The development of numerical methods for the solution of partial differential equations (PDEs) with nonlocal boundary conditions is important, since such type of problems arise as mathematical models of various real-world processes. We use radial basis function (RBF) collocation technique for the solution of a multidimensional linear elliptic equation with classical Dirichlet boundary condition...

Over the last few years there have been dramatic advances in the area of uncertainty quantification. In particular, we have seen a surge of interest in developing efficient, scalable, stable, and convergent computational methods for solving differential equations with random inputs. Stochastic collocation (SC) methods, which inherit both the ease of implementation of sampling methods like Monte...

We introduce a novel technique of uncertainty quantification using polynomial regression with derivative information and apply it to analyze the performance of a model of a sodium-cooled fast reactor. We construct a surrogate model as a goal-oriented projection onto an incomplete space of polynomials, find coordinates of projection by collocation, and use derivative information to reduce the nu...

In this paper, second order non-linear ordinary differential equations of Lane-Emden type are solved using the boundary value technique. A class of second derivative backward differentiation formula is derived from some continuous multistep schemes using the multistep collocation technique. The technique transforms the numerical methods to a system of non-linear equations represented as a tridi...

This paper presents a new technique for numerical treatments of Volterra delay integro-differential equations that have many applications in biological and physical sciences. The technique is based on the mono-implicit Runge — Kutta method (described in [12]) for treating the differential part and the collocation method (using Boole’s quadrature rule) for treating the integral part. The efficie...

Keywords: Volterra integro-differential equations Multistep collocation Superconvergence Stability a b s t r a c t Multistep collocation methods for Volterra integro-differential equations are derived and analyzed. They increase the order of convergence of classical one-step collocation methods, at the same computational cost. The numerical stability analysis is carried out and classes of A 0-s...

In this paper, collocation methods are used for detecting blow-up solutions of nonlinear homogeneous Volterra-Hammerstein integral equations. To do this, we introduce the concept of “blow-up collocation solution” and analyze numerically some blow-up time estimates using collocation methods in particular examples where previous results about existence and uniqueness can be applied. Finally, we d...

1Department of Mathematics, Alzahra University, Tehran, Iran 2Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan 87317-51167, Iran (Received November07, 2011, accepted March 1, 2012) Abstract. The sinc-collocation method is presented for solving second-order boundary value problems of nonlinear integro-differential equation. The method is effective...

Abstract. A local convergence rate is established for an orthogonal collocation method based on Radau quadrature applied to an unconstrained optimal control problem. If the continuous problem has a sufficiently smooth solution and the Hamiltonian satisfies a strong convexity condition, then the discrete problem possesses a local minimizer in a neighborhood of the continuous solution, and as the...

A numerical method for solving second order, transient, parabolic partial differential equation is presented. The spatial discretization is based on Hermite collocation method (HCM). It is a combination of orthogonal collocation method and piecewise cubic Hermite interpolating polynomials. The solution is obtained in terms of cubic Hermite interpolating basis. Numerical results have been plotte...