In this paper, the Legendre spectral-collocation method was applied to obtain approximate solutions for some types of fractional optimal control problems (FOCPs). The fractional derivative was described in the Caputo sense. Two different approaches were presented, in the first approach, necessary optimality conditions in terms of the associated Hamiltonian were approximated. In the second appro...

We present two different approaches for the numerical solution of fractional optimal control problems (FOCPs) based on a spectral method using Chebyshev polynomials. The fractional derivative is described in the Caputo sense. The first approach follows the paradigm "optimize first, then discretize" and relies on the approximation of the necessary optimality conditions in terms of the associated...

We analyze the FEAST method for computing selected eigenvalues and eigenvectors of large sparse matrix pencils. After establishing the close connection between FEAST and the well-known Rayleigh–Ritz method, we identify several critical issues that influence convergence and accuracy of the solver: the choice of the starting vector space, the stopping criterion, how the inner linear systems impac...

A theoretical analysis of the buckling mono-axially compressed rectangular grid is carried out. The composed two orthogonal orders continuous beams, simply supported at ends. critical load determined by different models: (i) Kirchhoff’s homogeneous orthotropic plate, and (ii) a system solved Rayleigh-Ritz method. Parametric studies are out numerical examples developed. analytical solutions vali...

In this article we further develop a perturbation approach to the Rayleigh– Ritz approximations from our earlier work. We both sharpen the estimates and extend the applicability of the theory to nonnegative definite operators . The perturbation argument enables us to solve two problems in one go: We determine which part of the spectrum of the operator is being approximated by the Ritz values an...

The Rayleigh-Ritz (RR) procedure, including orthogonalization, constitutes a major bottleneck in computing relatively high-dimensional eigenspaces of large sparse matrices. Although operations involved in RR steps can be parallelized to a certain level, their parallel scalability, which is limited by some inherent sequential steps, is lower than dense matrix-matrix multiplications. The primary ...

Beginning with the exact Kolmogorov power spectrum for an extended random medium (such as interstellar space), the structure function is first expressed as an inverse Fourier-Bessel transform. An approximate series representation involving fractional powers of the transverse coordinate is then optimized by an equivalent Gaussian expansion up to two terms using the Rayleigh-Ritz technique. The d...