SUMUDU DECOMPOSITION METHOD ON A DISCONTINUED PROBLEM

Published 30 September 2020 •  vol 142  • 


Authors:

 

Jamshad Ahmad, Department of Mathematics, Faculty of Sciences, University of Gujrat, Pakistan
Sundas Rubab, Department of Mathematics, NCBA&E (Gujrat Campus), Pakistan

Abstract:

 

In this paper, a reliable algorithm based on new Sumudu Decomposition Method (SDM) is proposed to solve a nonlinear differential-difference equation arising in nanotechnology and engineering phenomena. Continuum hypothesis on nanoscales is invalid, and a differential-difference model is considered as an alternative approach to describing discontinued problems. The SDM is a combined form of Sumudu transform, decomposition method and He’s polynomials. The technique finds the solution without any discretization or restrictive assumptions and avoids the round-off errors. The numerical solutions show that the proposed method is very efficient and computationally attractive, more realistic.

Keywords:

 

Sumudu Transform, Decomposition Method, Nonlinear Differential–Difference Equations; Discretized M KDV Lattice Equation; Nanotechnology

References:

 

[1] Ablowitz, M. J. and Ladik, J. F., “Nonlinear differential-difference equations and Fourier analysis”, Journal of Mathematical Physics, vol. 17, no. (6), (1976), pp. 1011-1018.
[2] Su, W. P., Schrieffer, J. R. and Heege, A. J., “Solitons in polyacetylene”, Physical Review Letters, vol. 42, (1979), pp. 1698-1701.
[3] Marquii, P., Bilbault, J. M. and Rernoissnet, M., “Observation of nonlinear localized modes in an electrical lattice”, Physical Review E 51, (1995), pp. 6127-6133.
[4] Davydov, A. S., “The theory of contraction of proteins under their excitation”, Journal of Theoretical Biology, vol. 38, (1973), pp. 559-569.
[5] Eisenberg, H. S., Silberberg, Y., Morandotti, R., Boyd, A. R. and Aitchison, J. S., “Discrete Spatial Optical Solitons in Waveguide Arrays”, Physical Review Letters, vol. 81, (1998), pp. 3383.
[6] Morandotti, R., Peschel, U., Aitchison, J. S., Eisenberg, H. S. and Silberberg, Y., “Dynamics of Discrete Solitons in Optical Waveguide Arrays”, Physical Review Letters, vol. 83, (1999), pp. 2726.
[7] Gökdoğan, A., Yildirim, A. and Merdan, M., “Solving a fractional order model of HIV infection of CD4+ T cells”, Mathematical and Computer Modelling, vol. 54, no. 9-10, (2011), pp. 2132-2138.
[8] Zhu, S. D., “Exp-function method for the hybrid-lattice system”, Internat. J. Nonlinear Sci., vol. 8, (2007), pp. 461-46.
[9] Mokhtari, R., “Variational iteration method for solving nonlinear differential-difference equations”, Int. J. Nonlinear Sci., vol. 9, no. 1, (2008), pp. 19-24.
[10] He, J. H., “Homotopy perturbation method: a new nonlinear analytic technique for nonlinear equations”, Int. J. Mod. Phys. B, vol. 20, no. 10, (2006), pp. 1141-1199.
[11] He, J. H., “An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering”, Int. J. Mod. Phys. B, vol. 22, no. 21, (2008), pp. 3487-3578.
[12] Naschie, E., “Deterministic quantum mechanics versus classical mechanical indeterminism”, International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 1, (2007), pp. 5-10.
[13] Naschie, E., “A review of applications and results of E-infinity theory”, International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 1, (2007), pp. 11-20.
[14] Naschie, E., “Probability set particles”, International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 1, (2007), pp. 117-119.
[15] Naschie, E., “Nanotechnology for the developing world”, Chaos, Solitons & Fractals, vol. 30, (2006), pp. 769-773.
[16] Liu, Y. and He, J. H., “Bubble electrospinning for mass production of nanofibers”, International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, (2007), pp. 393-396.
[17] He, J. H., Wan, Y. Q. and Xu, L., “Nano-effects, quantum-like properties in electrospun nanofibers”, Chaos, Solitons & Fractals, vol. 33, (2007), pp. 26-37.
[18] He, J. H., Liu, Y.Y., Xu, L. and Yu, J. Y., “Micro sphere with nanoporosity by electrospinning”, Chaos, Solitons & Fractals, vol. 32, (2007), pp. 1096-1100.
[19] He, J. H. and Zhu, S. D., “Differential-difference model for nanotechnology”, Journal of Physics: Conference Series, vol. 96, (2008), p. 012189.
[20] He, J. H. and Zhu, S. D., “Differential-difference model for nanotechnology”, Journal of Physics: Conference Series, vol. 96, (2008), pp. 012189.
[21] Suris, Y. B., “Miura transformation for toda-type integrable system with applications to the problem of integrable discretizations”, In Fachbereich Mathematik, Technische University Press, Berlin, (1998).
[22] Zhu, S. D., “Exp-function method for the Hybrid-Lattice system”, International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 3, (2007), pp. 461-464.
[23] Zhu, S. D., “Exp-function method for the discrete mKdV lattice”, International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 3, (2007), pp. 465-469.
[24] Zhu, S. D., “Discrete (2+1)-dimensional Toda lattice equation via Expfunction method”, Physics Letters A, vol. 372, (2008), pp. 654-657.
[25] Mokhtari, R., “Variational iteration method for solving nonlinear differential–difference equations”, International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, no. 1, (2008), pp. 19-24.
[26] Zhu, S. D., Chu, Y. M. and Qiu, S. L., “The homotopy perturbation method for discontinued problems arising in nanotechnology”, Computers and Mathematics with Applications, vol. 58, (2009), pp. 2398-2401.
[27] Nik, H. S. and Golchaman, M., “The homotopy analysis method for solving discontinued problems arising in nanotechnology”, World Academy of Science, Engineering and Technology, vol. 76, (2011), pp. 891-894.
[28] Weerakoon, S., “Application of Sumudu transform to partial differential equations”, International Journal of Mathematical Education in Science and Technology, vol. 25, (1994), pp. 277-283.
[29] Asiru, M. A., “Classroom note: Application of the Sumudu transform to discrete dynamic systems”, International Journal of Mathematical Education in Science and Technology, vol. 34, no. 6, (2003), pp. 944-949.
[30] Airu, M. A., “Further properties of the Sumudu transform and its applications”, International Journal of Mathematical Education in Science and Technology, vol. 33, no. 3, (2002), pp. 441-449.
[31] Singh, J., Kumar, D. and Kılıçman, A., “Homotopy Perturbation Method for Fractional Gas Dynamics Equation Using Sumudu Transform”, Abstract and Applied Analysis, vol. 2013, (2013), pp. 8.
[32] Ghorbani, A., “Beyond adomian’s polynomials: He polynomials”, Chaos, Solitons & Fractals, vol. 39, (2009), pp. 1486-1492.
[33] Mohyud-Din, S. T., Noor, M. A. and Noor, K. I., “Traveling wave solutions of seventh-order generalized KdV equation using He’s polynomials”, International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, (2009), pp. 227-233.

Citations:

 

APA:
Ahmad, J., and Rubab, S. (2020). Sumudu Decomposition Method on a Discontinued Problem. International Journal of Advanced Science and Technology (IJAST), ISSN: 2005-4238(Print); 2207-6360 (Online), NADIA, 142, 15-20. doi: 10.33832/ijast.2020.142.02.

MLA:
Ahmad, Jamshad, et al. “Sumudu Decomposition Method on a Discontinued Problem.” International Journal of Advanced Science and Technology, ISSN: 2005-4238(Print); 2207-6360 (Online), NADIA, vol. 142, 2020, pp. 15-20. IJAST, http://article.nadiapub.com/IJAST/Vol142/2.html.

IEEE:
[1] J. Ahmad, and S. Rubab, "Sumudu Decomposition Method on a Discontinued Problem." International Journal of Advanced Science and Technology (IJAST), ISSN: 2005-4238(Print); 2207-6360 (Online), NADIA, vol. 142, pp. 15-20, September 2020.