During the last decade, three main variations have been proposed for solving elliptic PDEs by means of collocation with radial basis functions (RBFs). In this study, we have implemented them for infinitely smooth RBFs, and then compared them across the full range of values for the shape parameter of the RBFs. This was made possible by a recently discovered numerical procedure that bypasses the ...

The problem of the boundary layer flow of an incompressible viscous fluid over a non-linear stretching sheet is considered. A spectral collocation method is performed in order to find an analytical solution of the governing nonlinear differential equations. The obtained results are finally compared through the illustrative graphs and tables with the exact solution and some well-known results ob...

Burgers’ equation is a fundamental partial differential equation in fluid mechanics. This paper reports a new space-time spectral algorithm for obtaining an approximate solution for the space-time fractional Burgers’ equation (FBE) based on spectral shifted Legendre collocation (SLC) method in combination with the shifted Legendre operational matrix of fractional derivatives. The fractional der...

An accurate and efficient solution method using spectral collocation method with domain decomposition is proposed for computing optical waveguides with discontinuous refractive index profiles. The use of domain decomposition divides the usual single domain into a few subdomains at the interfaces of discontinuous refractive index profiles. Each subdomain can be expanded by a suitable set of orth...

The aim of this paper is to design a new family of numerical methods of arbitrarily high order for systems of rst-order diierential equations which are to be termed pseudo two-step Runge-Kutta methods. By using collocation techniques, we can obtain an arbitrarily high-order stable pseudo two-step Runge-Kutta method with any desired number of implicit stages in retaining the two-step nature. In ...

We study the spherical collapse of non-top-hat matter fluctuations in presence dark energy with arbitrary sound speed. The model is described by a system partial differential equations solved using pseudo-spectral method collocation points. This can reproduce known analytical solutions linear regime an accuracy better than $10^{-6}\%$ and $10^{-2}\%$ for virialization threshold given usual mode...

on the interval [0, 2π]. Spectral discretizations of this equation have been in use ever since the work of Camassa and Holm [2] and Camassa, Holm and Hyman [3]. However, to the knowledge of the authors, no proof that such a discretization actually converges has appeared heretofore. Therefore, this issue is taken up here. Our method of proof is related to the work of Maday and Quarteroni on the ...

and Applied Analysis 3 with the initial conditions