A general Gauss divergence theorem with applications to convolution integrals of the form ∫ f(x̄)h(|x̄− ā|)dVn, where the integration extends over an n-dimensional polyhedral domain, is presented. The kernel h(|x̄ − ā|) may be singular, but the given integral must remain integrable. As a result of the Gauss theorem, the given integral is reduced to an integral over the boundary of the n-dimensiona...

We obtain new numerical schemes for weakly singular integrals of convolution type called Caputo fractional order using Taylor and series expansions grouping terms in a novel manner. A expansion argument is utilized to provide fractional-order approximations functions with minimal regularity. The resulting allow the approximation Cγ[0,T], where 0<γ≤5. mild invertibility criterion provided implic...

In the paper we study stochastic convolution appearing in Volterra equation driven by so called Lévy process. By Lévy process we mean a process with homogeneous independent increments, continuous in probability and cadlag.

Simulation of beam cooling usually requires performing certain integral transformations every time step or so, which is a significant burden on the CPU. Examples are the dispersion integrals (Hilbert transforms) in the stochastic cooling, wake fields and IBS integrals. An original method is suggested for fast and sufficiently accurate computation of the integrals. This method is applied for the...