Comparison of the Nodal Integral Method and Nonstandard Finite-Difference Schemes for the Fisher Equation

Uddin, Rizwan

doi:10.1137/s1064827597325463

Cited by 29 publications

(12 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Otherwise, a rational function approximation to the solution of (3.25) is done. The motivations for presenting this method are the following (i) The numerical methods; finite difference, finite elements ,· · · etc for treating reaction diffusion equations [37,30,38] work with initial-boundary value problems only. Numerical solutions may not be stable for large value of time .…”

Section: Formulation Of the Methodsmentioning

confidence: 99%

See 1 more Smart Citation

Criterion of Existence of Realistic Permanent Travelling Wave Solutions in Reaction Diffusion Systems

Abdel‐Gawad¹,

El-Shrae²,

Saad³

2017

Communications in Numerical Analysis

View full text Add to dashboard Cite

In this paper, we aim to postulate the conditions of existence of permanent travelling wave solution in reaction diffusion systems subjected to initial, boundary or initial-boundary conditions. In a concomitant way we present for a method that allows us to treat initial, boundary or initial-boundary value problems. It is based on finding approximate analytical solutions starting from the formal exact ones. That is by constructing the Picard iterative sequence of solutions and proving a theorem for,the uniform convergence of this sequence. This sequence is then truncated at first, second or higher approximations. The relative error estimate between approximate analytical solutions and some known exact solutions are of the same order as the error between numerical solutions and the exact ones. We should mention that numerical schemes treat only initial-boundary value problem. It is found that the necessary conditions for the presence of travelling wave in the form of u = u(x − ct) is that the initial (or boundary) conditions at the extreme points of domain of definition of the problem have to be different. It is also shown that the sufficient condition is the presence of an advection term, with coefficient which is a constant or a function in the dependent variable, in the reaction diffusion equation.

show abstract

Section: Formulation Of the Methodsmentioning

confidence: 99%

“…We develop an approach similar to that proposed in [26,27,30,31,28,29] to find approximate analytical solutions to the equations ( 1.1) for the initial value problems (IVP). The approach is based on finding the formal exact solution for the IVP.…”

Section: Formulation Of the Methodsmentioning

confidence: 99%

Criterion of Existence of Realistic Permanent Travelling Wave Solutions in Reaction Diffusion Systems

Abdel‐Gawad¹,

El-Shrae²,

Saad³

2017

Communications in Numerical Analysis

View full text Add to dashboard Cite

show abstract

“…Our contribution is to consider the dispersion relation through the construction of finite difference schemes. Fortunately, exponentially fitted methods and nonstandard finite differences provide us with a very useful tool: denominators in terms of non-polynomial functions [17,18]. We propose forward time derivative and the second space derivative in (2) to obtain an explicit EF scheme.…”

Section: Pure Diffusionmentioning

confidence: 99%

“…Therefore, an implicit version of EF finite differences should be used to deal with stiffness. By employing backward time difference in (2), one obtains an implicit version of (17). Local error based parameter selection strategy gives…”

Section: Reaction-diffusionmentioning

confidence: 99%

Exponentially Fitted Finite Difference Schemes for Reaction-Diffusion Equations

Erdoğan

Akarbulut

Tan

2016

MCA

View full text Add to dashboard Cite

Abstract:The purpose of this work is to introduce a new kind of finite difference formulation inspired from Fourier analysis, for reaction-diffusion equations. Compared to classical schemes, the proposed scheme is much more accurate and has interesting stability properties. Convergence properties and stability of the scheme are discussed. Numerical examples are provided to show better performance of the method, compared with other existing methods in the literature.

show abstract

“…For numerical research on Eq. (1), we can refer to works of Al-Khaled [3] by the Sinc collocation method, Uddin [40] using the nodal integral scheme, Chen et al [11] by nonstandard finite difference methods, Dehghan and Fakhar-Izadi [13] using pseudospectral methods, Bastani and Salkuyeh [4] by a sixth-order compact finite difference (CFD6) scheme, Gorder and Vajravelu [23] by a variational technique, Bhrawy [9] by a Jacobi-Gauss-Lobatto collocation method, Li and Ding [28] by a higher order finite difference method and Ramos [34] using exponential methods.…”

Section: Introductionmentioning

confidence: 99%

Stability and convergence of radial basis function finite difference method for the numerical solution of the reaction–diffusion equations

Golbabai

Nikpour

2015

Applied Mathematics and Computation

View full text Add to dashboard Cite

Comparison of the Nodal Integral Method and Nonstandard Finite-Difference Schemes for the Fisher Equation

Cited by 29 publications

References 17 publications

Criterion of Existence of Realistic Permanent Travelling Wave Solutions in Reaction Diffusion Systems

Criterion of Existence of Realistic Permanent Travelling Wave Solutions in Reaction Diffusion Systems

Exponentially Fitted Finite Difference Schemes for Reaction-Diffusion Equations

Stability and convergence of radial basis function finite difference method for the numerical solution of the reaction–diffusion equations

Contact Info

Product

Resources

About