Boundedness of solutions of difference equations and application to numerical solution of Volterra integral equations of the second kind
Abstract We study the difference equations obtained when some numerical methods for Volterra integral equations of the second kind are applied to the linear test problem y(t) = 1 + ∝0t (λ + μt + vs) y(s) ds, t ⩾ 0, with fixed stepsize h. The resulting difference equations are of Poincare type and we formulate a criterion for boundedness of solutions of these equations if the associated characteristic polynomial is a simple von Neumann polynomial. This result is then used in stability analysis of reducible quadrature methods for Volterra integral equations.
We consider the Cauchy problem for the Korteweg-de Vries equation driven by a cylindrical fractional Brownian motion (fBm) in this paper. With Hurst parameter $H\geq\frac{7}{16}$ of the fBm, we obtain the local existence results with initial value in classical Sobolev spaces $H^s$ with $s\geq -\frac{9}{16}$. Furthermore, we give the relation between the Hurst parameter $H$ and the index $s$ to the Sobolev spaces $H^s$, which finds out the regularity between the driven term fBm and the initial value for the stochastic Korteweg-de Vries equation.