Linear hyperbolic partial differential equations in a homogeneous medium, e.g., the wave equation describing the propagation and scattering of acoustic waves, can be rewritten as a time-domain boundary integral equation. We propose an efficient implementation of a numerical discretization of such equations when the strong Huygens’ principle does not hold. For the numerical discretization, we ma...

This work devotes to developing a systematic and convenient approach based on the celebrated convolution quadrature theory design analyze difference formulas for fractional calculus at an arbitrary shifted point $$x_{n-\theta }$$ . The developed theory, called (SCQ), covers most from aspects of characterizing formation related generating functions which are convergent with integer orders. For s...

Abstract. We evaluate the fractional integral Iα[f ](t) = 1 Γ(α) ∫ t 0 (t− τ)α−1 f(τ) dτ , 0 < α < 1, at time steps t = Δt, 2Δt, . . . , NΔt by making use of the integral representation of the convolution kernel tα−1 = 1 Γ(1−α) ∫∞ 0 e −ξ t ξ−α dξ. We construct an efficient Q-point quadrature of this integral representation and use it as a part of a fast time stepping method. The new method has ...

Quadrature filters are a well known means for local spectral analysis of images and to extract relevant structure. Recently, there has been the discovering of an isotropic quadrature filter for images that does not need steering with respect to orientation and provides the user with geometric (local orientation) and structural (local phase) information. Here, we present a further extension of t...

An integro-differential equation involving a convolution integral with a weakly singular kernel is considered. The kernel can be that of a fractional integral. The integro-differential equation is discretized using the discontinuous Galerkin method with piecewise constant basis functions. Sparse quadrature is introduced for the convolution term to overcome the problem with the growing amount of...

A fast and accurate method for pricing early exercise and certain exotic options in computational finance is presented. The method is based on a quadrature technique and relies heavily on Fourier transformations. The main idea is to reformulate the well-known risk-neutral valuation formula by recognising that it is a convolution. The resulting convolution is dealt with numerically by using the ...

We consider the efficient numerical solution of coupled dynamical systems, consisting a low dimensional nonlinear part and high linear time invariant part, e.g., stemming from spatial discretization an underlying partial differential equation. The subsystem can be eliminated in frequency domain for resulting integro-differential algebraic equations, we propose combination Runge–Kutta or multist...