An asymptotic-numerical hybrid method for singularly perturbed system of two-point reaction-diffusion boundary-value problems

This article focuses on the numerical approximate solution of singularly perturbed systems of secondorder reaction-diffusion two-point boundary-value problems for ordinary differential equations. To handle these types of problems, a numerical-asymptotic hybrid method has been used. In this hybrid approach, an efficient asymptotic method, the so-called successive complementary expansion method (SCEM) is employed first, and then a numerical method based on finite differences is applied to approximate the solution of corresponding singularly perturbed reactiondiffusion systems. Two illustrative examples are provided to demonstrate the efficiency, robustness, and easy applicability of the present method with convergence propertiesNo sponso

On an asymptotic-numerical hybrid method for solving singularly perturbed nonlinear delay differential equations

International Conference on Computational and Experimental Science and Engineering (4. : 2017 : Antalya, Turkey)Modelling automatic systems often involves the idea of control because feedback is necessary in order to maintain a stable state. But much of this feedback require a finite time to sense information and react to it. A general way for describing this process is to formulate a delay differential equation (difference-differential equation). Delay differential equations (DDE) are widely used for modelling problems in population dynamics, nonlinear optics, fluid mechanics, mechanical engineering, evolutionary biology and even in modelling of (HIV) infection and human pupil-light reflex. Almost all physical phenomena in nature are modelled using differential equations, and singularly perturbed problems are vital class of these kind of problems. In general, a singular perturbation problem defined as a differential equation that is controlled by a positive small parameter 0<ε≪1 that exists as multiplier to the highest derivative term in the differential equation. As ε tends to zero, the solution of problem exhibits interesting behaviours since the order of the equation reduces. The region where these rapid changes occur is called inner region and the region in which the solution changes mildly is called outer region. In this work, approximate solutions to singularly perturbed two-point nonlinear delay differential equations will be considered. An asymptotic-numerical hybrid method will be proposed to achieve this. This hybrid method consists of an efficient asymptotic method so-called Successive Complementary Expansion Method and an easy-applicable numerical procedure based on finite difference approximations. Numerical results show that the present method is well-suited for solving singularly perturbed nonlinear delay differential equations.No sponso

A micro-macro parareal algorithm: application to singularly perturbed ordinary differential equations

We introduce a micro-macro parareal algorithm for the time-parallel integration of multiscale-in-time systems. The algorithm first computes a cheap, but inaccurate, solution using a coarse propagator (simulating an approximate slow macroscopic model), which is iteratively corrected using a fine-scale propagator (accurately simulating the full microscopic dynamics). This correction is done in parallel over many subintervals, thereby reducing the wall-clock time needed to obtain the solution, compared to the integration of the full microscopic model. We provide a numerical analysis of the algorithm for a prototypical example of a micro-macro model, namely singularly perturbed ordinary differential equations. We show that the computed solution converges to the full microscopic solution (when the parareal iterations proceed) only if special care is taken during the coupling of the microscopic and macroscopic levels of description. The convergence rate depends on the modeling error of the approximate macroscopic model. We illustrate these results with numerical experiments

On an efficient hybrid method for solving singularly perturbed difference-differential equations exhibiting turning layer behavior

International Conference on Computational and Experimental Science and Engineering (4. : 2017 : Antalya, Turkey)Singularly perturbed differential equations that involve positive small perturbation parameter(s) 0<ɛ≪1 as the multiplier to the highest order derivative term are important concepts of mathematical and engineering sciences. As ɛ→0, solution of this kind of problems exhibits rapid changes that we call boundary layer behavior since the order of the equation reduces and it is a well-known fact that classical numerical methods are often insufficient to handle them. One may encounter with singular perturbation problems in almost all science branches. Some application areas may be given as modelling of fluid flow problems at high Reynold numbers, electrical and electronic circuits/systems, nuclear reactors, astrophysics problems, control theory problems, combustion theory, quantum mechanics, signal/image processing, etc. This study concerns with finding approximations to the solution of singularly perturbed two-point boundary value problems that exhibit interior layer (turning point) behavior. To achieve this, an efficient and easy-applicable asymptotic-numerical hybrid method is employed. The asymptotic part of the method is based on Successive Complementary Expansion Method (SCEM) and the numerical part is based on finite difference approximations that applies a Lobatto IIIa formula. As the first stage of present method, an asymptotic approximation to the solution of the singularly perturbed problem is proposed using SCEM with the help of stretching variable transformation and later the resulting two-point boundary value problems that come from the SCEM procedure are solved using the numerical procedure. Numerical experiments show that the present method is well-suited for solving this type of problems.No sponso

Sixth-order compact finite difference method for singularly perturbed 1D reaction diffusion problems

AbstractIn this paper, the sixth-order compact finite difference method is presented for solving singularly perturbed 1D reaction–diffusion problems. The derivative of the given differential equation is replaced by finite difference approximations. Then, the given difference equation is transformed to linear systems of algebraic equations in the form of a three-term recurrence relation, which can easily be solved using a discrete invariant imbedding algorithm. To validate the applicability of the proposed method, some model examples have been solved for different values of the perturbation parameter and mesh size. Both the theoretical error bounds and the numerical rate of convergence have been established for the method. The numerical results presented in the tables and graphs show that the present method approximates the exact solution very well