A spectral collocation method based on rational interpolants and adaptive grid points is presented. The rational interpolants approximate analytic functions with exponential accuracy by using prescribed barycentric weights and transformed Chebyshev points. The locations of the grid points are adapted to singularities of the underlying solution, and the locations of these singularities are appro...

A collocation-//"'-Galerkin method is defined for some elliptic boundary value problems on a rectangle. The method uses tensor products of discontinuous piecewise polynomial spaces and collocation based on Jacobi points with weight function >c2(l x)2. Optimal order of L2 rates of convergence is established for the approximation solution. A numerical example which confirms these results is prese...

This paper presents a numerical matrix method based on Bernstein polynomials (BPs) for approximate the solution of a system of m-th order nonlinear Volterra integro-differential equations under initial conditions. The approach is based on operational matrices of BPs. Using the collocation points,this approach reduces the systems of Volterra integro-differential equations associated with the giv...

This paper reviews some developments in error estimation and mesh selection for collocation methods for ordinary differential equations. The basic idea of collocation has great generality and simplicity. Given some operator equation we look for an approximate solution in the form of a linear combination of some fixed basis functions. The coefficients in the linear combination are found by subst...

This paper presents a computational technique that called Tau-collocation method for the developed solution of non-linear integro-differential equations which involves a population model. To do this, the nonlinear integro-differential equations are transformed into a system of linear algebraic equations in matrix form without interpolation of non-poly-nomial terms of equations. Then, using coll...

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...

In this paper, an effective procedure based on coordinate stretching and radial basis functions (RBFs) collocation method is applied to solve singularly perturbed differential-difference equations with layer behavior. It is well known that if the boundary layer is very small, for good resolution of the numerical solution at least one of the collocation points must lie in the boundary layer. In ...

This paper investigates an adaptive wavelet collocation time domain method for the numerical solution of Maxwell’s equations. In this method a computational grid is dynamically adapted at each time step by using the wavelet decomposition of the field at that time instant. In the regions where the fields are highly localized, the method assigns more grid points; and in the regions where the fiel...

The problem of solving pseudodiierential equations on spheres by collocation with zonal kernels is considered and bounds for the approximation error are established. The bounds are given in terms of the maximum separation distance of the collocation points, the order of the pseudodiierential operator, and the smoothness of the employed zonal kernel. A by-product of the results is an improvement...

The effect of diverse sources of uncertainties and the intrinsically multi-scale nature of real-world physical systems poses a considerable challenge to analyze such systems. Such phenomena are particularly critical in material systems wherein microstructural variability and randomness at lower scales have a significant impact on the behavior of the system. Towards this goal, some of our accomp...