Exact Solutions and Linearization of Modified Emden Equation

Year 2020, Volume: 4 Issue: 2, 273 - 282, 30.12.2020

Özlem Orhan , Bahar Milani , Muhammed Milani

Abstract

In this study, we present that the modified Emden equation has invariant solutions for arbitrary coefficients α and β. Firstly, we demonstrated that modified Emden equation can be linearized. The symmetries of the equation can be derived using a feasible algorithm after this equation is linearized. The exact solutions of the equation are found using a new algorithm and with helping these symmetries. Additionally, finding solutions are classified with respect to the physical meaning of arbitrary coefficients. Finally, all graphics of solutions have been presented with Mathematica and Matlab.

Keywords

Feasible algorithm, Differential equations, Symmetries, Modified Emden equation

Modife Edilmiş Emden Denkleminin Tam Çözümleri ve Lineerleştirilmesi

Abstract

Bu çalışmada, keyfi α ve β katsayılarını içeren modife edilmiş Emden denklemi ele alınmıştır. Öncelikle, modife edilmiş Emden denkleminin lineerleştirilebildiği gösterilmiştir. Bu denklemi lineerleştirdikten sonra, elverişli bir algoritma kullanılarak denklemin simetrileri elde edilmiştir. Bu elde edilen simetriler ve yeni algoritma kullanılarak modife edilmiş Emden denkleminin kesin çözümleri bulunmuştur. Ek olarak, bulunan çözümler içerdikleri keyfi kaysayıların fiziksel anlamlarına göre sınıflandırılmıştır. Son olarak, bulunan çözümler kullanılarak bu çözümlerin zamana göre grafikleri Mathematica ve Matlab programları ile elde edilmiştir.

Keywords

Diferansiyel denklemler, Elverişli Algoritma, Simetriler, Modife edilmiş Emden denklemi

References

• Bluman, S. and G.W. Kumei. 1989. Symmetries and Differential Equations, Springer-Verlag, New York.
• Chandrasekhar, S. 1957. An Introduction to the Study of Stellar Structure (New York: Dover); Dixon J M and Tuszynski J A 1990 Phys. Rev. A 41 4166.
• Chandrasekhar, V.K., M. Senthilvelan, and M. Lakshmanan. 2005. On the complete integrability and linearization of certain second-order nonlinear ordinary differential equations, Proceedings of the Royal Society A, 461, 2451-2476.
• Chandrasekhar V.K., M. Senthilvelan, and M. Lakshmanan. 2007. On the General Solutions for the Modified Emden Type Equation, Journal of Physics A Mathematical and Theoretical, 40(18), 4717.
• Duarte L.G.S., I.C. Moreira, and F.C. Santos. 1994. “Linearization under non-point transformation”, J. Phys. A:Math. Gen, vol. 27, pp. 739-743.
• Hosseinpour S., M. Alavi Milani, and H. Pehlivan. 2018. Step by step solution methodology for mathematical expressions, Symmetry, 10(7), 285.
• Ince E.L. 1956. Ordinary Differential Equations, Dover, New York.
• Lie S. 1883. Klassifikation and integration von gewhnlichen differentialgleichungen zwischen x; y, eine gruppe von transformationen gestatten, III, Arch. Mat. Naturvidenskab. Cambridge, 8, 371-458.
• Moreira I.C. 1984. Hadronic Journal, 7, 475.
• Muriel M. and J.L. Romero. 2001. New methods of reduction for ordinary differential equations, IMA Journal of Applied Mathematics, 66, 111-125.
• Muriel, M. and J.L. Romero. 2009. Second-Order Ordinary Differential Equations and First Integrals of The Form A(t; x) +B(t; x), Journal of Nonlinear Mathematical Physics, 16, 209-222.
• Noether, E. 1971. Invariante Variationsprobleme, Nachr. König. Gesell. Wissen. Göttingen, Math.-Phys. Kl. Heft, 2, 235-257, 1918. English translation in Transport Theory and Statistical Physics, 13, pp. 186-207.
• Orhan O. 2019. The Modeling of Psychotropic Bacteria Affecting Milk Products. Electronic Letters on Science and Engineering, 15(3), 95-100.
• Painleve P. 1902. Sur les équations différentielles du second ordre et d’ordre supérieur dont l’intégrale générale est uniforme, Acta Mathematica. 25, 1-85.
• Stephani H. 1989. Differential Equations and Their Solutions Using Symmetries, Cambridge University Press, Cambridge.