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Modified Differential Transform Method for Singular Lane-Emden Equations in Integer and Fractional Order

Year 2015, Volume: 5 Issue: 1, 124 - 131, 01.06.2015

Abstract

In the present work the modiŞed differential transform method, incorporating the Adomian polynomials into the differential transform method DTM , is used tosolve the nonlinear and singular Lane-Emden equations in integer and fractional order.Numerical examples with different types are solved. The results show that this methodis very effective and simple

References

  • Agarwal, R. P., Regan, D. O. and Lakshmikanthamr, V., (2001), Quadratic forms and nonlinear non- resonant singular second order boundary value problems of limit circle type, Zeitschrift fur Analysis und ihre Anwendungen, 20, pp. 727-737.
  • Agarwal, R. P. and Regan, D. O., (2001), Existence theory for single and multiple solutions to singular positone boundary value problems, J. Diff. Equ., 175(2), pp. 393-414.
  • Lane, J. H., (1870), On the theoretical temperature of the sun under the hypothesis of a gaseous mass maintaining its volume by its internal heat and depending on the laws of gases known to terrestrial experiment, The American Journal of Science and Arts, 50, pp. 57-74.
  • Emden, R., (1907), Gaskugeln, Teubner, Leipzig and Berlin.
  • Marasi, H. R. and Nikbakht, M., (2011), Adomian decompositiom method for boundary value prob- lems, Aus. J. Basic. Appl. Sci., 5, pp. 2106-2111.
  • Adomian, G., (1994), Solving frontier problems of physics: The decomposition method, Kluwer Aca- demic, Dordrecht.
  • Marasi, H.R. and Karimi, S., (2014), Convergence of variational iteration method for solving fractional Klein-Gordon equation, J. Math. Comp. Sci., 4, pp. 257-266.
  • Assas, L. M. B., (2008), Variational iteration method for solving coupled-KdV equations, Chaos Solitons Fractals., 38(4), pp. 1225-1228.
  • Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J., (2006), Theory and applications of fractional differential equations, North-Holland Mathematics Studies., 204, pp. 7-10.
  • Podlubny, I., (1999), Fractional differential equations, Academic Press, New york.
  • Marasi, H. R. and Jodayree Akbarfam, A., (2007), On the canonical solution of indeŞnite problem with m turning points of even order, J. Math. Anal. Appl., 332, pp. 1071-1086
  • Marasi, H. R., (2011), Asymptotic form and inŞnite product representation of solution of a second order initial value problem with a complex parameter and a Şnite number of turning points, J. Cont. Math. Anal., 4, pp. 57-76
  • Marasi, H. R. and Jodayree Akbarfam, A., (2012), Dual equation and inverse problem for an indefnite Sturm-Liouville problem with m turning points of even order, Math. Modell. Anal., 17(5), pp. 618-629. [14] Chowdhury, M. and Hashim, I., (2009), Solutions of Emden-Fowler equations by homotopy perturba- tion method, Non-Linear Analysis: Real World Application., 101, pp. 104-115.
  • Yildirim, A. and Ozi, T., (2009), Solutions of singular IVPs of Lane-Emden type by the variational iteration method, Nonlinear Analysis: Theory, Methods Applications, 70(6), pp. 2480-2484.
  • Parand, K. and Pirkhedri, A., (2010), Sinc-collocation method for solving astrophysics equations, New Astronomy, 15(6), pp. 533-537.
  • He, J., (2006), Homotopy perturbation method for solving boundary value problems, Phys. Lett. A., 350, pp. 87-88.
  • Zhou, J. K., (1986), Deferential transformation and its application for electrical circuits, Huazhong University Press, Wuhan China.
  • Nazari-Golshan, A., Nourazar, S. S., Ghafoori-Fard, H., Yildirim, A. and Campo, A., (2013), A modiŞed homotopy perturbation method coupled with the Fourier transform for nonlinear and singular Lane-Emden equations, Appl. Math. Lett., http://dx.doi.org/10.1016/j.aml., pp. 2013.05.010.
  • Elsaid, A., (2012), Fractional differential transform method combined with the Adomian polynomials, Apll. Math. Comput., 218, pp. 6899-6911.
Year 2015, Volume: 5 Issue: 1, 124 - 131, 01.06.2015

Abstract

References

  • Agarwal, R. P., Regan, D. O. and Lakshmikanthamr, V., (2001), Quadratic forms and nonlinear non- resonant singular second order boundary value problems of limit circle type, Zeitschrift fur Analysis und ihre Anwendungen, 20, pp. 727-737.
  • Agarwal, R. P. and Regan, D. O., (2001), Existence theory for single and multiple solutions to singular positone boundary value problems, J. Diff. Equ., 175(2), pp. 393-414.
  • Lane, J. H., (1870), On the theoretical temperature of the sun under the hypothesis of a gaseous mass maintaining its volume by its internal heat and depending on the laws of gases known to terrestrial experiment, The American Journal of Science and Arts, 50, pp. 57-74.
  • Emden, R., (1907), Gaskugeln, Teubner, Leipzig and Berlin.
  • Marasi, H. R. and Nikbakht, M., (2011), Adomian decompositiom method for boundary value prob- lems, Aus. J. Basic. Appl. Sci., 5, pp. 2106-2111.
  • Adomian, G., (1994), Solving frontier problems of physics: The decomposition method, Kluwer Aca- demic, Dordrecht.
  • Marasi, H.R. and Karimi, S., (2014), Convergence of variational iteration method for solving fractional Klein-Gordon equation, J. Math. Comp. Sci., 4, pp. 257-266.
  • Assas, L. M. B., (2008), Variational iteration method for solving coupled-KdV equations, Chaos Solitons Fractals., 38(4), pp. 1225-1228.
  • Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J., (2006), Theory and applications of fractional differential equations, North-Holland Mathematics Studies., 204, pp. 7-10.
  • Podlubny, I., (1999), Fractional differential equations, Academic Press, New york.
  • Marasi, H. R. and Jodayree Akbarfam, A., (2007), On the canonical solution of indeŞnite problem with m turning points of even order, J. Math. Anal. Appl., 332, pp. 1071-1086
  • Marasi, H. R., (2011), Asymptotic form and inŞnite product representation of solution of a second order initial value problem with a complex parameter and a Şnite number of turning points, J. Cont. Math. Anal., 4, pp. 57-76
  • Marasi, H. R. and Jodayree Akbarfam, A., (2012), Dual equation and inverse problem for an indefnite Sturm-Liouville problem with m turning points of even order, Math. Modell. Anal., 17(5), pp. 618-629. [14] Chowdhury, M. and Hashim, I., (2009), Solutions of Emden-Fowler equations by homotopy perturba- tion method, Non-Linear Analysis: Real World Application., 101, pp. 104-115.
  • Yildirim, A. and Ozi, T., (2009), Solutions of singular IVPs of Lane-Emden type by the variational iteration method, Nonlinear Analysis: Theory, Methods Applications, 70(6), pp. 2480-2484.
  • Parand, K. and Pirkhedri, A., (2010), Sinc-collocation method for solving astrophysics equations, New Astronomy, 15(6), pp. 533-537.
  • He, J., (2006), Homotopy perturbation method for solving boundary value problems, Phys. Lett. A., 350, pp. 87-88.
  • Zhou, J. K., (1986), Deferential transformation and its application for electrical circuits, Huazhong University Press, Wuhan China.
  • Nazari-Golshan, A., Nourazar, S. S., Ghafoori-Fard, H., Yildirim, A. and Campo, A., (2013), A modiŞed homotopy perturbation method coupled with the Fourier transform for nonlinear and singular Lane-Emden equations, Appl. Math. Lett., http://dx.doi.org/10.1016/j.aml., pp. 2013.05.010.
  • Elsaid, A., (2012), Fractional differential transform method combined with the Adomian polynomials, Apll. Math. Comput., 218, pp. 6899-6911.
There are 19 citations in total.

Details

Primary Language English
Journal Section Research Article
Authors

H. R. Marasi This is me

N. Sharifi This is me

H. Piri This is me

Publication Date June 1, 2015
Published in Issue Year 2015 Volume: 5 Issue: 1

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