Analytical solutions of Kolmogorov–Petrovskii–Piskunov equation

Abstract

In the current study, analytical solutions are constructed by applying (1/G') -expansion method to the Kolmogorov–Petrovskii–Piskunov (KPP) equation. Hyperbolic type exact solutions of the KPP equation are presented with the successfully applied method. 3D, 2D and contour graphics are presented by giving special values to the parameters in the solutions obtained. This article explores the applicability and effectiveness of this method on nonlinear evolution equations (NLEEs).

Keywords

• Kolmogorov–Petrovskii–Piskunov equation,traveling wave solutions,exact solution,(1/G')- expansion method

Kolmogorov – Petrovskii – Piskunov denkleminin analitik çözümleri

Abstract

Bu çalışmada, Kolmogorov – Petrovskii –Piskunov (KPP) denkleminin analitik çözümleri (1/G')-açılım yöntemi uygulanarak elde edilmiştir. Başarılı bir şekilde uygulanan yöntem ile KPP denkleminin hiperbolik tipte tam çözümleri sunulmuştur. Elde edilen çözümlerdeki parametrelere özel değerler verilerek 3 boyutlu, 2 boyutlu ve kontur grafikleri sunulmuştur. Bu makalede, bu yöntemin doğrusal olmayan evrim denklemleri (NLEE'ler) üzerindeki uygulanabilirliği ve etkinliği araştırılmaktadır.

Keywords

• Kolmogorov–Petrovskii–Piskunov denklemi,yürüyen dalga çözümleri,tam çözüm,(1/G')-açılım metodu

References

1. Yavuz, M. and Özdemır, N., An Integral Transform Solution for Fractional Advection-Diffusion Problem, Mathematical Studies and Applications, 4-6 October, 442. (2018).
2. Evirgen, F. and Özdemir, N., A fractional order dynamical trajectory approach for optimization problem with HPM, In Fractional Dynamics and Control (pp. 145-155). Springer, New York, NY (2012).
3. Evirgen, F., Analyze the optimal solutions of optimization problems by means of fractional gradient based system using VIM, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 6(2), 75-83, (2016).
4. Yokuş, A. and Kaya, D., Traveling wave solutions of some nonlinear partial differential equations by using extended-expansion method, İstanbul Ticaret Üniversitesi Fen Bilimleri Dergisi, 28, 85-92 (2015).
5. Durur, H., Different types analytic solutions of the (1+ 1)-dimensional resonant nonlinear Schrödinger’s equation using (G′/G)-expansion method, Modern Physics Letters B, 34(03), 2050036, (2020).
6. Baskonus, H. M., Sulaiman, T. A., Bulut, H. and Aktürk, T., Investigations of dark, bright, combined dark-bright optical and other soliton solutions in the complex cubic nonlinear Schrödinger equation with δ-potential, Superlattices and Microstructures, 115, 19-29, (2018).
7. Cattani, C., Sulaiman, T. A., Baskonus, H. M. and Bulut, H., On the soliton solutions to the Nizhnik-Novikov-Veselov and the Drinfel’d-Sokolov systems, Optical and Quantum Electronics, 50(3), 138, (2018).
8. Durur, H., Taşbozan, O., Kurt, A. and Şenol, M. New Wave Solutions of Time Fractional Kadomtsev-Petviashvili Equation Arising In the Evolution of Nonlinear Long Waves of Small Amplitude, Erzincan University Journal of the Institute of Science and Technology, 12(2), 807-815.

9. Yokuş, A. and Durur, H., Complex hyperbolic traveling wave solutions of Kuramoto-Sivashinsky equation using (1/G') expansion method for nonlinear dynamic theory, Journal of Balıkesir University Institute of Science and Technology, 21(2), 590-599, (2019).
10. Yokuş, A. and Kaya, D., Conservation laws and a new expansion method for sixth order Boussinesq equation, In AIP Conference Proceedings (Vol. 1676, No. 1, p. 020062), (2015).
11. Durur, H. and Yokuş, A., (1/G')-Açılım Metodunu Kullanarak Sawada–Kotera Denkleminin Hiperbolik Yürüyen Dalga Çözümleri, Afyon Kocatepe Üniversitesi Fen ve Mühendislik Bilimleri Dergisi, 19(3), 615-619, (2019).
12. Su-Ping, Q. and Li-Xin, T., Modification of the Clarkson–Kruskal Direct Method for a Coupled System, Chinese Physics Letters, 24(10), 2720, (2007).
13. Kumar, D., Seadawy, A. R. and Joardar, A. K., Modified Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology, Chinese journal of physics, 56(1), 75-85, (2018).
14. Kaya, D. and Yokus, A., A numerical comparison of partial solutions in the decomposition method for linear and nonlinear partial differential equations, Mathematics and Computers in Simulation, 60(6), 507-512, (2002).
15. Kaya, D. and Yokus, A., A decomposition method for finding solitary and periodic solutions for a coupled higher-dimensional Burgers equations, Applied Mathematics and Computation, 164(3), 857-864, (2005).
16. Yavuz, M. and Özdemir, N., A quantitative approach to fractional option pricing problems with decomposition series, Konuralp Journal of Mathematics, 6(1), 102-109, (2018).
17. Yokus, A., Kuzu, B. and Demiroğlu, U., Investigation of solitary wave solutions for the (3+1)-dimensional Zakharov–Kuznetsov equation, International Journal of Modern Physics B, 33(29), 1950350, (2019).
18. Darvishi, M., Arbabi, S., Najafi, M. and Wazwaz, A., Traveling wave solutions of a (2+ 1)-dimensional Zakharov-like equation by the first integral method and the tanh method, Optik, 127(16), 6312-6321, (2016).
19. Aziz, I. and Šarler, B., The numerical solution of second-order boundary-value problems by collocation method with the Haar wavelets, Mathematical and Computer Modelling, 52(9-10), 1577-1590, (2010).
20. Kurt, A., Tasbozan, O. and Durur, H., The Exact Solutions of Conformable Fractional Partial Differential Equations Using New Sub Equation Method, Fundamental Journal of Mathematics and Applications, 2(2), 173-179, (2019).
21. Durur, H., Şenol, M., Kurt, A. and Taşbozan, O., Zaman-Kesirli Kadomtsev-Petviashvili Denkleminin Conformable Türev ile Yaklaşık Çözümleri, Erzincan University Journal of the Institute of Science and Technology, 12(2), 796-806, (2019).
22. Rady, A. A., Osman, E. S. and Khalfallah, M., The homogeneous balance method and its application to the Benjamin–Bona–Mahoney (BBM) equation, Applied Mathematics and Computation, 217(4), 1385-1390, (2010).
23. Feng, J., Li, W. and Wan, Q., Using G′ G-expansion method to seek the traveling wave solution of Kolmogorov–Petrovskii–Piskunov equation, Applied Mathematics and Computation, 217(12), 5860-5865, (2011).
24. Zayed, E. M. E. and Ibrahim, S. H., Exact solutions of Kolmogorov-Petrovskii-Piskunov equation using the modified simple equation method, Acta Mathematicae Applicatae Sinica, English Series, 30(3), 749-754, (2014).
25. Hariharan, G., The homotopy analysis method applied to the Kolmogorov–Petrovskii–Piskunov (KPP) and fractional KPP equations, Journal of Mathematical Chemistry, 51(3), 992-1000, (2013).
26. Unal, A. O. On the Kolmogorov–Petrovskii–Piskunov equation, Commun. Fac. Sci. Univ. Ank. Series A, 1, (2013).
27. Rouhparvar, H., Travelling wave solution of the Kolmogorov-Petrovskii-Piskunov equation by the first integral method, Bulletin of the Malaysian Mathematical Sciences Society, 37(1), (2014).
28. Kaya, D., Yokuş, A. and Demiroğlu, U., Comparison of Exact and Numerical Solutions for the Sharma–Tasso–Olver Equation, In Numerical Solutions of Realistic Nonlinear Phenomena, 53-65, (2020).
29. Ahmad, H., Seadawy, A. R., Khan, T. A. and Thounthong, P., Analytic approximate solutions for some nonlinear Parabolic dynamical wave equations, Journal of Taibah University for Science, 14(1), 346-358, (2020).
30. Durur, H., Tasbozan, O. and Kurt, A., New Analytical Solutions of Conformable Time Fractional Bad and Good Modified Boussinesq Equations, Applied Mathematics and Nonlinear Sciences, 5(1), 447-454, (2020).
31. Durur, H., Kurt, A. and Tasbozan, O., New Travelling Wave Solutions for KdV6 Equation Using Sub Equation Method, Applied Mathematics and Nonlinear Sciences, 5(1), 455-460, (2020).
32. Ahmad, H., Rafiq, M., Cesarano, C. and Durur, H., Variational Iteration Algorithm-I with an Auxiliary Parameter for Solving Boundary Value Problems, Earthline Journal of Mathematical Sciences, 3(2), 229-247, (2020).
33. Rezazadeh, H., Kumar, D., Neirameh, A., Eslami, M. and Mirzazadeh, M., Applications of three methods for obtaining optical soliton solutions for the Lakshmanan–Porsezian–Daniel model with Kerr law nonlinearity, Pramana, 94(1), 39. (2020).
34. Gao, W., Silambarasan, R., Baskonus, H. M., Anand, R. V. and Rezazadeh, H., Periodic waves of the non dissipative double dispersive micro strain wave in the micro structured solids, Physica A: Statistical Mechanics and its Applications, 545, 123772, (2020).
35. Avcı, D., Yavuz, M. and Özdemir, N., Fundamental solutions to the Cauchy and Dirich-let problems for a heat conduction equation equipped with the Caputo-Fabrizio differentiation. Heat conduction: methods, applications and research, Nova Science Publishers, 95-107, (2019).
36. Evirgen, F. and Yavuz, M., An alternative approach for nonlinear optimization problem with Caputo-Fabrizio derivative, In ITM Web of Conferences, 22, 01009, (2018).
37. Ismael, H. F., Bulut, H. and Baskonus, H. M., Optical soliton solutions to the Fokas–Lenells equation via sine-Gordon expansion method and (m+(G'/G))-expansion method, Pramana, 94(1), 35, (2020).
38. Yokus A., Solutions of some nonlinear partial differential equations and comparison of their solutions, PhD thesis, Fırat University, (2011).
39. Daghan, D., and Esen, R. K., Exact solutions for two different non-linear partial differential equations, New Trends in Mathematical Sciences, 6(3), 83-93, (2018).
40. Ali, K. K., Yilmazer, R., Yokus, A., and Bulut, H., Analytical solutions for the (3+ 1)-dimensional nonlinear extended quantum Zakharov–Kuznetsov equation in plasma physics, Physica A: Statistical Mechanics and its Applications, 124327, (2020).