Some Exact Solutions for a Klein Gordon Equation
Main Article Content
Keywords
Lie simmetries, Klein Gordon equation, invariant solutions.
Abstract
In solving practical problems in science and engineering arises as a direct consequence differential equations that explains the dynamics of the phenomena. Finding exact solutions to this equations provides importan information about the behavior of physical systems. The Lie symmetry method allows tofind invariant solutions under certain groups of transformations for differential equations.This method not very well known and used is of great importance in the scientific community. By this approach it was possible to find several exactinvariant solutions for the Klein Gordon Equation uxx − utt = k(u). A particularcase, The Kolmogorov equation uxx − utt = k1u + k2un was considered.These equations appear in the study of relativistic and quantum physics. The general solutions found, could be used for future explorations on the study for other specific K(u) functions.
MSC: 35A30
Downloads
References
[2] Abraham Cohen. An Introduction to the Lie Theory of one Parameter Groups, Kessinger Publishing, 2007. Referenced in 57
[3] R. Cherniha, V. Davydovych. “Conditional symmetries and exact solutions of the diffusive Lotka-Volterra system”. Mathematical and Computer Modellling, vol. 54, no 5-6, 1238-1251, 2011. Referenced in 57
[4] G. Emanuel. Solution of Ordinary Differential Equations by Continuous Groups, Chapman and Hall/CRC, 2000. Referenced in 64
[5] Liu Hanze, Li Jibin. “Lie symmetry analysis and exact solutions for the short pulse equation. Nonlinear Analysis”. Nonlinear Analysis: Theory, Methods and Application, vol. 71, no5-6, pp. 2126-2133, 2009. Referenced in 58
[6] W. Hereman. “Review of Symbolic Software for Lie Symmetrie Analysis”. Mathematical and Computer Modelling, vol. 25 no8-9, pp. 115-132, 1997. Referenced in 67
[7] Robert C. McOwen. Partial Differential Equations: Methods and Applications, Prentice Hall, 2002. Referenced in 63
[8] Peter J. Olver. Application of Lie Groups to Differential Equations, Springer Verlag, 1993. Referenced in 57, 58, 60, 61
[9] L. Ovsiannikov. Group Analysis of Differential Equations, Academic Press, 1978. Referenced in 58
[10] Andrei D. Polyanin, V. Zaitsev. Handbook of Nonlinear Partial Differential Equations, Chapman and Hall/CRC, 2004. Referenced in 65
[11] Nikolay Sukhomlin, Jan Marcos Ortiz. “Equivalence and new exact solutions to the Black-Scholes and diffusion equations”. Applied Mathematics E-Notes, vol. 7, no 2, pp. 206-213, 2007. Referenced in 57