Solution of Modified Equations of Emden-Type by Differential Transform Method

ABSTRACT

In this paper the Modified Equations of Emden type (MEE), χ+αχχ+βχ^{3} is solved numerically by the differential transform method. This technique doesn’t require any discretization, linearization or small perturbations and therefore it reduces significantly the numerical computation. The current results of this paper are in excellent agreement with those provided by Chandrasekar et al. [1] and thereby illustrate the reliability and the performance of the differential transform method. We have also compared the results with the classical Runge-Kutta 4 (RK4) Method.

In this paper the Modified Equations of Emden type (MEE), χ+αχχ+βχ

KEYWORDS

Modified Equations of Emden Type, Differential Transforms Method, Runge-Kutta 4 (RK4) Method

Modified Equations of Emden Type, Differential Transforms Method, Runge-Kutta 4 (RK4) Method

Cite this paper

nullS. Mukherjee, B. Roy and P. Chatterjee, "Solution of Modified Equations of Emden-Type by Differential Transform Method,"*Journal of Modern Physics*, Vol. 2 No. 6, 2011, pp. 559-563. doi: 10.4236/jmp.2011.26065.

nullS. Mukherjee, B. Roy and P. Chatterjee, "Solution of Modified Equations of Emden-Type by Differential Transform Method,"

References

[1] V.K Chandrasekar, M.Senthilvelan and M.Lakshmanan, “On the general solution for the modified Emden-type equation ”, Journal of Physics A, Vol. 40, 2004, pp. 4717-4727.

[2] P Painleve, “Sur les équations différentielles du second ordre et d’ordre supérieure dont l’intégrale générale est uniforme”, Acta Math, Vol. 25, 1902, pp.1.

[3] E. L. Ince, “Ordinary Differential Equations”, Newyork: Dover, 1956.

[4] H. T. Davis, “Introduction to Nonlinear Differential and Integral Equations”, New York: Dover, 1962.

[5] E.Kamke, “Differential gleichungen Losungsmethoden und Losungen”, Stuggart: Teubner, 1983.

[6] G. M. Murphy, “Ordinary Differential Equations and their Solutions”, Newyork: Van Nostrand, 1960.

[7] V. V. Golubev, “Lectures on Analytical Theory of Differential Equations”, Moscow: Gostekhizdat, 1950.

[8] J.S. R.Chisholm and A. K. Common, “A class of second-order differential equations and related first-order systems”, Journal of Physics A, Vol. 20, No. 16, 1987, pp. 5459-5472.

[9] I.C Moreira, “Lie symmetries for the reduced three-wave”, Hadronic.J., Vol.7 ,1984, pp. 475. P G L Leach, “First integrals for the modified Emden equation ”, Journal of Physics, Vol. 26, 1985, pp. 2510.

[10] S.Chandrasekhar, “An Introduction to the study of Stellar Structure”, New york: Dover, 1957. J M Dixon and J A Tuszynski, “Solutions of a Generalized Emden Equation and Their Physical Significance”, Physical Review A, Vol. 41, 1990, pp. 4166-4173.

[11] G.C.McVittie, “The Mass-Particle in an Expanding Universe”, Mon. Not. R. Astron. Soc., Vol. 93, 1933, pp. 325.; Ann. Inst. H. Poincare, Vol. 6, 1967, pp1; Ann. Inst. H. Poincare, Vol. 40, No.3, 1984, pp.231.

[12] V.J.Erwin, W.F.Ames and E.Adams, “Wave Phenomenon: Modern Theory and Applications”, ed C Rogers and J B Moodie .Amsterdam: North-Holland, 1984.

[13] F.M.Mahomed and P.G.L.Leach, “The Lie Algebra SL (3,R) and Linearization”, Quaestiones Math. Vol. 12, 1989, No. 2, pp.121-139.

[14] L.G.S.Duarte, S.E.S.Duarte and I.C.Moreira, “One Dimensional Equations with the maximum number of symmetry generators”, J. Phys. A: Math. Gen. Vol. 20, 1987, pp.L701.

[15] S.E.Bouquet, M.R.Feix and P.G.L.Leach, “Properties of second order ordinary differential equations invariant under time translation and self similar transformation”, J.Math.Phys., Vol. 32., No. 6, 1991, pp.1480.

[16] W. Sarlet, F.M. Mahomed and P.G.L.Leach, “Symmetries of nonlinear differential equations and linearization”, J. Phys. A: Math. Gen. Vol. 20, No. 2, 1987, pp.277-292.

[17] P.G. L. Leach, M. R. Feix and S. Bouquet, “Analysis and Solution of a Nonlinear Second-Order Equation through Rescaling and Through a Dynamical Point of View”, J. Math. Phys. 29, 1988, pp. 2563-2569. R. L. Lemmer and P. G. L. Leach, “The Painlev′e test, hidden symmetries and the equation ”, J. Phys. A: Math. Gen. Vol. 26, 1993. pp. 5017-5024.

[18] W.H. Steeb, “Invertible Point Transformations and Nonlinear Differential Equations” London: World Scientific, 1993.

[19] M. R. Feix, C. Geronimi, L. Cairo, P.G.L. Leach, R.L. Lemmer and S. Bouquet, “On the Singularity Analysis of Ordinary Differential Equation Invariant under Time Translation and Rescaling”, J. Phys. A: Math. Gen. Vol. 30, 1997, pp. 7437-7461.

[20] N. H. Ibragimov, “Elementary Lie Group Analysis and Ordinary Differential Equations” NewYork: John Wiley & Sons, 1999.

[21] P. G. L. Leach, S. Cotsakis and G. P. Flessas, “Symmetries, Singularities and Integrability in Complex Dynamics II: Rescalings and Time-Translations in 2D Systems”, J. Math. Anal. Appl., Vol. 251, 2000, pp. 587-608.

[22] V. K. Chandrasekar, M. Senthilvelan and M. Lakshmanan, “On the complete integrability and linearization of certain second order nonlinear ordinary differential equations”, Proc. R. Soc. London A461, 2005, pp. 2451.

[23] V.K. Chandrasekar, S. N. Pandey, M. Senthilvelan and M. Lakshmanan, “A simple and unified approach to identify integrable nonlinear oscillators and systems”, J. Math. Phys. Vol. 47, No. 2, 2006, pp.023508.

[24] J.K. Zhou, “Differential Transformation and its Application in Electrical Circuits”, Huazhong University Press, Wuhan, China, 1986.

[25] S. Mukherjee, B.Roy, S. Dutta, “Solution of Duffing-Van der Pol oscillator equation by a Differential Transform Method”. Physica Scripta , Vol. 83, No. 1, 2010, 015006.

[26] V.K. Chandrasekar, M. Senthilvelan and M.Lakshmanan, “New aspects of integrability of force-free Duffing-Van der Pol oscillator and related nonlinear system”, Journal of Physics A, Vol. 37, No. 16, 2004, pp.4527.

[27] M. Euler, N.Euler and P.G.L.Leach, “The Riccati and Ermakov-Pinney hierarchies”, Report no. 08, Institut Mittag-Leffler, Sweden, 2005/2006.

[28] M.Abramowitz and I.A.Stegun, “Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables”, New York: Dover, eds., 1972.

[1] V.K Chandrasekar, M.Senthilvelan and M.Lakshmanan, “On the general solution for the modified Emden-type equation ”, Journal of Physics A, Vol. 40, 2004, pp. 4717-4727.

[2] P Painleve, “Sur les équations différentielles du second ordre et d’ordre supérieure dont l’intégrale générale est uniforme”, Acta Math, Vol. 25, 1902, pp.1.

[3] E. L. Ince, “Ordinary Differential Equations”, Newyork: Dover, 1956.

[4] H. T. Davis, “Introduction to Nonlinear Differential and Integral Equations”, New York: Dover, 1962.

[5] E.Kamke, “Differential gleichungen Losungsmethoden und Losungen”, Stuggart: Teubner, 1983.

[6] G. M. Murphy, “Ordinary Differential Equations and their Solutions”, Newyork: Van Nostrand, 1960.

[7] V. V. Golubev, “Lectures on Analytical Theory of Differential Equations”, Moscow: Gostekhizdat, 1950.

[8] J.S. R.Chisholm and A. K. Common, “A class of second-order differential equations and related first-order systems”, Journal of Physics A, Vol. 20, No. 16, 1987, pp. 5459-5472.

[9] I.C Moreira, “Lie symmetries for the reduced three-wave”, Hadronic.J., Vol.7 ,1984, pp. 475. P G L Leach, “First integrals for the modified Emden equation ”, Journal of Physics, Vol. 26, 1985, pp. 2510.

[10] S.Chandrasekhar, “An Introduction to the study of Stellar Structure”, New york: Dover, 1957. J M Dixon and J A Tuszynski, “Solutions of a Generalized Emden Equation and Their Physical Significance”, Physical Review A, Vol. 41, 1990, pp. 4166-4173.

[11] G.C.McVittie, “The Mass-Particle in an Expanding Universe”, Mon. Not. R. Astron. Soc., Vol. 93, 1933, pp. 325.; Ann. Inst. H. Poincare, Vol. 6, 1967, pp1; Ann. Inst. H. Poincare, Vol. 40, No.3, 1984, pp.231.

[12] V.J.Erwin, W.F.Ames and E.Adams, “Wave Phenomenon: Modern Theory and Applications”, ed C Rogers and J B Moodie .Amsterdam: North-Holland, 1984.

[13] F.M.Mahomed and P.G.L.Leach, “The Lie Algebra SL (3,R) and Linearization”, Quaestiones Math. Vol. 12, 1989, No. 2, pp.121-139.

[14] L.G.S.Duarte, S.E.S.Duarte and I.C.Moreira, “One Dimensional Equations with the maximum number of symmetry generators”, J. Phys. A: Math. Gen. Vol. 20, 1987, pp.L701.

[15] S.E.Bouquet, M.R.Feix and P.G.L.Leach, “Properties of second order ordinary differential equations invariant under time translation and self similar transformation”, J.Math.Phys., Vol. 32., No. 6, 1991, pp.1480.

[16] W. Sarlet, F.M. Mahomed and P.G.L.Leach, “Symmetries of nonlinear differential equations and linearization”, J. Phys. A: Math. Gen. Vol. 20, No. 2, 1987, pp.277-292.

[17] P.G. L. Leach, M. R. Feix and S. Bouquet, “Analysis and Solution of a Nonlinear Second-Order Equation through Rescaling and Through a Dynamical Point of View”, J. Math. Phys. 29, 1988, pp. 2563-2569. R. L. Lemmer and P. G. L. Leach, “The Painlev′e test, hidden symmetries and the equation ”, J. Phys. A: Math. Gen. Vol. 26, 1993. pp. 5017-5024.

[18] W.H. Steeb, “Invertible Point Transformations and Nonlinear Differential Equations” London: World Scientific, 1993.

[19] M. R. Feix, C. Geronimi, L. Cairo, P.G.L. Leach, R.L. Lemmer and S. Bouquet, “On the Singularity Analysis of Ordinary Differential Equation Invariant under Time Translation and Rescaling”, J. Phys. A: Math. Gen. Vol. 30, 1997, pp. 7437-7461.

[20] N. H. Ibragimov, “Elementary Lie Group Analysis and Ordinary Differential Equations” NewYork: John Wiley & Sons, 1999.

[21] P. G. L. Leach, S. Cotsakis and G. P. Flessas, “Symmetries, Singularities and Integrability in Complex Dynamics II: Rescalings and Time-Translations in 2D Systems”, J. Math. Anal. Appl., Vol. 251, 2000, pp. 587-608.

[22] V. K. Chandrasekar, M. Senthilvelan and M. Lakshmanan, “On the complete integrability and linearization of certain second order nonlinear ordinary differential equations”, Proc. R. Soc. London A461, 2005, pp. 2451.

[23] V.K. Chandrasekar, S. N. Pandey, M. Senthilvelan and M. Lakshmanan, “A simple and unified approach to identify integrable nonlinear oscillators and systems”, J. Math. Phys. Vol. 47, No. 2, 2006, pp.023508.

[24] J.K. Zhou, “Differential Transformation and its Application in Electrical Circuits”, Huazhong University Press, Wuhan, China, 1986.

[25] S. Mukherjee, B.Roy, S. Dutta, “Solution of Duffing-Van der Pol oscillator equation by a Differential Transform Method”. Physica Scripta , Vol. 83, No. 1, 2010, 015006.

[26] V.K. Chandrasekar, M. Senthilvelan and M.Lakshmanan, “New aspects of integrability of force-free Duffing-Van der Pol oscillator and related nonlinear system”, Journal of Physics A, Vol. 37, No. 16, 2004, pp.4527.

[27] M. Euler, N.Euler and P.G.L.Leach, “The Riccati and Ermakov-Pinney hierarchies”, Report no. 08, Institut Mittag-Leffler, Sweden, 2005/2006.

[28] M.Abramowitz and I.A.Stegun, “Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables”, New York: Dover, eds., 1972.