Generalization of the Global Error Minimization for Constructing Analytical Solutions to Nonlinear Evolution Equations

Serge Bruno Yamgoué^*, Bonaventure Nana

Department of Physics, Higher Teacher Training College-Bambili, The University of Bamenda, Bamenda, Cameroon.

DOI: 10.4236/jamp.2015.39142                

Abstract

The global error minimization is a variational method for obtaining approximate analytical solutions to nonlinear oscillator equations which works as follows. Given an ordinary differential equation, a trial solution containing unknowns is selected. The method then converts the problem to an equivalent minimization problem by averaging the squared residual of the differential equation for the selected trial solution. Clearly, the method fails if the integral which defines the average is undefined or infinite for the selected trial. This is precisely the case for such non-periodic solutions as heteroclinic (front or kink) and some homoclinic (dark-solitons) solutions. Based on the fact that these types of solutions have vanishing velocity at infinity, we propose to remedy to this shortcoming of the method by averaging the product of the residual and the derivative of the trial solution. In this way, the method can apply for the approximation of all relevant type of solutions of nonlinear evolution equations. The approach is simple, straightforward and accurate as its original formulation. Its effectiveness is demonstrated using a Helmholtz-Duffing oscillator.

Keywords

Global Error, Minimization, Heteroclinic Solution, Homoclinic Solution, Front/Kink, Dark/Anti-Dark Soliton

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Griffiths, G.W. and Schiesser, W.E. (2012) Traveling Wave Analysis of Partial Differential Equations: Numerical and Analytical Methods with Matlab and Maple. Academic Press, New York.
[2]	Li, Z.B. and He, J.H. (2010) Fractional Complex Transformation for Fractional Differential Equations. Computer and Mathematics with Applications, 15, 970-973.
[3]	Wu, B.S. and Li, P.S. (2001) A Method for Obtaining Approximate Analytical Periods for a Class of Nonlinear Oscillators. Meccanica, 36, 167-176. http://dx.doi.org/10.1023/A:1013067311749
[4]	Wu, B.S. and Lim, C.W. (2004) Large Amplitude Non-Linear Oscillations of a General Conservative System. International Journal of Non-Linear Mechanics, 39, 859-870. http://dx.doi.org/10.1016/S0020-7462(03)00071-4
[5]	Wu, B.S., Sun, W.P. and Lim, C.W. (2006) An Analytical Technique for a Class of Strongly Non-Linear Oscillators. International Journal of Non-Linear Mechanics, 41, 766-774. http://dx.doi.org/10.1016/j.ijnonlinmec.2006.01.006
[6]	Yamgoué, S.B. and Kofané, T.C. (2008) Linearized Harmonic Balance Based Derivation of the Slow Flow for Some Class of Autonomous Single Degree of Freedom Oscillators. International Journal of Non-Linear Mechanics, 43, 993-999. http://dx.doi.org/10.1016/j.ijnonlinmec.2008.05.001
[7]	Yamgoué, S.B. (2012) On the Harmonic Balance with Linearization for Asymmetric Single Degree of Freedom Non-Linear Oscillators. Nonlinear Dynamics, 69, 1051-1062. http://dx.doi.org/10.1007/s11071-012-0326-1
[8]	Yamgoué, S.B., Nana, B. and Lekeufack, O.T. (2015) Improvement of Harmonic Balance Using Jacobian Elliptic Functions. Journal of Applied Mathematics and Physics, 3, 680-690. http://dx.doi.org/10.4236/jamp.2015.36081
[9]	Amore, P. and Aranda, A. (2003) Presenting a New Method for the Solution of Nonlinear Problems. Physics Letters A, 316, 218-225. http://dx.doi.org/10.1016/j.physleta.2003.08.001
[10]	Liao, S.J. (2003) Beyond Perturbation: Introduction to Homotopy Analysis Method. Chapman & Hall/CRC Press, Boca Raton. http://dx.doi.org/10.1201/9780203491164
[11]	Yuste Bravo, S. (1991) Comments on the Method of Harmonic Balance in Which Jacobian Elliptic Functions Are Used. Journal of Sound Vibration, 145, 381-390. http://dx.doi.org/10.1016/0022-460X(91)90109-W
[12]	Yuste Bravo, S. (1992) “Cubication” of Non-Linear Oscillators Using the Principle of Harmonic Balance. International Journal of Non-Linear Mechanics, 27, 347-356. http://dx.doi.org/10.1016/0020-7462(92)90004-Q
[13]	Beléndez, A., Méndez, D.I., Fernandez, E., Marini, S. and Pascual, I. (2009) An Approximate Solution to the Duffing-Harmonic Oscillator by a Cubication Method. Physics Letters A, 373, 2805-2809. http://dx.doi.org/10.1016/j.physleta.2009.05.074
[14]	Yamgoué, S.B., Bogning, J.R., Kenfack Jiotsa, A. and Kofané, T.C. (2010) Rational Harmonic Balance-Based Approximate Solutions to Nonlinear Single-Degree-of-Freedom Oscillator Equations. Physica Scripta, 81, Article ID: 035003. http://dx.doi.org/10.1088/0031-8949/81/03/035003
[15]	Farzaneh, Y. and Tootoonchi, A.A. (2010) Global Error Minimization Method for Solving Strongly Nonlinear Oscillator Differential Equations. Computers and Mathematics with Applications, 59, 2887-2895. http://dx.doi.org/10.1016/j.camwa.2010.02.006
[16]	Elias-Zuniga, A. (2014) “Quintication” Method to Obtain Approximate Analytical Solutions of Non-Linear Oscillators. Applied Mathematics and Computations, 243, 849-855. http://dx.doi.org/10.1016/j.amc.2014.05.085
[17]	He, J.H. (2002) Preliminary Report on the Energy Balance for Nonlinear Oscillators. Mechanics Research Communications, 29, 107-111. http://dx.doi.org/10.1016/S0093-6413(02)00237-9
[18]	He, J.H. (2010) Hamiltonian Approach to Nonlinear Oscillators. Physics Letters A, 374, 2312-2314. http://dx.doi.org/10.1016/j.physleta.2010.03.064
[19]	Mickens, R.E. (2002) Fourier Representations for Periodic Solutions of Odd-Parity Systems. Journal of Sound and Vibration, 258, 398-401. http://dx.doi.org/10.1006/jsvi.2002.5200
[20]	Baldwin, D., Göktas, ü., Hereman, W., Hong, L. Martino, R.S. and Miller, J.C. (2004) Symbolic Computation of Exact Solutions Expressible in Hyperbolic and Elliptic Functions for Nonlinear PDEs. Journal of Symbolic Computations, 37, 669-705. http://dx.doi.org/10.1016/j.jsc.2003.09.004