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June 2019, 12(3): 487-502. doi: 10.3934/dcdss.2019032

## The first integral method for two fractional non-linear biological models

 1 Department of Physics, Adeyemi College of Education, Ondo, Nigeria 2 Department of Mathematics Education, University of Education, Winneba, (Kumasi campus), Ghana 3 Department of Physical Sciences, Al-Hikmah University, Ilorin, Nigeria

* Corresponding author: olusolakolebaje2008@gmail.com

Received  June 2017 Revised  November 2017 Published  September 2018

Travelling wave solutions of the space and time fractional models for non-linear blood flow in large vessels and Deoxyribonucleic acid (DNA) molecule dynamics defined in the sense of Jumarie's modified Riemann-Liouville derivative via the first integral method are presented in this study. A fractional complex transformation was applied to turn the fractional biological models into an equivalent integer order ordinary differential equation. The validity of the solutions to the fractional biological models obtained with first integral method was achieved by putting them back into the models. We observed that introducing fractional order to the biological models changes the nature of the solution.

Citation: Olusola Kolebaje, Ebenezer Bonyah, Lateef Mustapha. The first integral method for two fractional non-linear biological models. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 487-502. doi: 10.3934/dcdss.2019032
##### References:
 [1] S. Abbasbandy and A. Shirzadi, The first integral method for modified benjamin-bona-mahony equation, Communications in Nonlinear Science and Numerical Simulation, 15 (2010), 1759-1764. doi: 10.1016/j.cnsns.2009.08.003. [2] M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, Pa, USA, 1981. [3] M. Aguero, M. Najera and M. Carrillo, Non classic solitonic structures in dna vibrational dynamics, Int. J. Modern Phys. B., 22 (2008), 2571-2582. [4] W. Alka, A. Goyal and C. N. Kumar, Nonlinear dynamics of DNA-Riccati generalized solitary wave solutions, Physics Letters A, 375 (2011), 480-483. doi: 10.1016/j.physleta.2010.11.017. [5] A. Bekir and O. Unsal, Analytic treatment of nonlinear evolution equations using first integral method, Pramana, 79 (2012), 3-17. doi: 10.1007/s12043-012-0282-9. [6] N. Bourbaki, Commutative Algebra, Addison-Wesley, Paris, 1972. [7] H. Demiray, Weakly nonlinear waves in a viscous fluid contained in a viscoelastic tube with variable cross-section, Eur J. Mech. A. Solid, 24 (2005), 337-347. doi: 10.1016/j.euromechsol.2004.12.002. [8] H. Demiray, Variable coefficient modified kdv equation in fluid-filled elastic tubes with stenosis: Solitary waves, Chaos, Solitons and Fractals, 42 (2009), 358-364. doi: 10.1016/j.chaos.2008.12.014. [9] T. R. Ding and C. Z. Li, Ordinary Differential Equations, Peking University Press, Peking, 1996. [10] B. Eliasson and P. K. Shukla, Formation and dynamics of finite amplitude localized pulses in elastic tubes, Phys. Rev. E. , 71 (2005), 067302. doi: 10.1103/PhysRevE.71.067302. [11] E. Fan, Extended tanh-function method and its applications to nonlinear equations, Physics Letters A, 277 (2000), 212-218. doi: 10.1016/S0375-9601(00)00725-8. [12] E. G. Fan, Two new applications of the homogeneous balance method, Physics Letters A, 265 (2000), 353-357. doi: 10.1016/S0375-9601(00)00010-4. [13] Z. S. Feng, Exact solution to an approximate sine-gordon equation in (n+1)-dimensional space, Physics Letters A, 302 (2002), 64-76. doi: 10.1016/S0375-9601(02)01114-3. [14] X. Gong, J. Tian and J. Wang, First integral method for an oscillator system, Electronic Journal of Differential Equations, 96 (2013), 1-12. [15] Y. Hashimuze, Nonlinear pressure waves in a fluid-filled elastic tube, J. Phys. Soc. Jpn., 54 (1985), 3305-3312. [16] J. H. He and X. H. Wu, Exp-function method for nonlinear wave equations, Chaos, Solitons and Fractals, 30 (2006), 700-708. doi: 10.1016/j.chaos.2006.03.020. [17] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing, New Jersey, NJ, USA, 2000. doi: 10.1142/9789812817747. [18] R. Hirota, Exact envelope-soliton solutions of a nonlinear wave equation, Journal of Mathematical Physics, 14 (1973), 805-809. doi: 10.1063/1.1666399. [19] A. J. M. Jawad, M. D. Petković and A. Biswas, Modified simple equation method for nonlinear evolution equations, Applied Mathematics and Computation, 217 (2010), 869-877. doi: 10.1016/j.amc.2010.06.030. [20] G. Jumarie, Modified riemann-liouville derivative and fractional taylor series of nondifferentiable functions further results, Computers and Mathematics with Applications, 51 (2006), 1367-1376. doi: 10.1016/j.camwa.2006.02.001. [21] G. Jumarie, Fractional partial differential equations and modified riemann-liouville derivative new methods for solution, Journal of Applied Mathematics and Computing, 24 (2007), 31-48. doi: 10.1007/BF02832299. [22] M. A. Knyazev and D. M. Knyazev, New kink-like solutions for nonlinear equation describing the dynamics of dna, Journal of Physical Studies, 16 (2012), 1001-1004. [23] G. R. Kol and C. B. Tabi, Application of the g'/g expansion method to nonlinear blood flow in large vessels, IC, 26 (2010), 1-9. [24] D. X. Kong, S. Y. Lou and J. Zeng, Nonlinear dynamics in a new double chain-model of DNA, Commun. Theor. Phys., 36 (2001), 737-742. [25] Z. B. Li and J. H. He, Fractional complex transform for fractional differential equations, Mathematical and Computational Applications, 15 (2010), 970-973. [26] W. Malfliet, The tanh method: A tool for solving certain classes of non-linear pdes, Mathematical Methods in the Applied Sciences, 28 (2005), 2031-2035. doi: 10.1002/mma.650. [27] V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer, Berlin, Germany, 1991. doi: 10.1007/978-3-662-00922-2. [28] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, NY, USA, 1993. [29] M. R. Miura, Backlund Transformation, Springer-Verlag, Berlin-New York, 1976. [30] S. Noubissie and P. Woafo, Dynamics of solitary blood waves in arteries with prostheses, Phys. Rev. E. , 67 (2003), 0419111. doi: 10.1103/PhysRevE.67.041911. [31] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974. [32] M. Peyrard and A. Bishop, Statistical mechanics of a nonlinear model of dna denaturation, Phys. Rev. Lett., 62 (1989), 2755-2758. doi: 10.1103/PhysRevLett.62.2755. [33] K. R. Raslan, The first integral method for solving some important nonlinear partial differential equations, Nonlinear Dynamics, 53 (2008), 281-286. doi: 10.1007/s11071-007-9262-x. [34] C. Rogers and W. F. Shadwick, Backlund Transformation, Academic Press, New York, NY, USA, 1982. [35] M. L. Wang, X. Li and J. Zhang, The (frac(g'/g))-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Physics Letters A, 372 (2008), 417-423. doi: 10.1016/j.physleta.2007.07.051. [36] B. J. West, M. Bologna and P. Grigolini, Physics of Fractal Operators, Springer, New York, NY, USA, 2003. doi: 10.1007/978-0-387-21746-8. [37] L. V. Yakushevich, Nonlinear dna dynamics: A new model, Physics Letters A, 136 (1989), 413-417. doi: 10.1016/0375-9601(89)90425-8. [38] C. T. Yan, A simple transformation for nonlinear waves, Physics Letters A, 224 (1996), 77-84. doi: 10.1016/S0375-9601(96)00770-0. [39] Z. Y. Yan and H. Q. Zhang, New explicit solitary wave solutions and periodic wave solutions for whitham-broer-kaup equation in shallow water, Physics Letters A, 285 (2001), 355-362. doi: 10.1016/S0375-9601(01)00376-0. [40] S. Yomosa, Solitary waves in large blood vessels, J. Phys. Soc. Jpn., 56 (1987), 506-520. doi: 10.1143/JPSJ.56.506. [41] E. M. E. Zayed, Traveling wave solutions for higher dimensional nonlinear evolution equations using the (g'/g)- expansion method, J. Phys. A, 42 (2009), 195202, 13 pp. doi: 10.1088/1751-8113/42/19/195202. [42] E. M. E. Zayed, A note on the modified simple equation method applied to sharma-tasso-olver equation, Applied Mathematics and Computation, 218 (2011), 3962-3964. doi: 10.1016/j.amc.2011.09.025. [43] E. M. E. Zayed and A. H. Arnous, Many exact solutions for nonlinear dynamics of dna model using the generalized riccati equation mapping method, Scientific Research and Essays, 8 (2013), 340-346. [44] E. M. E. Zayed and S. A. H. Ibrahim, Exact solutions of nonlinear evolution equations in mathematical physics using the modified simple equation method, Chinese Physics Letters, 29 2012. doi: 10.1088/0256-307X/29/6/060201.

show all references

##### References:
 [1] S. Abbasbandy and A. Shirzadi, The first integral method for modified benjamin-bona-mahony equation, Communications in Nonlinear Science and Numerical Simulation, 15 (2010), 1759-1764. doi: 10.1016/j.cnsns.2009.08.003. [2] M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, Pa, USA, 1981. [3] M. Aguero, M. Najera and M. Carrillo, Non classic solitonic structures in dna vibrational dynamics, Int. J. Modern Phys. B., 22 (2008), 2571-2582. [4] W. Alka, A. Goyal and C. N. Kumar, Nonlinear dynamics of DNA-Riccati generalized solitary wave solutions, Physics Letters A, 375 (2011), 480-483. doi: 10.1016/j.physleta.2010.11.017. [5] A. Bekir and O. Unsal, Analytic treatment of nonlinear evolution equations using first integral method, Pramana, 79 (2012), 3-17. doi: 10.1007/s12043-012-0282-9. [6] N. Bourbaki, Commutative Algebra, Addison-Wesley, Paris, 1972. [7] H. Demiray, Weakly nonlinear waves in a viscous fluid contained in a viscoelastic tube with variable cross-section, Eur J. Mech. A. Solid, 24 (2005), 337-347. doi: 10.1016/j.euromechsol.2004.12.002. [8] H. Demiray, Variable coefficient modified kdv equation in fluid-filled elastic tubes with stenosis: Solitary waves, Chaos, Solitons and Fractals, 42 (2009), 358-364. doi: 10.1016/j.chaos.2008.12.014. [9] T. R. Ding and C. Z. Li, Ordinary Differential Equations, Peking University Press, Peking, 1996. [10] B. Eliasson and P. K. Shukla, Formation and dynamics of finite amplitude localized pulses in elastic tubes, Phys. Rev. E. , 71 (2005), 067302. doi: 10.1103/PhysRevE.71.067302. [11] E. Fan, Extended tanh-function method and its applications to nonlinear equations, Physics Letters A, 277 (2000), 212-218. doi: 10.1016/S0375-9601(00)00725-8. [12] E. G. Fan, Two new applications of the homogeneous balance method, Physics Letters A, 265 (2000), 353-357. doi: 10.1016/S0375-9601(00)00010-4. [13] Z. S. Feng, Exact solution to an approximate sine-gordon equation in (n+1)-dimensional space, Physics Letters A, 302 (2002), 64-76. doi: 10.1016/S0375-9601(02)01114-3. [14] X. Gong, J. Tian and J. Wang, First integral method for an oscillator system, Electronic Journal of Differential Equations, 96 (2013), 1-12. [15] Y. Hashimuze, Nonlinear pressure waves in a fluid-filled elastic tube, J. Phys. Soc. Jpn., 54 (1985), 3305-3312. [16] J. H. He and X. H. Wu, Exp-function method for nonlinear wave equations, Chaos, Solitons and Fractals, 30 (2006), 700-708. doi: 10.1016/j.chaos.2006.03.020. [17] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing, New Jersey, NJ, USA, 2000. doi: 10.1142/9789812817747. [18] R. Hirota, Exact envelope-soliton solutions of a nonlinear wave equation, Journal of Mathematical Physics, 14 (1973), 805-809. doi: 10.1063/1.1666399. [19] A. J. M. Jawad, M. D. Petković and A. Biswas, Modified simple equation method for nonlinear evolution equations, Applied Mathematics and Computation, 217 (2010), 869-877. doi: 10.1016/j.amc.2010.06.030. [20] G. Jumarie, Modified riemann-liouville derivative and fractional taylor series of nondifferentiable functions further results, Computers and Mathematics with Applications, 51 (2006), 1367-1376. doi: 10.1016/j.camwa.2006.02.001. [21] G. Jumarie, Fractional partial differential equations and modified riemann-liouville derivative new methods for solution, Journal of Applied Mathematics and Computing, 24 (2007), 31-48. doi: 10.1007/BF02832299. [22] M. A. Knyazev and D. M. Knyazev, New kink-like solutions for nonlinear equation describing the dynamics of dna, Journal of Physical Studies, 16 (2012), 1001-1004. [23] G. R. Kol and C. B. Tabi, Application of the g'/g expansion method to nonlinear blood flow in large vessels, IC, 26 (2010), 1-9. [24] D. X. Kong, S. Y. Lou and J. Zeng, Nonlinear dynamics in a new double chain-model of DNA, Commun. Theor. Phys., 36 (2001), 737-742. [25] Z. B. Li and J. H. He, Fractional complex transform for fractional differential equations, Mathematical and Computational Applications, 15 (2010), 970-973. [26] W. Malfliet, The tanh method: A tool for solving certain classes of non-linear pdes, Mathematical Methods in the Applied Sciences, 28 (2005), 2031-2035. doi: 10.1002/mma.650. [27] V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer, Berlin, Germany, 1991. doi: 10.1007/978-3-662-00922-2. [28] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, NY, USA, 1993. [29] M. R. Miura, Backlund Transformation, Springer-Verlag, Berlin-New York, 1976. [30] S. Noubissie and P. Woafo, Dynamics of solitary blood waves in arteries with prostheses, Phys. Rev. E. , 67 (2003), 0419111. doi: 10.1103/PhysRevE.67.041911. [31] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974. [32] M. Peyrard and A. Bishop, Statistical mechanics of a nonlinear model of dna denaturation, Phys. Rev. Lett., 62 (1989), 2755-2758. doi: 10.1103/PhysRevLett.62.2755. [33] K. R. Raslan, The first integral method for solving some important nonlinear partial differential equations, Nonlinear Dynamics, 53 (2008), 281-286. doi: 10.1007/s11071-007-9262-x. [34] C. Rogers and W. F. Shadwick, Backlund Transformation, Academic Press, New York, NY, USA, 1982. [35] M. L. Wang, X. Li and J. Zhang, The (frac(g'/g))-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Physics Letters A, 372 (2008), 417-423. doi: 10.1016/j.physleta.2007.07.051. [36] B. J. West, M. Bologna and P. Grigolini, Physics of Fractal Operators, Springer, New York, NY, USA, 2003. doi: 10.1007/978-0-387-21746-8. [37] L. V. Yakushevich, Nonlinear dna dynamics: A new model, Physics Letters A, 136 (1989), 413-417. doi: 10.1016/0375-9601(89)90425-8. [38] C. T. Yan, A simple transformation for nonlinear waves, Physics Letters A, 224 (1996), 77-84. doi: 10.1016/S0375-9601(96)00770-0. [39] Z. Y. Yan and H. Q. Zhang, New explicit solitary wave solutions and periodic wave solutions for whitham-broer-kaup equation in shallow water, Physics Letters A, 285 (2001), 355-362. doi: 10.1016/S0375-9601(01)00376-0. [40] S. Yomosa, Solitary waves in large blood vessels, J. Phys. Soc. Jpn., 56 (1987), 506-520. doi: 10.1143/JPSJ.56.506. [41] E. M. E. Zayed, Traveling wave solutions for higher dimensional nonlinear evolution equations using the (g'/g)- expansion method, J. Phys. A, 42 (2009), 195202, 13 pp. doi: 10.1088/1751-8113/42/19/195202. [42] E. M. E. Zayed, A note on the modified simple equation method applied to sharma-tasso-olver equation, Applied Mathematics and Computation, 218 (2011), 3962-3964. doi: 10.1016/j.amc.2011.09.025. [43] E. M. E. Zayed and A. H. Arnous, Many exact solutions for nonlinear dynamics of dna model using the generalized riccati equation mapping method, Scientific Research and Essays, 8 (2013), 340-346. [44] E. M. E. Zayed and S. A. H. Ibrahim, Exact solutions of nonlinear evolution equations in mathematical physics using the modified simple equation method, Chinese Physics Letters, 29 2012. doi: 10.1088/0256-307X/29/6/060201.
Figure showing $P(z,t)$ with $\chi = z^{\sigma}/\Gamma(1+\sigma)+2 t^{\gamma}/\Gamma(1+\gamma)\sqrt{1-4 B_0^2}$ and $0\leq z, t \leq 20$ for (a) $\sigma = 1$ , $\gamma = 1$ , (b) $\sigma = 1$ , $\gamma = 0.5$ , (c) $\sigma = 0.5$ , $\gamma = 1$ , (d) $\sigma = 0.5$ , $\gamma = 0.5$
Figure showing $P(z,t)$ with $\chi = z^{\sigma}/\Gamma(1+\sigma)-2 t^{\gamma}/\Gamma(1+\gamma)\sqrt{1-4 B_0^2}$ and $0\leq z, t \leq 20$ for (a) $\sigma = 1$ , $\gamma = 1$ , (b) $\sigma = 1$ , $\gamma = 0.5$ , (c) $\sigma = 0.5$ , $\gamma = 1$ , (d) $\sigma = 0.5$ , $\gamma = 0.5$
Figure showing $\phi(x,t)$ (Eq. 65) with $B_0, l, k, Y, \lambda, \mu = 1$ , $h = 3.33$ , $\rho = 0.85$ , and $0\leq x, t \leq 5$ for (a) $\sigma = 1$ , $\gamma = 1$ , (b) $\sigma = 1$ , $\gamma = 0.5$ , (c) $\sigma = 0.5$ , $\gamma = 1$ , (d) $\sigma = 0.5$ , $\gamma = 0.5$
Figure showing $\phi(x,t)$ (Eq. 66) with $B_0, l, k, Y, \lambda, \mu = 1$ , $h = 3.33$ , $\rho = 0.85$ , and $0\leq x, t \leq 5$ for (a) $\sigma = 1$ , $\gamma = 1$ , (b) $\sigma = 1$ , $\gamma = 0.5$ , (c) $\sigma = 0.5$ , $\gamma = 1$ , (d) $\sigma = 0.5$ , $\gamma = 0.5$
Figure showing $\phi(x,t)$ (Eq. 78) with $B_0, l, k, Y, \lambda, \mu = 1$ , $h = 3.33$ , $\rho = 0.85$ , and $0\leq x, t \leq 5$ for (a) $\sigma = 1$ , $\gamma = 1$ , (b) $\sigma = 1$ , $\gamma = 0.5$ , (c) $\sigma = 0.5$ , $\gamma = 1$ , (d) $\sigma = 0.5$ , $\gamma = 0.5$
Figure showing $\phi(x,t)$ (Eq. 79) with $B_0, l, k, Y, \lambda, \mu = 1$ , $h = 3.33$ , $\rho = 0.85$ , and $0\leq x, t \leq 5$ for (a) $\sigma = 1$ , $\gamma = 1$ , (b) $\sigma = 1$ , $\gamma = 0.5$ , (c) $\sigma = 0.5$ , $\gamma = 1$ , (d) $\sigma = 0.5$ , $\gamma = 0.5$
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