http://hdl.handle.net/10034/31839 Sat, 25 May 2013 10:10:02 GMT 2013-05-25T10:10:02Z http://hdl.handle.net/10034/231920 Title: A finite element method for time fractional partial differential equations Authors: Atallah, Samia A Abstract: Fractional differential equations, particularly fractional partial differential equations (FPDEs) have many applications in areas such as diffusion processes, electromagnetics, electrochemistry, material science and turbulent flow. There are lots of work for the existence and uniqueness of the solutions for fractional partial differential equations. In recent years, people start to consider the numerical methods for solving fractional partial differential equation. The numerical methods include finite difference method, finite element method and the spectral method. In this dissertation, we mainly consider the finite element method, for the time fractional partial differential equation. We consider both time discretization and space discretization. We obtain the optimal error estimates both in time and space. The numerical examples demonstrate that the numerical results are consistent with the theoretical results. Thu, 01 Sep 2011 00:00:00 GMT http://hdl.handle.net/10034/231920 2011-09-01T00:00:00Z http://hdl.handle.net/10034/128957 Title: Stabilizing a nonlinear system by using feedback control Authors: Kareem, Rasaq A Abstract: 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. Fri, 01 Jan 2010 00:00:00 GMT http://hdl.handle.net/10034/128957 2010-01-01T00:00:00Z http://hdl.handle.net/10034/124826 Title: Analytical and numerical solution methods for integral equations using Maple symbolic algebraic package Authors: Little, Alexander John Martyn Tue, 01 Dec 1998 00:00:00 GMT http://hdl.handle.net/10034/124826 1998-12-01T00:00:00Z http://hdl.handle.net/10034/123074 Title: Tensor decomposition and its applications Authors: Tock, Daniel Abstract: This dissertation reviews classical vector - tensor analysis, building up to the necessary techniques required to decompose a tensor into a tensor train and to reconstruct it back into the original tensor with minimal error. The tensor train decomposition decomposes a tensor of dimensionality d into a train of d third order tensors, whose sizes are dependent upon the rank and chosen error bound. I will be reviewing the required operations of matricization, tensor - matrix, vector and tensor multiplication to be able to compute this decomposition. I then move onto analysing the tensor train decomposition by ap-plying it to different types of tensor, of differing dimensionality with a variety of accuracy bounds to investigate their influence on the time taken to complete the decomposition and the final absolute error. Finally I explore a method to compute a d-dimensional integration from the tensor train, which will allow larger tensors to be integrated with the memory required dramatically reduced after the tensor is decomposed. I will be applying this technique to two tensors with different ranks and compare the efficiency and accuracy of integrating directly from the tensor to that of the tensor train decomposition. Wed, 01 Sep 2010 00:00:00 GMT http://hdl.handle.net/10034/123074 2010-09-01T00:00:00Z