• Finite difference approximation for stochastic parabolic partial differential equations

      Yan, Yubin; Patel, Babubhai M. (University of Chester, 2009-09)
      Differential equations, especially partial differential equations (PDES) have wide range of applications in sciences, finance (economics), Engineering and so forth. In last decade, substantial amount of work has been done in studying stochastic partial differential equations (SPDES). A SPDE is a PDE containing a random ‘noise’ term. SPDES have no analytical solutions. Various numerical methods have been developed from time to time and tested for their validity using Matlab program. In this thesis, the author will discuss the finite difference method for stochastic parabolic partial differential equations. Matlab software is used for simulation of the solution of this equation. The main objective of this thesis is to investigate the finite difference approximation of a stochastic parabolic partial differential equation with white noise. The author discusses alternative proof for error bounds using Green function in support of this method.
    • Stabilizing a nonlinear system by using feedback control

      Yan, Yubin; Kareem, Rasaq (University of Chester, 2010)
      This dissertation considers how to stabilize a nonlinear system by using feedback control. To stabilize a nonlinear system, we first need to find the unstable steady state. Then we consider the linearized problem at this steady state and solve the Riccati equation using the linear quadratic regulator (lqr). We then design the feedback controller on the linearized system,. Finally, we apply the feedback controller on the original nonlinear system. We use the forward Euler method, backward Euler method and Trapezoidal method to consider the discretization of the nonlinear system. We design the algorithm and consider two numerical examples of ecological models and verify that the results obtained are in accordance with theoretical results.