(1) A one-dimensional minimization problem and the Ritz method. (2) Weak formulation and the Galerkin method. Abstract error estimates. (3) The lemma of variations. Euler-Lagrange equations. Weak formulation, again. (4) The Ritz-Galerkin finite element method: philosophy. (5) The piecewise linear finite element space and basis functions. The linear system. (6) The piecewise linear finite elemen...

Strictly positive operators are often encountered in applications, e.g., vibrational modes and associated energies of a mechanical system (e.g., a vibrating rod or plate), Hamiltonian operators for quantum mechanical systems (the eigenvalues correspond to energies). In these cases, the Hilbert space H of concern is infinite dimensional and therefore difficult to work with practically. A goal of...

We present a detailed investigation of atmospheres around accreting neutron stars with high magnetic field (B ∼> 1012 G) and low luminosity (L ∼< 1033 erg/s). We compute the atmospheric structure, intensity and emergent spectrum for a plane–parallel, pure hydrogen medium by solving the transfer equations for the normal modes coupled to the hydrostatic and energy balance equations. The hard tail...

The presence of the delamination causes reductions in the bending stiffness which in turn leads to the undesirable loss in the compressive buckling and post-buckling strength. Thus, it is of chief importance to investigate the compressive behavior of composites with delaminations. Anastasiadis et al. analyzed the buckling and postbuckling behavior of delaminated composite laminates with a throu...

The problem of the out of plane buckling of polar orthotropic armuli under uniform internal or external pressure loading has been investigated with the assumption of axisymmetry by numerous researchers. In this paper, the buckling of polar and rectilinearly orthotropic annuli subjected to internal or external pressure loading is examined without restriction to axisymmetric buckling modes. The b...

Harmonic extraction methods for the multiparameter eigenvalue problem willbe presented. These techniques are generalizations of their counterparts forthe standard and generalized eigenvalue problem. The methods aim to ap-proximate interior eigenpairs, generally more accurately than the standardextraction does. The process can be combined with any subspace expansionapproa...

The uniform formulation of dynamic vibration analysis multispan beams is presented by using an efficient domain decomposition method in this paper. Firstly, the structure divided into several equal sections based on method. Next, artificial spring used to simulate complex boundaries and continuity condition beam. Finally, admissible displacement functions are expanded through Jacobi orthogonal ...

In this paper, an approximate formula of the fractional derivatives (Caputo sense) is derived. The proposed formula is based on the generalized Laguerre polynomials. Special attention is given to study the convergence analysis of the presented formula. The spectral Laguerre collocation method is presented for solving a class of fractional optimal control problems (FOCPs). The properties of Lagu...

In this paper, the compressive behavior of composite laminates with multiple through-the-width delaminations is investigated analytically. The analytical method is based on the CLPT theory and its formulation is developed on the basis of the Rayleigh-Ritz approximation technique to analyze the buckling and post-buckling behavior of the delaminated laminates. The method can handle both local buc...