We discuss the numerical solution of a class of second-kind integral equations in which the integral operator is not compact. Such equations arise, for example, when boundary integral methods are applied to potential problems in a two-dimensional domain with corners in the boundary. We are able to prove the optimal orders of convergence for the usual collocation and product integration methods ...

In the present paper the authors propose two numerical methods to approximate Hadamard transforms of the type Hp( f wβ , t) = ∫ = R f (x) (x − t)p+1 wβ (x)d x , where p is a nonnegative integer and wβ (x) = e−|x | β , β > 1, is a Freud weight. One of the procedures employed here is based on a simple tool like the “truncated” Gaussian rule conveniently modified to remove numerical cancellation a...

This study presents a Feynman–Kac path-integral implementation for solving the Dirichlet problem for Poisson’s equation. The algorithm is a modified “walk on spheres” (WOS) that includes the Feynman–Kac path-integral contribution for the source term. In our approach, we use an h-conditioned Green’s function instead of simulating Brownian trajectories in detail to implement this path-integral co...

in this paper, we conduct a comparative study between the homotopy perturbation method (hpm) and adomian’s decomposition method (adm) for analytic treatment of nonlinear volterra integral equations, and we show that the hpm with a specific convex homotopy is equivalent to the adm for these type of equations.

We demonstrate an alternative method for calculating the asymptotic behaviour of the discrete one-coin quantum walk on the infinite line, via the Jacobi polynomials that arise in the path integral representation. We calculate the asymtotics using a method that is significantly easier to use than the Darboux method. It also provides a single integral representation for the wavefunction that work...

In this paper a new method for computation of higher order corrections to the saddle point approximation of the Feynman path integral is introduced. The saddle point approximation leads to local Schrödinger problems around classical orbits. Especially, the saddle point approximation leads to Schrödinger problems around classical periodic orbits when it is applied to the trace of Green’s functio...

We study counting statistics of number of transitions in a stochastic process. For mesoscopic systems, a path integral formulation for the counting statistics has already been derived. We here show that it is also possible to derive the similar path integral formulation without the assumption of mesoscopic systems. It has been clarified that the saddle point method for the path integral is not ...

Accurate path integral Monte Carlo or molecular dynamics calculations of isotope effects have until recently been expensive because of the necessity to reduce three types of errors present in such calculations: statistical errors due to sampling, path integral discretization errors, and thermodynamic integration errors. While the statistical errors can be reduced with virial estimators and path...