We investigate some basic properties of the proper orthogonal decomposition (POD) method as it is applied to data compression and model reduction of finite dimensional nonlinear systems. First we provide an analysis of the errors involved in solving a nonlinear ODE initial value problem using a POD reduced order model. Then we study the effects of small perturbations in the ensemble of data fro...

The present study is an attempt to find a solution for Volterra's Population Model by utilizing Spectral methods based on Rational Christov functions. Volterra's model is a nonlinear integro-differential equation. First, the Volterra's Population Model is converted to a nonlinear ordinary differential equation (ODE), then researchers solve this equation (ODE). The accuracy of method is tested i...

We present a family of numerical implementations of Kato’s ODE propagating global bases of analytically varying invariant subspaces, of which the first-order version is a surprising simple “greedy algorithm” that is both stable and easy to program and the second-order version a relaxation of a first-order scheme of Brin and Zumbrun. The method has application to numerical Evans function computa...

-Three methods for numerical transient analysis of Markov chains, the modified Jensen's method (Jensen's method with steady-state detection of the underlying DTMC and computation of Poisson probabilities using the method of Fox and Glynn [1]), a third-order L-stable implicit Runge-Kutta method, and a second-order L-stable method, TR-BDF2, are compared. These methods are evaluated on the basis o...

In this paper, we introduce SODES (Stepwise Ordinary Differential Equations Solver) a new solver for (ODE). can optionally provide the solution displaying all steps needed to obtain it. This way, is an important tool not only researchers who need solving ODE but also constitutes teaching and learning process of ODE. has been developed using programming with Computer Algebra System (CAS). Specif...

We analyse the most general bosonic supersymmetric solutions of type IIB supergravity whose metrics are warped products of five-dimensional anti-de Sitter space (AdS5) with a five-dimensional Riemannian manifold M5. All fluxes are allowed to be non-vanishing consistent with SO(4, 2) symmetry. We show that the necessary and sufficient conditions can be phrased in terms of a local identity struct...