This paper uses a spectral algorithm for the numerical solution of unsteady periodic problems. This algorithm which is based on discrete Fourier transforms uses a Fourier representation for periodic conditions of problems and hence solves the periodic state directly, without resolving transients because the discrete Fourier transforms have the periodic property to confirm physics of flow. The algorithm has been proposed for the fast and efficient computation of periodic flows. The algorithm has been validated with Stokes’ second problem as a linear problem and Burgers' equation as a nonlinear problem. The same numerical results are compared with an analytical solution, second-order Backward Difference Formula (BDF) and finite difference method (FDM) results. By enforcing periodicity by using Fourier representation that has a spectral accuracy, a tremendous increase of accuracy has been obtained compared to using the conventional numerical methods like BDF and FDM. Results verify the small number of time intervals per oscillating cycle required to capture the flow physics in Stokes’ second problem. Moreover, they show that a small number of points in a computational grid are required to capture the flow physics in Burgers' equation. Furthermore, this algorithm is more able than a finite difference method to capture shock.

The solution of the financial application, be it asset pricing, portfolio allocation or risk management, relies on the simulation of discretized versions of the stochastic differential equations(SDEs). The simplest way to confront SDEs in numerical situations is to discretize them and use monte carlo simulation. The Euler scheme is most often used for discretization of SDEs. This discretization involves an approximation error. In this topic at the first we recall an introduction to SDEs and Monte carlo simulation. Then, we study the asymptotic error distribution of Euler approximations of solutions of SDEs. We also study the error distribution associated with a Doss transformation of the state variables. Convergence results for Euler schemes with and without doss transformation and the comparison of them with Milshtein scheme are presented at the end.

In this paper, a new method is suggested for image hiding using chaos signals. The combination of movement of pixels and adjusting quantity of gray level is simultaneously used in this method. The process of moving pixels is done by an order which is taken from logistic map and for adjusting gray level the order of standing of bits of pixel's quantity of gray level is changed by the means of chaos signal as well. Experimental results show that this method has a proper efficiency for prevalent terms. For example the amount of entropy obtained by this method is around 7.9949 that is so near to 8 its ideal amount.