We present in this paper several extremely efficient and accurate spectral-Galerkin methods for secondand fourth-order equations in polar and cylindrical geometries. These methods are based on appropriate variational formulations which incorporate naturally the pole condition(s). In particular, the computational complexities of the Chebyshev–Galerkin method in a disk and the Chebyshev–Legendre–...

I apply Fourier and Chebyshev spectral methods to derive accurate and efficient algorithms for velocity continuation. As expected, the accuracy of the spectral methods is noticeably superior to that of the finite-difference approach. Both methods apply a transformation of the time axis to squared time. The Chebyshev method is slightly less efficient than the Fourier method, but has less problem...

In this paper, a Chebyshev finite difference method has been proposed in order to solve nonlinear two-point boundary value problems for second order nonlinear differential equations. A problem arising from chemical reactor theory is then considered. The approach consists of reducing the problem to a set of algebraic equations. This method can be regarded as a non-uniform finite difference schem...

A method for computing highly accurate numerical solutions of 1D convection–diffusion equations is proposed. In this method, the equation is first discretized with respect to the spatial variable, transforming the original problem into a set of ordinary differential equations, and then the resulting system is integrated in time by the fourth-order Runge–Kutta method. Spatial discretization is d...

The numerical solution of a model describing a two-dimensional fluidized bed by a Chebyshev super spectral viscosity (SSV) method is considered. The model is in the form of a hyperbolic system of conservation laws with a source term, coupled with an elliptic equation for determining a stream function. The coupled elliptic equation is solved by a finite difference method. The mixed SSV/finite di...

It is well known that high-order finite-difference methods may become unstable due to the presence of boundaries and the imposition of boundary conditions. For uniform grids, Gustafsson, Kreiss, and Sundstrom theory and the summation-by-parts method provide sufficient conditions for stability. For non-uniform grids, clustering of nodes close to the boundaries improves the stability of the resul...

In chemical engineering, several processes are represented by singular boundary value problems. In general, classical numerical methods fail to produce good approximations for the singular boundary value problems. In this paper, Chebyshev finite difference (ChFD) method and DTM-Pad´e method, which is a combination of differential transform method (DTM) and Pad´e approximant, are applied for sol...

Abstract: In this paper, we are implemented the Chebyshev spectral method for solving the non-linear fractional Klein-Gordon equation (FKGE). The fractional derivative is considered in the Caputo sense. We presented an approximate formula of the fractional derivative. The properties of the Chebyshev polynomials are used to reduce FKGE to the solution of system of ordinary differential equations...

It is well known that the classical families of orthogonal polynomials are characterized as eigenfunctions of a second order linear differential/difference operator with polynomial coefficients. In this paper we present a study of classical orthogonal polynomials in a more general framework by using the differential (or difference) calculus and Operator Theory. In such a way we obtain a unified...