR]

The Kohlrausch functions $\exp (- {t}^{\beta } )$, with $\beta \in (0, 1)$, which are important in a wide range of physical, chemical and biological applications, correspond to specific realizations of completely monotone functions. In this paper, using nonuniform grids and midpoint estimates, constructive procedures are formulated and analysed for the Kohlrausch functions. Sharper estimates are discussed to improve the approximation results. Numerical results and representative approximations are presented to illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

We derive asymptotic expansions of the Poisson transform of a locally integrable function f( t): $(\sqrt{4\pi t})^{-1}\int_{-\infty}^{\infty}\exp \lbrace{-(x-y)^{2}/(4t)}\rbrace f(y) \,dy$, for small and large t. All the expansions are accompanied by error bounds for the remainder at any order of the approximation. [ABSTRACT FROM AUTHOR]

In this paper, we introduce efficient methods for the approximation of solutions to weakly singular Volterra integral equations of the second kind with highly oscillatory Bessel kernels. Based on the asymptotic analysis of the solution, we derive corresponding convergence rates in terms of the frequency for the Filon method, and for piecewise constant and linear collocation methods. We also present asymptotic schemes for large values of the frequency. These schemes possess the property that the numerical solutions become more accurate as the frequency increases. [ABSTRACT FROM AUTHOR]

In this paper, we study the approximation of the inversion of windowed Fourier transforms using Riemannian sums. We show that for certain window functions, the Riemannian sums are well defined on L(ℝ), 1 < p < ∞, and tend to the function to be reconstructed as the sampling density tends to infinity. [ABSTRACT FROM AUTHOR]

Starting from piecewise constant functions, a novel family of generalized symmetric B-splines, with realizable ideal low-pass filters, are constructed. The first order generalized B-spline low-pass filter is closely related to functions analytic in a neighborhood of the unit disc and the generalized sinc functions. The properties of this kind of low-pass filters are investigated. The behavior of the generalized B-spline low-pass filter related to normalized Gaussian distribution is considered. [ABSTRACT FROM AUTHOR]

This paper offers a characterization of amplitude functions in $${L^2(\mathbb R)}$$ satisfying the Bedrosian identity in the case that the phase functions are determined by the boundary value on the unit circle of a function analytic in the neighborhood of the unit circle and real on the real axis. [ABSTRACT FROM AUTHOR]

This paper studies new inversion methods for the Laplace transform of vector-valued functions arising from a combination of A-stable rational approximation schemes to the exponential and the shift operator semigroup. Each inversion method is provided in the form of a (finite) linear combination of the Laplace transform of the function and a finite amount of its derivatives. Seven explicit methods arising from A-stable schemes are provided, such as the Backward Euler, RadauIIA, Crank-Nicolson, and Calahan scheme. The main result shows that, if a function has an analytic extension to a sector containing the nonnegative real line, then the error estimate for each method is uniform in time. [ABSTRACT FROM AUTHOR]

Asymptotic expansions for oscillatory integrals typically depend on the values and derivatives of the integrand at a small number of critical points. We show that using values of the integrand at certain complex points close to the critical points can actually yield a higher asymptotic order approximation to the integral. This superinterpolation property has interesting ramifications for numerical methods based on exploiting asymptotic behaviour. The asymptotic convergence rates of Filon-type methods can be doubled at no additional cost. Numerical steepest descent methods already exhibit this high asymptotic order, but their analyticity requirements can be significantly relaxed. The method can be applied to general oscillators with stationary points as well, through a simple change of variables. [ABSTRACT FROM AUTHOR]

This paper offers a characterization of amplitude functions in $${L^2(\mathbb R)}$$ satisfying the Bedrosian identity in the case that the phase functions are determined by the boundary value on the unit circle of finite Blaschke products. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the numerical simulation of the generalized Fisher–Kolmogorov–Petrovskii–Piskunov equation. After introducing a new variable, the integro-differential equation is transformed into an equivalent coupled system of first-order differential equations. A second-order accurate difference scheme is constructed for the new system of equations, which is proved to be local uncoupled by separation of variables. It is also proved that the scheme is uniquely solvable and second-order convergent in both time and space in L 2-norm. A numerical example is given to demonstrate the theoretical results. [ABSTRACT FROM PUBLISHER]

In earlier work we have studied a method for discretization in time of a parabolic problem, which consists of representing the exact solution as an integral in the complex plane and then applying a quadrature formula to this integral. In application to a spatially semidiscrete finite-element version of the parabolic problem, at each quadrature point one then needs to solve a linear algebraic system having a positive-definite matrix with a complex shift. We study iterative methods for such systems, considering the basic and preconditioned versions of first the Richardson algorithm and then a conjugate gradient method. [ABSTRACT FROM AUTHOR]

In this work we give sufficient conditions on the symbol of a pseudo-differential operator on $${\mathbb Z^n}$$ ensuring the L-boundedness. We give a sufficient condition for ( p, q)-boundedness where $${\frac{1}{p}+ \frac{1}{q} =1}$$ . A sufficient condition for the L-boundedness, 1 ≤ p < ∞, for a pseudo-differential operator whose symbol has certain differentiability properties with respect to the Fourier variable is also given. Other results establish a explicit relation between the ( p, q)-weak continuity and the ( p, q)-boundedness of a pseudo-differential operator. Some necessary conditions for boundedness and compactness are also given at the end. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present an efficient numerical procedure for solving the two-dimensional nonlinear Volterra integro-differential equations (2-DNVIDE) by two-dimensional differential transform method (2-DDTM). The technique that we used is the differential transform method, which is based on Taylor series expansion. Using the differential transform, 2-DNVIDE can be transformed to algebraic equations, and the resulting algebraic equations are called iterative equations. New theorems for the transformation of integrals and partial differential equations are introduced and proved. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments. [ABSTRACT FROM AUTHOR]

The aim of the article is to find an efficient algorithm for local numerical evaluation of Fn (p) by expressing the corresponding finite Hankel transform Fˆn (p) in Haar series and truncating it. [ABSTRACT FROM AUTHOR]

We present a quadrature-based method to evaluate exponential-like operators required by different kinds of exponential integrators. The method approximates these operators by means of a quadrature formula that converges like O( e), with K the number of quadrature nodes, and it is useful when solving parabolic equations. The approach allows also the evaluation of the associated scalar mappings. The method is based on numerical inversion of sectorial Laplace transforms. Several numerical illustrations are provided to test the algorithm, including examples with a mass matrix and the application of the method inside the MATLAB package EXP4, an adaptive solver based on an exponential Runge–Kutta method. [ABSTRACT FROM AUTHOR]

For parallel-ray transmission tomographic image reconstruction, a hybrid simulated annealing (HSA)-based statistical algorithm has been developed which utilizes the preprocessing template system. The proposed methodology has been applied for the reconstruction of 8 × 8, 16 × 16, 32 × 32, 64 × 64 and 128 × 128 head and lung phantoms to enhance the resolution. The effect of generic parameters including the initial temperature, final temperature, annealing profile has been analysed. The sensitivity of reconstruction quality has also been evaluated for problem specific parameters including the number of projections and size of reconstruction. Image quality is measured using root-mean-squared error, Euclidean error and peak signal-to-noise ratio (PSNR). The results of computer simulations have been compared to the standard deterministic techniques including filtered back-projection (FBP) and algebraic reconstruction technique. Hybrid simulated annealing has been found to be superior yielding the minimum error and the maximum SNR value, as compared to the Matlab® functions, 'radon' and 'iradon', with PSNR for HSA and FBP as 44.92, 33.21 and 13.61, 14.59 dB for the 8 × 8 head and lung phantoms. The parametric optimization of transmission tomography-based image reconstruction provides a scope for achieving greater dose reduction even for much larger images with parallel platforms. [ABSTRACT FROM AUTHOR]

Laplace transforms are used to derive an exact expression for the cdf of the sum of n i.i.d. Pareto random variables with common pdf f(x) = (α/β)(1 + x/β)-α-1 for x > 0, where α > 0 and is not an integer, and β > 0. An attractive feature of this expression is that it involves an integral of non oscillating real-valued functions on the positive real line. Examples of values of cdfs are provided and are compared to those determined via simulations. [ABSTRACT FROM AUTHOR]

A new analytical model was established to describe the complex behavior of ceramic/metal armor under impact of deformable projectile by assuming some hypotheses. Three aspects were taken into account: the mushrooming deformation of the projectile, the fragment of ceramic tile and the formation and change of ceramic conoid and the deformation of the metal backup plate. Solving the set of equations, all the variables were obtained for the different impact velocities: the extent and particle velocity in rigid zone; the extent, cross-section area and particle velocity in plastic zone; the velocity and depth of penetration of projectile to the target; the reduction in volume and compressive strength of the fractured ceramic conoid; the displacement and movement velocity of the effective zone of backup plate. Agreement observed among analytical result, numerical simulation and experimental result confirms the validity of the model, suggesting the model developed can be a useful tool for ceramic/metal armor design. [ABSTRACT FROM AUTHOR]

Wavelet–type transform associated with singular Laplace–Bessel differential operator $\Delta _\nu = \sum\limits_{k = 1}^n {\frac{{\partial ^2 }} {{\partial x_k^2 }}} + \frac{{2\nu}} {{x_n }}\frac{\partial } {{\partial x_n }}$ is introduced and the relevant Calderón–type reproducing formula is established. Representations of the generalized Bessel potentials $(E - \Delta _\nu )^{ - \alpha /2} f,\quad (Re \alpha > 0)$ and their inverses via the wavelet–type transform are obtained. [ABSTRACT FROM AUTHOR]

We propose an adaptive algorithm which extends Chebyshev series approximation to bivariate functions, on domains which are smooth transformations of a square. The method is tested on functions with different degrees of regularity and on domains with various geometries. We show also an application to the fast evaluation of linear and nonlinear bivariate integral transforms. [ABSTRACT FROM AUTHOR]

When a Nyström-Broyden method is used to solve numerically Urysohn integral equations, the main problem is to evaluate the action of the integral operator at a low cost. Here we suitably approximate the relevant discrete integral operator of dimensionn by itsm-degree truncated Chebyshev series expansion (withm≪n), reducing the complexity from the basic to . A technique to evaluate cheaply suchm is presented. Several linear and nonlinear examples are considered. [ABSTRACT FROM AUTHOR]

On the basis of Terzaghi’s one-dimensional consolidation theory, the variation of effective stress ratio in layered saturated soils with impeded boundaries under time-dependent loading was studied. By the method of Laplace transform, the solution was presented. Influences of different kinds of cyclic loadings and impeded boundaries conditions were discussed. Through numerical inversion of Laplace transform, useful illustrations were given considering several common time-dependent loadings. Pervious or impervious boundary condition is just the special case of the problem considered here. Compared with average index method, the results from the method illustrated are more accurate. [ABSTRACT FROM AUTHOR]

In the investigation on fracture mechanics, the potential function was introduced, and the moving differential equation was constructed. By making Laplace and Fourier transformation as well as sine and cosine transformation to moving differential equations and various responses, the dual equation which is constructed from boundary conditions lastly was solved. This method of investigating dynamic crack has become a more systematic one that is used widely. Some problems are encountered when the dynamic crack is studied. After the large investigation on the problems, it is discovered that during the process of mathematic derivation, the method is short of precision, and the derived results in this method are accidental and have no credibility. A model for example is taken to explain the problems existing in initial deriving process of the integral-transformation method of dynamic crack. [ABSTRACT FROM AUTHOR]

A class of regularization methods using unbounded regularizing operators is considered for obtaining stable approximate solutions for ill-posed operator equations. With an a posteriori as well as an a priori parameter choice strategy, it is shown that the method yields the optimal order. Error estimates have also been obtained under stronger assumptions on the generalized solution. The results of the paper unify and simplify many of the results available in the literature. For example, the optimal results of the paper include, as particular cases for Tikhonov regularization, the main result of Mair (1994) with an a priori parameter choice, and a result of Nair (1999) with an a posteriori parameter choice. Thus the observations of Mair (1994) on Tikhonov regularization of ill-posed problems involving finitely and infinitely smoothing operators is applicable to various other regularization procedures as well. Subsequent results on error estimates include, as special cases, an optimal result of Vainikko (1987) and also some recent results of Tautenhahn (1996) in the setting of Hilbert scales. [ABSTRACT FROM AUTHOR]

The cascade algorithm with mask a and dilation M generates a sequence $\phi_n, n=1,2,\ldots, $ by the iterative process \[ \phi_n(x) = \sum_{\alpha\in{\mathbb Z}^s} a(\alpha) \phi_{n-1}(Mx - \alpha) \quad x\in{\mathbb R}^s, \] from a starting function $\phi_0,$ where M is a dilation matrix. A complete characterization is given for the strong convergence of cascade algorithms in Sobolev spaces for the case in which M is isotropic. The results on the convergence of cascade algorithms are used to deduce simple conditions for the computation of integrals of products of derivatives of refinable functions and wavelets. [ABSTRACT FROM AUTHOR]

The mathematical foundation of an algorithm for fast and accurate evaluation of singular integral transforms was given by Daripa [9,10,12]. By construction, the algorithm offers good parallelization opportunities and a lower computational complexity when compared with methods based on quadrature rules. In this paper we develop a parallel version of the fast algorithm by redefining the inherently sequential recurrences present in the original sequential formulation. The parallel version only utilizes a linear neighbor-to-neighbor communication path, which makes the algorithm very suitable for any distributed memory architecture. Numerical results and theoretical estimates show good parallel scalability of the algorithm. [ABSTRACT FROM AUTHOR]

Multidimensional (MD) systems describe relations between signals depending on two or more independent variables. They are also called distributed parameter systems, if the independent variables are time and space. The only conventional models for their description are partial differential equations. This is in contrast to onedimensional (lumped parameter) systems, where a variety of different models including transfer functions is used. This paper extends the concept of transfer function models to multidimensional systems with bounded spatial domains, i.e., systems which can be described as initial-boundary-value problems. These transfer function models are useful for system analysis and as a starting point for the derivation of numerically efficient discrete simulation models. [ABSTRACT FROM AUTHOR]

We prove convergence results and error estimates for interpolatory product quadrature formulas for Cauchy principal value integrals on the real line with Freud–type weight functions. The formulas are based on polynomial interpolation at the zeros of orthogonal polynomials associated with the weight function under consideration. As a by–product, we obtain new bounds for the derivative of the functions of the second kind for these weight functions. [ABSTRACT FROM AUTHOR]