Hybrid Algorithm for Singularly Perturbed Delay Parabolic Partial Differential Equations

This study aims at constructing a numerical scheme for solving singularly perturbed parabolic delay differential equations. Taylor’s series expansion is applied to approximate the shift term. The obtained result is approximated by using the implicit Euler method in the temporal discretization on a uniform step size with the hybrid numerical scheme consisting of the midpoint upwind method in the outer layer region and the cubic spline method in the inner layer region on a piecewise uniform Shishkin mesh in the spatial discretization. The constructed scheme is an ε−uniformly convergent accuracy of order one. Some test examples are considered to testify the theoretical investigations

A hybrid approximation scheme for 1-D singularly perturbed parabolic convection-diffusion problems

Our study is concerned with a hybrid spectral collocation approach to solving singularly perturbed 1-D parabolic convection-diffusion problems. In this approach, discretization in time is carried out with the help of Taylor series expansions before the spectral based on novel special polynomials is applied to the spatial operator in the time step. A detailed error analysis of the presented technique is conducted with regard to the space variable. The advantages of this attempt are presented through comparison of our results in the model problems obtained by this technique and other existing schemes

A hybrid approximation scheme for 1-D singularly perturbed parabolic convection-diffusion problems

Our study is concerned with a hybrid spectral collocation approach to solving singularly perturbed 1-D parabolic convection-diffusion problems. In this approach, discretization in time is carried out with the help of Taylor series expansions before the spectral based on novel special polynomials is applied to the spatial operator in the time step. A detailed error analysis of the presented technique is conducted with regard to the space variable. The advantages of this attempt are presented through comparison of our results in the model problems obtained by this technique and other existing schemes

Fitted numerical methods for delay differential equations arising in biology

Philosophiae Doctor - PhDFitted Numerical Methods for Delay Di erential Equations Arising in Biology E.B.M. Bashier PhD thesis, Department of Mathematics and Applied Mathematics,Faculty of Natural Sciences, University of the Western Cape. This thesis deals with the design and analysis of tted numerical methods for some delay di erential models that arise in biology. Very often such di erential equations are very complex in nature and hence the well-known standard numerical methods seldom produce reliable numerical solutions to these problems. Ine ciencies of these methods are mostly accumulated due to their dependence on crude step sizes and unrealistic stability conditions.This usually happens because standard numerical methods are initially designed to solve a class of general problems without considering the structure of any individual problems. In this thesis, issues like these are resolved for a set of delay di erential equations. Though the developed approaches are very simplistic in nature, they could solve very complex problems as is shown in di erent chapters.The underlying idea behind the construction of most of the numerical methods in this thesis is to incorporate some of the qualitative features of the solution of the problems into the discrete models. Resulting methods are termed as tted numerical methods. These methods have high stability properties, acceptable (better in many cases) orders of convergence, less computational complexities and they provide reliable solutions with less CPU times as compared to most of the other conventional solvers. The results obtained by these methods are comparable to those found in the literature. The other salient feature of the proposed tted methods is that they are unconditionally stable for most of the problems under consideration.We have compared the performances of our tted numerical methods with well-known software packages, for example, the classical fourth-order Runge-Kutta method, standard nite di erence methods, dde23 (a MATLAB routine) and found that our methods perform much better. Finally, wherever appropriate, we have indicated possible extensions of our approaches to cater for other classes of problems. May 2009

A reproducing kernel method for solving singularly perturbed delay parabolic partial differential equations

In this article, we put forward an efficient method on the foundation of a few reproducing kernel spaces(RK-spaces) and the collocation method to seek the solution of delay parabolic partial differential equations(PDEs) with singular perturbation. The approximated solution  to the equations is formulated and proved the exact solution is uniformly convergent by the solution. Furthermore, the partial differentiation of the approximated solution is also proved the partial derivatives of the exact solution is uniformly convergent by the solution. Meanwhile, we show that the accuracy of our method is in the order of T/n where T is the final time and n is the number of spatial (and time) discretization in the domain of interests. Three numerical examples are put forward to demonstrate the effectiveness of our presented scheme

Dynamical problems and phase transitions

Issued as Financial status report, Technical reports [nos. 1-12], and Final report, Project B-06-68

SciCADE 95: International conference on scientific computation and differential equations

This report consists of abstracts from the conference. Topics include algorithms, computer codes, and numerical solutions for differential equations. Linear and nonlinear as well as boundary-value and initial-value problems are covered. Various applications of these problems are also included

Convergence analysis for forward and inverse problems in singularly perturbed time-dependent reaction-advection-diffusion equations

In this paper, by employing the asymptotic expansion method, we prove the existence and uniqueness of a smoothing solution for a time-dependent nonlinear singularly perturbed partial differential equation (PDE) with a small-scale parameter. As a by-product, we obtain an approximate smooth solution, constructed from a sequence of reduced stationary PDEs with vanished high-order derivative terms. We prove that the accuracy of the constructed approximate solution can be in any order of this small-scale parameter in the whole domain, except a negligible transition layer. Furthermore, based on a simpler link equation between this approximate solution and the source function, we propose an efficient algorithm, called the asymptotic expansion regularization (AER), for solving nonlinear inverse source problems governed by the original PDE. The convergence-rate results of AER are proven, and the a posteriori error estimation of AER is also studied under some a priori assumptions of source functions. Various numerical examples are provided to demonstrate the efficiency of our new approach