Kailash C. Patidar

A survey of numerical techniques for solving singularly perturbed ordinary differential equations

This survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of the ideas and methods of singular perturbation theory. Starting from Prandtls work a large amount of work has been done in the area of singular perturbations. This paper limits its coverage to some standard singular perturbation models considered by various workers and the numerical methods developed by numerous researchers after 1984-2000. The work done in this area during the period 1905-1984 has already been surveyed by the first author of this paper, see [Appl. Math. Comput. 30 (1989) 223] for details. Due to the space constraints we have covered only singularly perturbed one-dimensional problems.

On the use of nonstandard finite difference methods

Many real life problems are modelled by differential equations, for which analytical solutions are not always easy to find. One of the most difficult problems is how to solve these differential equations efficiently. Several researchers have tried to do this in various different ways (e.g. via Finite Element Methods, Standard Finite Difference Methods, Spline Approximation Methods, etc.). In recent years, to get reliable results with less effort, researchers have applied nonstandard finite difference methods (NSFDMs) and obtained competitive results to those obtained with other methods. In this survey article, the author tries to provide as much stimulating information as available regarding these NSFDMs to the researchers, which will be helpful for them as the research proceeds in this direction. While the author made the utmost efforts to include whatever he could, he would like to apologize if there are any omissions which are totally unintentional.

Singularly perturbed problems in partial differential equations: a survey

This survey paper contains a surprisingly large amount of material on singularly perturbed partial differential equations and indeed can serve as an introduction to some of the ideas and methods of the singular perturbation theory. Starting from Prandtls work a large amount of work has been done in the area of singular perturbations. This paper limits its coverage to some standard singular perturbation models considered by various workers and the methods developed by numerous researchers after 1980-2000. In this review we have covered singularly perturbed partial differential equations. About ODEs the survey has already been done by us [see M.K. Kadalbajoo, K.C. Patidar, Appl. Math. Comput. 130 (2002) 457-510].

Higher-order time-stepping methods for time-dependent reaction–diffusion equations arising in biology☆

Abstract This paper demonstrates the use of higher order methods to solve some time-dependent stiff PDEs. In the past, the most popular numerical methods for solving system of reaction–diffusion equations was based on the combination of low order finite difference method with low order time-stepping method. We extend in this report the compatibility of fourth-order finite difference scheme (in space) coupled with fourth-order time-stepping methods such as IMEXLM4, IMEXPC4, IMEXRK4 and ETDRK4-B (in time), for direct integration of reaction–diffusion equations in one space dimension. Some interesting numerical anomaly phenomenons associated with steady state solutions of the examples chosen from the literature are well presented to address the naturally arising points and queries. Our findings have led to the understanding of pattern formation such as spiral waves and patchy structures as well as some spatio-temporal dynamical structures.

ε-Uniformly convergent non-standard finite difference methods for singularly perturbed differential difference equations with small delay

Abstract Non-standard finite difference methods (NSFDMs), now-a-days, are playing an important role in solving the real life problems governed by ODEs and/or by PDEs. Many differential models of sciences and engineerings for which the existing methodologies do not give reliable results, these NSFDMs are solving them competitively. To this end, in this paper we consider, second order, linear, singularly perturbed differential difference equations. Using the second of the five non-standard modeling rules of Mickens [R.E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994], the new finite difference methods are obtained for the particular cases of these problems. This rule suggests us to replace the denominator function of the classical second order derivative with a positive function derived systematically in such a way that it captures most of the significant properties of the governing differential equation(s). Both theoretically and numerically, we show that these NSFDMs are e-uniformly convergent.

Numerical simulations of multicomponent ecological models with adaptive methods

BackgroundThe study of dynamic relationship between a multi-species models has gained a huge amount of scientific interest over the years and will continue to maintain its dominance in both ecology and mathematical ecology in the years to come due to its practical relevance and universal existence. Some of its emergence phenomena include spatiotemporal patterns, oscillating solutions, multiple steady states and spatial pattern formation.MethodsMany time-dependent partial differential equations are found combining low-order nonlinear with higher-order linear terms. In attempt to obtain a reliable results of such problems, it is desirable to use higher-order methods in both space and time. Most computations heretofore are restricted to second order in time due to some difficulties introduced by the combination of stiffness and nonlinearity. Hence, the dynamics of a reaction-diffusion models considered in this paper permit the use of two classic mathematical ideas. As a result, we introduce higher order finite difference approximation for the spatial discretization, and advance the resulting system of ODE with a family of exponential time differencing schemes. We present the stability properties of these methods along with the extensive numerical simulations for a number of multi-species models.ResultsWhen the diffusivity is small many of the models considered in this paper are found to exhibit a form of localized spatiotemporal patterns. Such patterns are correctly captured in the local analysis of the model equations. An extended 2D results that are in agreement with Turing typical patterns such as stripes and spots, as well as irregular snakelike structures are presented. We finally show that the designed schemes are dynamically consistent.ConclusionThe dynamic complexities of some ecological models are studied by considering their linear stability analysis. Based on the choices of parameters in transforming the system into a dimensionless form, we were able to obtain a well-balanced system that is biologically meaningful. The accuracy and reliability of the schemes are justified via the computational results presented for each of the diffusive multi-species models.

Numerical Solution of Singular Patterns in One-dimensional Gray-Scott-like Models

Abstract In this paper, numerical simulations of coupled one-dimensional Gray-Scott model for pulse splitting process, self-replicating patterns and unsteady oscillatory fronts associated with autocatalytic reaction-diffusion equations as well as homoclinic stripe patterns, self-replicating pulse and other chaotic dynamics in Gierer-Meinhardt equations [12] are investigated. Our major approach is the use of higher order exponential time differencing Runge-Kutta (ETDRK) scheme that was earlier proposed by Cox and Matthews [5], which was later presented as a result of instability in a modified form by Krogstad [24] to solve stiff semi-linear problems. The semi-linear problems under consideration in this context are split into linear, which harbors the stiffest part of the dynamical system and nonlinear part that varies slowly than the linear part. For the spatial discretization, we employ higher-order symmetric finite difference scheme and solve the resulting system of ODEs with higher-order ETDRK method. Numerical examples are given to illustrate the accuracy and implementation of the methods, results and error comparisons with other standard schemes are well presented.

Numerical solution of singularly perturbed two-point boundary value problems by spline in tension

Some difference schemes for singularly perturbed two-point boundary value problems are derived using spline in tension. These schemes are second-order accurate. Numerical examples are given in support of the theoretical results.

Comparison of some recent numerical methods for initial-value problems for stiff ordinary differential equations

We consider the combustion equation as one of the candidates from the class of stiff ordinary differential equations. A solution over a length of time that is inversely proportional to @d>0 (where @d>0 is a small disturbance of the pre-ignition state) is sought. This problem has a transient at the midpoint of the integration interval. The solution changes from being non-stiff to stiff, and afterwards becomes non-stiff again. We provide its asymptotic and numerical solution obtained via a variety of methods. Comparisons are made for the numerical results which we obtain with the MATLAB ode solvers (ode45, ode15s and ode23s) and some nonstandard finite difference methods. Results corresponding to standard finite difference method are also presented. Furthermore, the discussion on these approaches along with the others, provides several open problems for new and young researchers.

High order parameter uniform numerical method for singular perturbation problems

In this paper, we systematically describe how to derive a method of higher order for the numerical solution of singularly perturbed ordinary differential equations. First we apply this idea to derive a fourth-order method for a self-adjoint singularly perturbed two point boundary value problem. This method is uniformly convergent on a piecewise uniform mesh of Shishkin type. After we have developed and analyzed a fourth-order method, we explain with appropriate details, how can one obtain the methods of order higher than four which looks straightforward but has not been seen in the literature so far. Besides these, the fourth-order e-uniformity in the theoretical estimate has been justified by some numerical experiments.