Numerical solution of a boundary value problem including both delay and boundary layer

Difference method on a piecewise uniform mesh of Shishkin type, for a singularly perturbed boundary-value problem for a nonlinear second order delay differential equation is analyzed. Also, the method is proved that it gives essentially first order parameter-uniform convergence in the discrete maximum norm. Furthermore, numerical results are presented in support of the theory

A posteriori mesh method for a system of singularly perturbed initial value problems

A system of singularly perturbed initial value problems with weak constrained conditions on the coefficients is considered. First the system of second-order singularly perturbed problems is transformed into a system of first-order singularly perturbed problems with integral terms, which facilitates the subsequent stability and a posteriori error analyses. Then a hybrid difference method with the use of interpolating quadrature rules is utilized to approximate the transformed system. Next a posteriori error analysis for the discretization scheme on an arbitrary mesh is presented. A solution-adaptive algorithm based on a posteriori error estimation is devised to generate a posteriori mesh and obtain approximation solution. Finally numerical experiments show a uniform convergence behavior of second-order for the scheme, which improves the previous results and achieves the optimal convergence order under the given discrete scheme

A nonstandard fitted operator finite difference method for two-parameter singularly perturbed time-delay parabolic problems

In this article, a class of singularly perturbed time-delay two-parameter second-order parabolic problems are considered. The presence of the two small parameters attached to the derivatives causes the solution of the given problem to exhibit boundary layer(s). We have developed a uniformly convergent nonstandard fitted operator finite difference method (NSFOFDM) to solve the considered problems. The Crank-Nicolson scheme with a uniform mesh is used for the discretization of the time derivative, while for the spatial discretization, we have applied a fitted operator finite difference method following the nonstandard methodology of Mickens. Moreover, the solution bounds of the governing equation are shown by asymptotic analysis. The convergence of the proposed numerical scheme is investigated using truncation error and the barrier function approach. The study shows that our proposed scheme is uniformly convergent independent of the perturbation parameters, quadratically in time, and linearly in space. Numerical experiments are carried out, and the results are presented in tables and graphically

On the design and implementation of a hybrid numerical method for singularly perturbed two-point boundary value problems

>Magister Scientiae - MScWith the development of technology seen in the last few decades, numerous solvers have been developed to provide adequate solutions to the problems that model different aspects of science and engineering. Quite often, these solvers are tailor-made for specific classes of problems. Therefore, more of such must be developed to accompany the growing need for mathematical models that help in the understanding of the contemporary world. This thesis treats two point boundary value singularly perturbed problems. The solution to this type of problem undergoes steep changes in narrow regions (called boundary or internal layer regions) thus rendering the classical numerical procedures inappropriate. To this end, robust numerical methods such as finite difference methods, in particular fitted mesh and fitted operator methods have extensively been used. While the former consists of transforming the continuous problem into a discrete one on a non-uniform mesh, the latter involves a special discretisation of the problem on a uniform mesh and are known to be more accurate. Both classes of methods are suitably designed to accommodate the rapid change(s) in the solution. Quite often, finite difference methods on piece-wise uniform meshes (of Shishkin-type) are adopted. However, methods based on such non-uniform meshes, though layer-resolving, are not easily extendable to higher dimensions. This work aims at investigating the possibility of capitalising on the advantages of both fitted mesh and fitted operator methods. Theoretical results are confirmed by extensive numerical simulations

Uniform numerical approximation for parameter dependent singularly perturbed problem with integral boundary condition

WOS: 000441460300026In this paper, a parameter-uniform numerical method for a parameterized singularly perturbed ordinary differential equation containing integral boundary condition is studied. Asymptotic estimates on the solution and its derivatives are derived. A numerical algorithm based on upwind finite difference operator and an appropriate piecewise uniform mesh is constructed. Parameter-uniform error estimate for the numerical solution is established. Numerical results are presented, which illustrate the theoretical results

Robust computational methods for two-parameter singular perturbation problems

Magister Scientiae - MScThis thesis is concerned with singularly perturbed two-parameter problems. We study a tted nite difference method as applied on two different meshes namely a piecewise mesh (of Shishkin type) and a graded mesh (of Bakhvalov type) as well as a tted operator nite di erence method. We notice that results on Bakhvalov mesh are better than those on Shishkin mesh. However, piecewise uniform meshes provide a simpler platform for analysis and computations. Fitted operator methods are even simpler in these regards due to the ease of operating on uniform meshes. Richardson extrapolation is applied on one of the tted mesh nite di erence method (those based on Shishkin mesh) as well as on the tted operator nite di erence method in order to improve the accuracy and/or the order of convergence. This is our main contribution to this eld and in fact we have achieved very good results after extrapolation on the tted operator finitete difference method. Extensive numerical computations are carried out on to confirm the theoretical results.South Afric

Estimation and control of non-linear and hybrid systems with applications to air-to-air guidance

Issued as Progress report, and Final report, Project no. E-21-67

Health Monitoring of Nonlinear Systems with Application to Gas Turbine Engines

Health monitoring and prognosis of nonlinear systems is mainly concerned with system health tracking and its evolution prediction to future time horizons. Estimation and prediction schemes constitute as principal components of any health monitoring framework. In this thesis, the main focus is on development of novel health monitoring techniques for nonlinear dynamical systems by utilizing model-based and hybrid prognosis and health monitoring approaches. First, given the fact that particle filters (PF) are known as a powerful tool for performing state and parameter estimation of nonlinear dynamical systems, a novel dual estimation methodology is developed for both time-varying parameters and states of a nonlinear stochastic system based on the prediction error (PE) concept and the particle filtering scheme. Estimation of system parameters along with the states generate an updated model that can be used for a long-term prediction problem. Next, an improved particle filtering-based methodology is developed to address the prediction step within the developed health monitoring framework. In this method, an observation forecasting scheme is developed to extend the system observation profiles (as time-series) to future time horizons. Particles are then propagated to future time instants according to a resampling algorithm in the prediction step. The uncertainty in the long-term prediction of the system states and parameters are managed by utilizing dynamic linear models (DLM) for development of an observation forecasting scheme. A novel hybrid architecture is then proposed to develop prognosis and health monitoring methodologies for nonlinear systems by integration of model-based and computationally intelligent-based techniques. Our proposed hybrid health monitoring methodology is constructed based on a framework that is not dependent on the structure of the neural network model utilized in the implementation of the observation forecasting scheme. Moreover, changing the neural network model structure in this framework does not significantly affect the prediction accuracy of the entire health prediction algorithm. Finally, a method for formulation of health monitoring problems of dynamical systems through a two-time scale decomposition is introduced. For this methodology the system dynamical equations as well as the affected damage model, are investigated in the two-time scale system health estimation and prediction steps. A two-time scale filtering approach is developed based on the ensemble Kalman filtering (EnKF) methodology by taking advantage of the model reduction concept. The performance of the proposed two-time scale ensemble Kalman filters is shown to be more accurate and less computationally intensive as compared to the well-known particle filtering approach for this class of nonlinear systems. All of our developed methods have been applied for health monitoring and prognosis of a gas turbine engine when it is affected by various degradation damages. Extensive comparative studies are also conducted to validate and demonstrate the advantages and capabilities of our proposed frameworks and methodologies