Travelling Wave Solution of the Fisher-Kolmogorov Equation with Non-Linear Diffusion

Shakeel, Muhammad

doi:10.4236/am.2013.48a021

Cited by 8 publications

(6 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Figures 1-4 show some novel analytical wave solutions with different values to the above-mentioned parameters. Comparing our solutions with those that have been obtained in a previously published paper [17,41] shows our solutions are completely different from those that have been evaluated in [41]. Still, some of our solutions match the obtained solutions in [17] when α = 1, where in that paper, the authors in that paper studied the fractional form of the considered model.…”

Section: Results' Interpretationcontrasting

confidence: 56%

Multiple Novels and Accurate Traveling Wave and Numerical Solutions of the (2+1) Dimensional Fisher-Kolmogorov- Petrovskii-Piskunov Equation

Khater

Alabdali

2021

Mathematics

View full text Add to dashboard Cite

The analytical and numerical solutions of the (2+1) dimensional, Fisher-Kolmogorov-Petrovskii-Piskunov ((2+1) D-Fisher-KPP) model are investigated by employing the modified direct algebraic (MDA), modified Kudryashov (MKud.), and trigonometric-quantic B-spline (TQBS) schemes. This model, which arises in population genetics and nematic liquid crystals, describes the reaction–diffusion system by traveling waves in population genetics and the propagation of domain walls, pattern formation in bi-stable systems, and nematic liquid crystals. Many novel analytical solutions are constructed. These solutions are used to evaluate the requested numerical technique’s conditions. The numerical solutions of the considered model are studied, and the absolute value of error between analytical and numerical is calculated to demonstrate the matching between both solutions. Some figures are represented to explain the obtained analytical solutions and the match between analytical and numerical results. The used schemes’ performance shows their effectiveness and power and their ability to handle many nonlinear evolution equations.

show abstract

Section: Results' Interpretationcontrasting

confidence: 56%

Multiple Novels and Accurate Traveling Wave and Numerical Solutions of the (2+1) Dimensional Fisher-Kolmogorov- Petrovskii-Piskunov Equation

Khater

Alabdali

2021

Mathematics

View full text Add to dashboard Cite

show abstract

“…The method is simple and convenient to use, but, unfortunately, the accuracy of the relevant approximate solutions has never been evaluated or discussed. To solve the Fisher-KPP equation in a traveling wave study, homotopy analysis was used (for more details, see, e.g., [22,23]).…”

Section: Introductionmentioning

confidence: 99%

An application of the Maslov complex germ method to the one-dimensional nonlocal Fisher–KPP equation

Шаповалов

Trifonov

2018

Int. J. Geom. Methods Mod. Phys.

View full text Add to dashboard Cite

A semiclassical approximation approach based on the Maslov complex germ method is considered in detail for the 1D nonlocal Fisher-Kolmogorov-Petrovskii-Piskunov equation under the supposition of weak diffusion. In terms of the semiclassical formalism developed, the original nonlinear equation is reduced to an associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation with a given accuracy of the asymptotic parameter. The solutions of the nonlinear equation are constructed from the solutions of both the linear equation and the algebraic equations. The solutions of the linear problem are found with the use of symmetry operators. A countable family of the leading terms of the semiclassical asymptotics is constructed in explicit form.The semiclassical asymptotics are valid by construction in a finite time interval. We construct asymptotics which are different from the semiclassical ones and can describe evolution of the solutions of the Fisher-Kolmogorov-Petrovskii-Piskunov equation at large times. In the example considered, an initial unimodal distribution becomes multimodal, which can be treated as an example of a space structure.

show abstract

“…By using infinitesimal methods, for this equation, we have obtained the group of equivalence transformations and the Lie group classification. The exponential form of diffusion used in this model produces the similar behavior as cell proliferation, and there is a great interest for the exponential form of diffusion in modeling the cell growth in vitro tissue engineering. Due to this fact, after having computed the one‐dimensional optimal systems of subalgebras, we have performed the corresponding similarity transformations for Equation .…”

Section: Discussionmentioning

confidence: 99%

“…Shakeel modeled cell growth in a bioreactor subject to uniform nutrient concentration; the diffusion is a function of cell density such that it is high in highly cell populated areas, and it is small in the regions of fewer cells. This exponential form of nonlinear diffusion is such that it produces similar behavior to cell proliferation.…”

Section: Introductionmentioning

confidence: 99%

Symmetry analysis for a Fisher equation with exponential diffusion

Gandarias

Rosa

Tracinà

2018

Math Methods in App Sciences

View full text Add to dashboard Cite

In this paper, we consider a generalized Fisher equation with exponential diffusion from the point of view of the theory of symmetry reductions in partial differential equations. The generalized Fisher‐type equation arises in the theory of population dynamics. These types of equations have appeared in many fields of study such as in the reaction‐diffusion equations, in heat transfer problems, in biology, and in chemical kinetics. By using the symmetry classification, simplified by equivalence transformations, for a special family of Fisher equations, all the reductions are derived from the optimal system of subalgebras and symmetry reductions are used to obtain exact solutions.

show abstract

Travelling Wave Solution of the Fisher-Kolmogorov Equation with Non-Linear Diffusion

Cited by 8 publications

References 21 publications

Multiple Novels and Accurate Traveling Wave and Numerical Solutions of the (2+1) Dimensional Fisher-Kolmogorov- Petrovskii-Piskunov Equation

Multiple Novels and Accurate Traveling Wave and Numerical Solutions of the (2+1) Dimensional Fisher-Kolmogorov- Petrovskii-Piskunov Equation

An application of the Maslov complex germ method to the one-dimensional nonlocal Fisher–KPP equation

Symmetry analysis for a Fisher equation with exponential diffusion

Contact Info

Product

Resources

About