Stochastic differential equation (SDE) and Fokker-Planck equation (FPE) are two general approaches to describe the stochastic drift-diffusion processes. Solving SDEs relies on the Monte Carlo samplings of individual system trajectory, whereas FPEs describe the time evolution of overall distributions via path integral alike methods. The large state space and required small step size are the majo...

We apply the Kurzweil-Henstock integral setting to prove a Fredholm Alternative-type result for the integral equation x (t)− K Z [a,b] α (t, s)x (s) ds = f (t) , t ∈ [a, b] , where x and f are Kurzweil integrable functions (possibly highly oscillating) defined on a compact interval [a, b] of the real line with values on Banach spaces. An application is given.

In this work, we present a computational method for solving second kindnonlinear Fredholm Volterra integral equations which is based on the use ofHaar wavelets. These functions together with the collocation method are thenutilized to reduce the Fredholm Volterra integral equations to the solution ofalgebraic equations. Finally, we also give some numerical examples that showsvalidity and applica...

Keywords: Approximate solutions to integral equations Radial and kernel-based networks Gaussian kernels Model complexity Analysis of algorithms a b s t r a c t Approximate solutions to inhomogeneous Fredholm integral equations of the second kind by radial and kernel networks are investigated. Upper bounds are derived on errors in approximation of solutions of these equations by networks with in...

In this work we consider the numerical computation of eigenpairs of a matrix derived from integral operators. The matrix is associated to a radiative transfer problem in stellar atmospheres that is formulated by means of a weakly singular Fredholm integral equation defined on a Banach space. We examine direct and iterative parallel strategies for the eigensolution phase, using state-of-the-art ...

In this paper, a method for finding an approximate solution of a class of two-dimensional nonlinear Volterra integral equations of the first-kind is proposed. This problem is transformedto a nonlinear two-dimensional Volterra integral equation of the second-kind. The properties ofthe bivariate shifted Legendre functions are presented. The operational matrices of integrationtogether with the produ...

The Karhunen–Loève series expansion (KLE) decomposes a stochastic process into an infinite of pairwise uncorrelated random variables and L2-orthogonal functions. For any given truncation order the basis is optimal in sense that total mean squared error minimized. orthogonal functions are determined as solution eigenvalue problem corresponding to homogeneous Fredholm integral equation second kin...

In this note, convex Homotopy perturbation method (HPM) is presented for the approximate solution of the linear Fredholm-Volterra integral and integro-differential equation. Convergence and rate of convergence of the HPM are proved for both equations. Five numerical examples are provided to verify the validity and accuracy of the proposed method. Example reveals that HPM is very accurate and si...

In this paper, Bernstein piecewise polynomials are used to solve the integral equations numerically. A matrix formulation is given for a non-singular linear Fredholm Integral Equation by the technique of Galerkin method. In the Galerkin method, the Bernstein polynomials are exploited as the linear combination in the approximations as basis functions. Examples are considered to verify the effect...