This work describes and analyzes a stochastic collocation method for stochastic Volterra integral equations (SVIEs) of the second kind with random forcing terms. A collocation method is used in temporal direction, and a spectral collocation method is used in the stochastic dimension, which lead to an uncoupled linear system associated with the collocation points. The convergence analysis is carried out and the optimal error estimates are obtained. Numerical experiments are conducted to verify the error estimates and demonstrate the effectiveness of the proposed numerical method.

We consider the asymptotic method designed by Olver \cite{olver} for linear differential equations of second order containing a large (asymptotic) parameter $\Lambda$, in particular, the second and third cases studied by Olver: differential equations with a turning point (second case) or a singular point (third case). It is well known that his method gives the Poincar\'e-type asymptotic expansion of two independent solutions of the equation in inverse powers of $\Lambda$. In this paper, we add initial conditions to the differential equation and consider the corresponding initial value problem. By using the Green's function of an auxiliary problem, we transform the initial value problem into a Volterra integral equation of the second kind. Then, using a fixed point theorem, we construct a sequence of functions that converges to the unique solution of the problem. This sequence also has the property of being an asymptotic expansion for large $\Lambda$ (not of Poincar\'e-type) of the solution of the problem. Moreover, we show that the technique also works for nonlinear differential equations with a large parameter.

The paper establishes a condition for the first order hyperbolic system (with a boundary condition at infinity) to have a solution that can be expressed by the second kind Volterra operator. On the basis of this condition, some canonical forms (for instance Dirac-type system) of the first order hyperbolic system can be sorted. The suitability of this condition in inverse scattering problem for the first order hyperbolic system on the half line is given as an application.

Consider a nonlinear operator equation $ x - K ( x )=f$, where $K$ is a Urysohn integral operator with a smooth kernel. Using the orthogonal projection onto a space of discontinuous piecewise polynomials of degree $\leq r$, previous authors have established an order $r + 1$ convergence for the Galerkin solution, $2 r + 2$ for the iterated Galerkin solution, $ 3 r + 3$ for the modified projection solution and $ 4 r + 4$ for the iterated modified projection solution. Equivalent results have also been established for the interpolatory projection at Gauss points. In this paper, the iterated Galerkin/iterated collocation solution and the iterated modified projection solution are shown to have asymptotic series expansions. The Richardson extrapolation can then be used to improve the order of convergence to $ 2 r + 4$ in the case of the iterated Galerkin/iterated collocation method and to $ 4 r + 6$ in the case of the iterated modified projection method. Numerical results are given to illustrate this improvement in the orders of convergence.

The evolution in time of a viscoelastic body is described by an equation with memory, which can be seen as a perturbation of the equations of elasticity. This observation is a useful tool in the study of control problems. In this paper, by using moment methods, we compare a viscoelastic system which fills a surface or a solid region (the string case has already been studied) with its elastic counterpart (which is a generalized telegrapher's equation) in order to prove exact controllability of the viscoelastic body as a consequence of the assumed controllability of the associated telegrapher's equation.

In a previous paper, the authors showed that the information complexity of the Fredholm problem of the second kind is essentially the same as that of the approximation problems over the spaces of kernels and right-hand sides. This allowed us to give necessary and sufficient conditions for the Fredholm problem to exhibit a particular level of tractability (for information complexity) over weighted tensor product (\textsc{wtp}) spaces, as well as over an important class of \textit{not} necessarily tensor product weighted Sobolev spaces. Furthermore, we addressed the overall complexity of this Fredholm problem for the case in which the kernels and right-hand sides belong to a \textsc{wtp} space. For this case, we showed that a nearly-minimal-error interpolatory algorithm is easily implementable, with cost very close (to within a logarithmic factor) to the information cost. As a result, tractability results, which had previously only held for the information complexity, now hold for the overall complexity--provided that our kernels and right-hand sides belong to \textsc{wtp} spaces. This result does not hold for the weighted Sobolev spaces mentioned above, since they are not necessarily tensor product spaces.

In this paper, we close this gap. We exhibit an easily implementable iterative approximation to a nearly minimal error interpolatory algorithm for this family of weighted Sobolev spaces. This algorithm exhibits the same good properties as the algorithm presented in the previous paper.