# The First Integrals of a Second Order Ordinary Differential Equation and Application

## Main Article Content

## Abstract

The first integrals of second order ordinary differential equations are considered. The necessary conditions of the existence of analytical first integrals for the equation are presented. Then, the first integrals of the equation are obtained using Lie symmetry method. The results of the first integrals are applied to certain classes of partial differential equations, the conditions of nonexistence of the traveling wave solutions of the partial differential equations are obtained, and traveling wave solutions of the equations under the certain parametric conditions are also obtained.

Keywords:

First integral, lie symmetry, traveling wave solutions, partial differential equations.

## Article Details

How to Cite

*Asian Journal of Research in Computer Science*,

*3*(3), 1-15. https://doi.org/10.9734/ajrcos/2019/v3i330095

Section

Original Research Article

## References

Li W, Shi S. Weak-Painlevse property and integrability of general dynamical system. Discrete and Continuous Dynamical Systems. 2014;34:3667-3681.

Shi S, Li Y. Non-integrability for general nonlinear systems. Z. angew. Math. Phys.2001;52:191-200.

Du Z, Romanovskib Valery G, Xiang Zhang. Varieties and analytic normalizations of partially integrable systems. J. Differential Equations. 2016;260:6855-6871.

Llibre J, Valls C, Zhang X. The completely integrable differential systems are essentially linear differential systems. J Nonlinear Sci. 2015;25:815-826.

Zhang X. Local first integrals for systems of differential equations. J. Phys. A: Math. Gen. 2003;36:12243-12253.

Colak IE, Llibre J, Valls C. Local analytic first integrals of planar analytic differential systems. Phys. Lett. 2013;377:1065-1069.

Llibre J, Zhang X. Polynomial first integrals for quasi-homogeneous polynomial differential systems. Nonlinearity. 2002;15:1269-1280.

Hu Y, Xue C. One-parameter Lie groups and inverse integrating factors of n-th order autonomous systems. J. Math. Anal. Appl. 2012;388:617-626.

Hu Y. On the first integrals of n-th order autonomous systems. J. Math. Anal. Appl. 2018;459:1062-1078.

Bluman G, Kumei W. Symmetrics and differential equations. 2nd ed. Springer-Verlag; 1989.

Nass A. Lie symmetry analysis and exact solutions of fractional ordinary differential equations with neutral delay. Applied Mathematics and Computation. 2018;347:370-380.

Nass A, Mpungu K, Nuruddeen R. Group classification of the time fractional nonlinear Poisson equation. Mathematical Communications. 2019;24:1-13.

Darboux G. Mmoire surles quations diffrentielles alg briques du premier ordre et du premier degr (mlanges). Bull. Sci. Math. 1878;2:60-96, 123-144, 151-200.

Bountis TC, Ramani A, Grammaticos B, Dorizzi B. On the complete and partial integrability of non-Hamiltonian systems. Phys. A. 1984;128:268-288.

Lax PD. Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math. 1968;21:467-490.

Nuruddeen R, Nass A M. Exact solitary wave solution for the fractional and classical GEW-Burgers equations: An application of Kudryashov method. Journal of Taibah University for Science. 2018;12:309-314.

Chen J, Yi Y, Zhang X. First integrals and normal forms for germs of analytic vector fields. J. Differential Equations. 2008;245:1167-1184.

Furta D. On non-integrability of general systems of differential equations. Z angew Math. Phys. 1996;47:112-131.

Llibre J, Zhang X. Polynomial first integrals of quadratic systems. Rocky Mountain Journal of Mathematics. 2001;31:1317-1371.

Hu YK, Guan K. Techniques for searching first integrals by Lie group and application to gyroscope system. Science in China Ser. A Mathematics. 2005;48:1135-1143.

Guan K, Liu S, Lei J. The lie algebra admitted by an ordinary differential equation system. Ann. of Diff. Eqs. 1998;14:131-142.

Liu M, Guan K. The Lie group and integrability of the Fisher type traveling wave equation. Acta Mathematicae Applicatae sinica. English Series. 2009;25:305-320.

Feng Z, Zhen S, Gao DY. Traveling wave solutions to a reaction-diffusion equation. Z. Angew. Math. Phys. 2009;60:756-773.

Feng Z. On traveling wave solutions of the Burgers-Korteweg-de-Vries equation. Nonlinearity. 2007;20:343-356.

Feng Z. A note on ”Explicit exact solutions to the compound Burgers-Korteweg-de Vries equation”. Phys. Lett. 2003;312:65-70.

Fisher RA. The wave of advance of advantageous genes. Ann. Eugenics. 1973;7:353-369.

Ma W, Fuchssteiner B. Explicit and exact solutions to a Kolmogorov-Petrovskii- Piskunov equation. International Journal of Non-Linear Mechanics. 1996;31:329-338.

Feng Z, Wang X. The first integral method to the two-dimensional Burgers-Korteweg-de Vries equation. Physics Letters A. 2003;308:173-178.

Ma W. An exact solution to two-dimensional Korteweg-de Vries-Burgers equation. J. Phys. A: Math. Gen. 1993;26:L17-L20.

Abramowitz M, Stegun I A. Handbook of Mathematical Functions. Dover. New York; 1970.

Almendral JA, Sanjuan MAF. Integrability and symmetries for the Helmholtz oscillator with friction. J. Phys. A(Math. Gen.).2003;36:695-710.

Shi S, Li Y. Non-integrability for general nonlinear systems. Z. angew. Math. Phys.2001;52:191-200.

Du Z, Romanovskib Valery G, Xiang Zhang. Varieties and analytic normalizations of partially integrable systems. J. Differential Equations. 2016;260:6855-6871.

Llibre J, Valls C, Zhang X. The completely integrable differential systems are essentially linear differential systems. J Nonlinear Sci. 2015;25:815-826.

Zhang X. Local first integrals for systems of differential equations. J. Phys. A: Math. Gen. 2003;36:12243-12253.

Colak IE, Llibre J, Valls C. Local analytic first integrals of planar analytic differential systems. Phys. Lett. 2013;377:1065-1069.

Llibre J, Zhang X. Polynomial first integrals for quasi-homogeneous polynomial differential systems. Nonlinearity. 2002;15:1269-1280.

Hu Y, Xue C. One-parameter Lie groups and inverse integrating factors of n-th order autonomous systems. J. Math. Anal. Appl. 2012;388:617-626.

Hu Y. On the first integrals of n-th order autonomous systems. J. Math. Anal. Appl. 2018;459:1062-1078.

Bluman G, Kumei W. Symmetrics and differential equations. 2nd ed. Springer-Verlag; 1989.

Nass A. Lie symmetry analysis and exact solutions of fractional ordinary differential equations with neutral delay. Applied Mathematics and Computation. 2018;347:370-380.

Nass A, Mpungu K, Nuruddeen R. Group classification of the time fractional nonlinear Poisson equation. Mathematical Communications. 2019;24:1-13.

Darboux G. Mmoire surles quations diffrentielles alg briques du premier ordre et du premier degr (mlanges). Bull. Sci. Math. 1878;2:60-96, 123-144, 151-200.

Bountis TC, Ramani A, Grammaticos B, Dorizzi B. On the complete and partial integrability of non-Hamiltonian systems. Phys. A. 1984;128:268-288.

Lax PD. Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math. 1968;21:467-490.

Nuruddeen R, Nass A M. Exact solitary wave solution for the fractional and classical GEW-Burgers equations: An application of Kudryashov method. Journal of Taibah University for Science. 2018;12:309-314.

Chen J, Yi Y, Zhang X. First integrals and normal forms for germs of analytic vector fields. J. Differential Equations. 2008;245:1167-1184.

Furta D. On non-integrability of general systems of differential equations. Z angew Math. Phys. 1996;47:112-131.

Llibre J, Zhang X. Polynomial first integrals of quadratic systems. Rocky Mountain Journal of Mathematics. 2001;31:1317-1371.

Hu YK, Guan K. Techniques for searching first integrals by Lie group and application to gyroscope system. Science in China Ser. A Mathematics. 2005;48:1135-1143.

Guan K, Liu S, Lei J. The lie algebra admitted by an ordinary differential equation system. Ann. of Diff. Eqs. 1998;14:131-142.

Liu M, Guan K. The Lie group and integrability of the Fisher type traveling wave equation. Acta Mathematicae Applicatae sinica. English Series. 2009;25:305-320.

Feng Z, Zhen S, Gao DY. Traveling wave solutions to a reaction-diffusion equation. Z. Angew. Math. Phys. 2009;60:756-773.

Feng Z. On traveling wave solutions of the Burgers-Korteweg-de-Vries equation. Nonlinearity. 2007;20:343-356.

Feng Z. A note on ”Explicit exact solutions to the compound Burgers-Korteweg-de Vries equation”. Phys. Lett. 2003;312:65-70.

Fisher RA. The wave of advance of advantageous genes. Ann. Eugenics. 1973;7:353-369.

Ma W, Fuchssteiner B. Explicit and exact solutions to a Kolmogorov-Petrovskii- Piskunov equation. International Journal of Non-Linear Mechanics. 1996;31:329-338.

Feng Z, Wang X. The first integral method to the two-dimensional Burgers-Korteweg-de Vries equation. Physics Letters A. 2003;308:173-178.

Ma W. An exact solution to two-dimensional Korteweg-de Vries-Burgers equation. J. Phys. A: Math. Gen. 1993;26:L17-L20.

Abramowitz M, Stegun I A. Handbook of Mathematical Functions. Dover. New York; 1970.

Almendral JA, Sanjuan MAF. Integrability and symmetries for the Helmholtz oscillator with friction. J. Phys. A(Math. Gen.).2003;36:695-710.