This paper investigates the stability of the Radial Basis Function (RBF) collocation method for convective problems on the circle and sphere. We prove that the RBF method is Lax-stable for problems on the circle when the collocation points are equispaced and the transport speed is constant. We also show that the eigenvalues of discretization matrices are purely imaginary in the case of variable...

In this paper, we develop a Jacobi-Gauss-Lobatto collocation method for solving the nonlinear fractional Langevin equation with three-point boundary conditions. The fractional derivative is described in the Caputo sense. The shifted Jacobi-Gauss-Lobatto points are used as collocation nodes. The main characteristic behind the Jacobi-Gauss-Lobatto collocation approach is that it reduces such a pr...

In this article, we have investigate a Taylor collocation method, which is based on collocation method for solving fractional pantograph equation. This method is based on first taking the truncated fractional Taylor expansions of the solution function in the mathematical model and then substituting their matrix forms into the equation. Using the collocation points, we have the system of nonline...

An algorithm is described for the efficient numerical solution of singular two-point boundary value problems. The algorithm is based on collocation at Gauss points, is applied directly to second order equations and uses a transformation of the independent variable to obtain extra smoothness if needed. Numerical comparisons between a code based on this approach and codes based on other options w...

Alocal convergence rate is established for an orthogonal collocationmethod based on Gauss quadrature applied to an unconstrained optimal control problem. If the continuous problem has a 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 number of collocation po...

The Falkner-Skan equation is a nonlinear third-order boundary value problem defined on the semi-infinite interval [0,∞). This equation plays an important role to illustrate the main physical features of boundary layer phenomena. This paper presents a new collocation method for solving the Falkner-Skan equation. The proposed approach is equipped by the orthogonal Chebyshev polynomials that have ...

We propose a method for determining an arbitrage-free density implied by the Hagan formula. (We use the wording “Hagan formula” as an abbreviation of the Hagan– Kumar–Leśniewski–Woodward model.) Our method is based on the stochastic collocation method. The principle is to determine a few collocation points on the implied survival distribution function and project them onto the polynomial of an ...

In this paper, we propose a Modified nested sparse grid based Adaptive Stochastic Collocation Method (MASCM) for block-based Statistical Static Timing Analysis (SSTA). The proposed MASCM employs an improved adaptive strategy derived from the existing Adaptive Stochastic Collocation Method (ASCM) to approximate the key operator MAX during timing analysis. In contrast to ASCM which uses non-neste...

A new approach for analyzing boundary value problems for linear and for integrable nonlinear PDEs was introduced in [1]. For linear elliptic PDEs, an important aspect of this approach is the characterization of a generalized Dirichlet to Neumann map: given the derivative of the solution along a direction of an arbitrary angle to the boundary, the derivative of the solution perpendicularly to th...

This work proposes and analyzes a Smolyak-type sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations with random coefficients and forcing terms (input data of the model). To compute solution statistics, the sparse grid stochastic collocation method uses approximate solutions, produced here by finite el...