New Exact Traveling Wave Solution of Fisher Kolmogorov-Petrovskii-Piskunov Equation for Favorite Genes Spreading by -expansion Method

الموضوعات :

Mohammad Gholami baladezaei ¹ , Morteza Gachpazan ² , Saedeh Foadian ³ , Hossein Mohammad-Pour Kargar ⁴

1 - Department of Applied Mathematics, School of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran
2 - Department of Applied Mathematics, School of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran
3 - Department of Applied Mathematics, Islamic Azad University, Damghan Branch, Damghan, Iran
4 - Department of Biology, Islamic Azad University, Damghan Branch, Damghan, Iran

تاريخ الإرسال : 24 الجمعة , جمادى الأولى, 1442 تاريخ التأكيد : 02 الإثنين , ذو الحجة, 1442 تاريخ الإصدار : 05 الخميس , صفر, 1444

الکلمات المفتاحية: Fisher KPP equation, Traveling wave solutions, Partial differential equation (PDE), expansion method, 2020 MSC: 35Q92, 35C05, 35C10,

ملخص المقالة :

Although designing and developing a mathematical model is extremely important in the mathematics but finding solution for designing model is essential as well. Thus one cannot propose a model without offering its solutions. In the mathematical modeling, there are many models based on nonlinear partial differential equations. In such models, there is no general method for solving any problem. However, numerical methods, approximate methods or analytical methods are available for some problems. It is clear that among the methods for solving a model based on partial differential equations, analytical methods are preferred, but for all problems, it is not possible to provide an exact solution. In this case, some methods can provide a class of solutions. In such methods, techniques that lead to more solutions are more important, but the use of different methods can provide a wide class of solutions. For this reason, various methods are used to find the possible solution of nonlinear partial differential equations. One of these methods is the expansion method. Since one of the well-known equations with wide application in genetics and gene mutation is the Fisher Kolmogorov-Petrovskii-Piskunov (Fisher KPP) equation, we applied expansion method, for finding exact traveling wave solutions which are based on the solutions of Bernoulli ordinary differential equation.

المصادر:

1. El-Hachem M., McCue S.W., Jin W., Du Y., Simpson M.J., 2019. Revisiting the Fisher-Kolmogorov-Petrovsky-Piskunov equation to interpret the spreading-extinction dichotomy. Proc Math Phys Eng Sci. 475(2229), 0378.

2. Loewe L., 2008. Genetic Mutation, Nature Education 1(1), 113.

3. Fisher R.A., 1937. The wave of advance of advantageous genes. Ann Eugen. 7, 355– 369.

4. Bacaer N., 2011. A Short History of Mathematical Population Dynamics, Springer-Verlag.

5. Kolmogorov A., Petrovskii I., Piskunov N., 1937. Study of a Diffusion Equation That Is Related to the Growth of a Quality of Matter and Its Application to a Biological Problem. Moscow University Mathematics Bulletin. 1, 1-26.

 6. Adomian G., 1995. Fisher-Kolmogorov equation. Appl Math Lett. 8(2), 51-52.

 7. Zheng B., 2011. A new Bernoulli Sub-ODE method for constructing traveling wave solutions for two nonlinear equations with any order, U.P.B.Sci. Bull A. 73(3), 85-94.

8. Kudryashov N.A., Zakharchenko A.S., 2014. A note on solutions of the generalized Fisher equation. Applied Mathematics Letters. 32, 53-56.

9. Malfliet W., Hereman W., 1996. The Tanh method: Exact solutions of nonlinear evolution and wave equations. Physica Scripta. 54, 563-568.

10. Tashin Khan A., Naher H., 2018. Solitons and periodic solutions of the Fisher equation with nonlinear ordinary differential equation as auxiliary equation. American Journal of Applied Mathematics and Statistics. 6(6), 244-252.

11. Taha W.M., Noorani M.S.M., 2014. Application of the -expansion method for the generalized Fisher’s equation and modified equal width equation. Journal of the Association of Arab Universities for Basic and Applied Sciences. 15, 82–89.

12. Zayed E.M.E., Amer Y.A., 2015. Exact solutions for the nonlinear KPP equation by using the Riccati equation method combined with the -expansion method. Scientific Research and Essays. 10(3), 86-96.

13. Yokus A., 2011. Solutions of some nonlinear partial differential equations and comparison of their solutions, Ph.D. Thesis, Firat Uni- versity, Elazig.

14. Zheng B., 2012. Application of a generalized Bernoulli Sub-ODE method for finding traveling solution of some nonlinear equations, Wseas Transactions on Mathematics. 11(7), 618-626.

15. Macías-Díaza J.E., Medina-Ramírezb I., A.Puric A., 2009. Numerical treatment of the spherically symmetric solutions of a generalized Fisher–Kolmogorov–Petrovsky–Piscounovequation. Journal of Computational and Applied Mathematics. 231, 851–868.

16. Unal A.O., 2013. On the Kolmogorov-Petrovskii-Piskunov equation. Commun Fac Sci Univ Ank Series A1. 62(1), 1-10.

17. Loyinmi A.C., Akinfe T.K., 2020. Exact solutions to the family of Fisher’s reaction-diffusion equation using Elzaki homotopy transformation perturbation method. Engineering Reports. 2, 1-32.