Abstract: The numerical computation of eigenvalues for large sparse matrices of structural mechanics and aerodynamics is a costly, but unreliable process. Earthquake safety of a building or structure depends heavily on an accurate determination of eigenvalues of the stiffness matrix, but round-off errors and ill-conditioning can destroy a spectrum completely. A lot of good ?eigenvalue-software? is currently available but the reliability of an eigenvalue solver can become questionable in the presence of rounding errors.
Abstract: In this article we show that the conditions imposed by Lagrange and other authors for finding a particular solution for a non-homogeneous linear differential equation of order n greater than 2 are convenient at times, but not always the best, and that a particular solution may be obtained by imposing much more general conditions.
Abstract: one-dimensional Fokker-Plank equation has been investigated with intent to devise the kernel method for it. The known fundamental solution of the FP equation, being non-symmetric, was not suitable for the direct application of the methods developed in phase I of the project [1-2], so a new approach has been designed. The original problem has been reduced to a new, intermediate one - a modified Fokker-Planck equation (modFP) with better properties. That equation allowed subsequent transition to a two-dimensional integral equation for the identification of the Green?s function. The integral equation has been investigated both numerically and via Picard?s method. We have identified a suitable regularized empirical error functional as a source of kernel functions for the solution of modFP and proved existence of a unique minimizer for this functional in the appropriate Reproducing Kernel Hilbert Space (RKHS, [1-2]). The minimization algorithm led to the solution of a matrix equation for a vector of coefficients required for the construction of the approximating modFP kernel function, which has been converted back into the FP environment, thus producing the approximation to the solution of the original problem.
Abstract: The method of Conjugate Gradients is known to converge for symmetric positive definite systems of equations. This paper applies it to non-symmetric and ill-conditioned matrices. In order to facilitate convergence, an approximate inverse is used to precondition the Conjugate Gradient method. This is achieved by applying Newton?s method. Three versions of Newton?s method are introduced to compute the approximate inverse. Convergence of each method is compared. Numerical experimentation is done for some known ?ill-conditioned? problems.
Abstract: Einstein-Maxwell Equations are derived.
Abstract: Eigensystem analysis techniques are applied to finite difference formulations of Euler and Navier-Stokes equations in two dimensions. Spectrums of the resulting implicit difference operators are computed. The convergence and stability properties of the iterative methods are studied by taking into account, the effect of grid geometry, time-step, numerical dissipation, viscosity, boundary conditions and the physics of the underlying flow. The largest eigenvalues are computed by using the Frechet Derivative of the operators and Arnoldi's method. The accuracy of Arnoldi's method is tested by comparing the dominant eigenvalues with the rate of convergence of the iterative method.
Abstract: Eigensystem analysis techniques are applied to finite difference formulations of Navier Stokes Equations in one dimension. Spectrums of the resulting implicit difference operators are computed. The largest eigenvalues are calculated by using a combination of the Frechet Derivative of the operators and Arnoldi's Method. The accuracy of Arnoldi's Method is tested by comparing the rate of convergence of the iterative method with the dominant eigenvalue of the original iteration matrix.
Abstract: Analysis of the convergence history of fluid flow through nozzles with shocks