An Anylatical approach for space–time fractal order nonlinear dynamics of microtubules

References

• Jumarie G. Modified riemann-liouville derivative and fractional taylor series of non-differentiable functions further results. Comput Math Appl. 2006;51:1367–1376. doi: 10.1016/j.camwa.2006.02.001
   Web of Science ®Google Scholar
• Jumarie G. Table of some basic fractional calculus formula derived from a modfied riemann-liouville derivative for nondifferentiable functions. Appl Maths Lett. 2009;22:378–385. doi: 10.1016/j.aml.2008.06.003
   Web of Science ®Google Scholar
• Abdou MA. On the fractional order space-time nonlinear equations arising in plasma physics. Indian J Phys. 2018; in press.
   Web of Science ®Google Scholar
• Abdou MA, Soliman AA. New exact travelling wave solutions for space-time fractional nonlinear equations describing nonlinear transmission lines. Results Phys. 2018;9:1497–1501. doi: 10.1016/j.rinp.2018.04.031
   Web of Science ®Google Scholar
• Abdou MA, Attia AT. New exact solutions for space- time fractal order on the ion acoustic waves in electron-positron-ion plasma. Nonlinear Sci Lett A. 2014;5:35–44.
   Google Scholar
• Abdou MA, Elgarayhi A, El-Shewy E. Fractional complex transform for space-time fractional nonlinear differential equations arising in plasma physics. Nonlinear Sci Lett A. 2014;5:31–34.
   Google Scholar
• Abdou MA, Elhanbaly A. New application of the fractional sub-equation method. Nonlinear Sci Lett A. 2015;6(1):10–18.
   Google Scholar
• Abdou MA, Soliman AA, Biswas A, et al. Dark–singular combo optical solitons with fractional complex ginzburg–landau equation. Optik. 2018;171:463–467. doi: 10.1016/j.ijleo.2018.06.076
   Web of Science ®Google Scholar
• Abdou MA, Yildirim A. Approximate analytical solution to time fractional nonlinear evolution equations. Int J Numer Meth Heat Fluid Fl. 2012;22:829–838. doi: 10.1108/09615531211255734
   Web of Science ®Google Scholar
• Abdou MA. An analytical method for space-time fractional nonlinear differentialequations arising in plasma physics. J Ocean Eng Sci. 2017;2:288–292. doi: 10.1016/j.joes.2017.09.002
   Web of Science ®Google Scholar
• Zhao J, Tang B, Kumar S, et al. The extended fractional subequation method for nonlinear fractional differential equations. Math Prob Eng. 2012. doi:10.1155/2012/924956.
   Web of Science ®Google Scholar
• Guo S, Mei L, Li Y, et al. The improved fractional subequation method and Its applications to the space-time fractional differential equations in fluid mechanics. Phys Lett A. 2012;376:407–411. doi: 10.1016/j.physleta.2011.10.056
   Web of Science ®Google Scholar
• Lu B. Backlund transformation of frcational riccati equation and its applications to nolinear fractional partial differential equations. Phys Lett A. 2012;376:2045–2048. doi: 10.1016/j.physleta.2012.05.013
   Web of Science ®Google Scholar
• AliAkbar M, Hj N, Ali M, Roy R. Closed form solutions of two time fractional nonlinear wave equations. Results in Phys. 2018;9:1031–1039. doi: 10.1016/j.rinp.2018.03.059
   Web of Science ®Google Scholar
• Feng Q. A new analytical method for seeking travelling wave solutions of space time fractional partial differential equations arising in math.physics. Optik Optical Inter J Light Electron Opt. 2017;130:310–323. doi: 10.1016/j.ijleo.2016.10.106
   Google Scholar
• He JH, Li ZB. Fractional complex transform for fractional differential equations. Math Comput Appl. 2010;15:970–973.
   Google Scholar
• Meng F, Feng Q. A New fractional subequation method and Its applications for space-time fractional partial differential equations. J App Math. 2013;2013:1–10. Article ID 481729. doi:10.1155/2013/481729.
   Google Scholar
• Zheng B, Wen C. Exact solutions for fractional partial differential equations by a new fractional sub-equation method. Adv Differ Equ. 2013;2013:199. doi:10.1186/1687-1847-2013-199.
   Web of Science ®Google Scholar
• Guner O, Bekir A. A novel method for nonlinear fractional differential equations using symbolic computation. Wave Random Complex Media. 2017;27(1):163–170. doi: 10.1080/17455030.2016.1213462
   Web of Science ®Google Scholar
• Guner O, Atik H. Soliton solution of fractional-order nonlinear differential equations based on the exp-function method. Optik. 2016;127:10076–10083. doi: 10.1016/j.ijleo.2016.07.070
   Web of Science ®Google Scholar
• Guner O, Bekir A. Exact solutions to the time-fractional differential equations via local fractional derivatives. Wave Random Complex Media. 2018;28(1):139–149. doi: 10.1080/17455030.2017.1332442
   Web of Science ®Google Scholar
• Guner O, Bekir A, Korkmaz A. Tanh-type and sech-type solitons for some space-time fractional PDE models. Eur Phys J Plus. 2017;132:92. doi:10.1140/epjp/i. doi: 10.1140/epjp/i2017-11370-7
   Web of Science ®Google Scholar
• Roshid H, Rahman N, Akbar MA. Traveling wave solutions of nonlinear Klein-Gordon equation by extended (G’/G)-expansion method. Annals Pure Appl Math. 2013;3(1):10–16.
   Google Scholar
• Alam MN, Akbar MA, Tauseef S. General traveling wave solutions of the strain wave equation in microstructured solids via the new approach of generalized (G’/G)-expansion method. Alexandria Eng J. 2014;53:233–241. doi: 10.1016/j.aej.2014.01.002
   Google Scholar
• Hafez MG, Alam MN, Akber MA. Exact traveling wave solutions of the Klein-Gordon equation using the novel (G’/G)-expansion method. Results Phys. 2014;4:177–184. doi: 10.1016/j.rinp.2014.09.001
   Web of Science ®Google Scholar
• Isalm R, Alam MN, Kazi AKM, et al. Traveling wave solutions of nonlinear evolution equations via Exp(–phi(η))-expansion method. Global J Sci Frontier Res. 2013;13(11):63–71.
   Google Scholar
• Alam MN, Akbar MA, Roshid H. Study of nonlinear evolution equations to construct traveling wave solutions via the new approach of the generalized (G’/G)-expansion method. Math Stat. 2013;1(3):102–112.
   Google Scholar
• Dustin P. Microtubules. Berlin: Springer; 1984.
   Google Scholar
• Zdravkpovic S. Kink and breathers in nonlinear dynamics of microtubules, AIP Conf. Proceeding 1618, 1021 (2014).
   Google Scholar
• Satric MV, Tuszynski JA. Nonlinear dynamics of microtubles: biophysical implications. J Biol Phys. 2005;31(3–4):487–500. doi: 10.1007/s10867-005-7288-1
   PubMed Web of Science ®Google Scholar
• Tuszyski JA, Brown JA, Crawford E, et al. Molecular dynamics simulations of tubulin structure and calculations of electrostatic properties of microtubules. Math Comput Model. 2005;41:1055–1070. doi: 10.1016/j.mcm.2005.05.002
   Google Scholar
• Zeković S, Muniyappan A, Zdravković S, Kavitha L. Employment of Jacobian elliptic functions for solving problems in nonlinear dynamics of microtubules. Chinese Phys B. 2014;23(2):020504.
   Google Scholar
• Zdravkovic S, Zekovic S. Nonlinear dynamics of microtubles and series expansion unknown function expansion. Chinese J Phys. doi:10.1016/Ciph.2017.10.009.
   Google Scholar
• Havelka D, Cifra M, Kucera O, et al. High -frequency electric field and radiation characteristics of cellular microtubule network. J Theor Biol. 2011;286:31–40. doi: 10.1016/j.jtbi.2011.07.007
   PubMed Web of Science ®Google Scholar
• Kuera O, Havelka D. Mechano-electrical vibrations of microtubules link to subcellular morphology. BioSystems. 2012;109:346–355. doi: 10.1016/j.biosystems.2012.04.009
   PubMed Web of Science ®Google Scholar
• Zdravkovi S. Microtubules: a network for solitary waves. J Serb Chem Soc. 2017;82(5):469–481. doi: 10.2298/JSC161118020Z
   Web of Science ®Google Scholar
• Zdravkovi S, Satari MV, Maluckov A, et al. A nonlinear model of the dynamics of radial dislocations in microtubules. Appl Math Comput. 2014;237:227–237.
   Web of Science ®Google Scholar
• Zdravkovic S, Bugay AN, Aru GF. Localized modulated wave in microtubles. Chaos. 2014;24:023139. doi:10.1063/1.4885777.
   PubMed Web of Science ®Google Scholar
• Alam N, Belgacem FBM. Mictotubles nonlinear model dynamics investigation tthrough the exp(–φ) method implementation. Math. 2016;4(1):6. doi:10.3390/math4010006.
   Web of Science ®Google Scholar