Finite dimensional approximation to fractional stochastic integro-differential equations with non-instantaneous impulses

TL;DR

This paper introduces a finite-dimensional approximation method for fractional stochastic integro-differential equations with non-instantaneous impulses, demonstrating convergence and existence of solutions using advanced mathematical tools.

Contribution

It develops a novel projection-based approximation scheme for FSIDE with non-instantaneous impulses and proves its convergence and existence of solutions in a Hilbert space setting.

Findings

Convergence of Faedo-Galerkin approximations to the mild solution.

Existence of solutions for the proposed FSIDE model.

Validation through a PDE example confirms theoretical results.

Abstract

This manuscript proposes a class of fractional stochastic integro-differential equation (FSIDE) with non-instantaneous impulses in an arbitrary separable Hilbert space. We use a projection scheme of increasing sequence of finite dimensional subspaces and projection operators to define approximations. In order to demonstrate the existence and convergence of an approximate solution, we utilize stochastic analysis theory, fractional calculus, theory of fractional cosine family of linear operators and fixed point approach. Furthermore, we examine the convergence of Faedo-Galerkin(F-G) approximate solution to the mild solution of our given problem. Finally, a concrete example involving partial differential equation is provided to validate the main abstract results.