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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

Olusola Kolebaje 1,, , Ebenezer Bonyah 2, and  Lateef Mustapha 3,

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.

Keywords: Travelling wave solutions, fractional calculus, blood flow, Deoxyribonucleic acid, soliton.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
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.  Google Scholar

[2]

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, Pa, USA, 1981.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

N. Bourbaki, Commutative Algebra, Addison-Wesley, Paris, 1972. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

T. R. Ding and C. Z. Li, Ordinary Differential Equations, Peking University Press, Peking, 1996. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

X. Gong, J. Tian and J. Wang, First integral method for an oscillator system, Electronic Journal of Differential Equations, 96 (2013), 1-12.   Google Scholar

[15]

Y. Hashimuze, Nonlinear pressure waves in a fluid-filled elastic tube, J. Phys. Soc. Jpn., 54 (1985), 3305-3312.   Google Scholar

[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.  Google Scholar

[17]

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing, New Jersey, NJ, USA, 2000. doi: 10.1142/9789812817747.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[25]

Z. B. Li and J. H. He, Fractional complex transform for fractional differential equations, Mathematical and Computational Applications, 15 (2010), 970-973.   Google Scholar

[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.  Google Scholar

[27]

V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer, Berlin, Germany, 1991. doi: 10.1007/978-3-662-00922-2.  Google Scholar

[28]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, NY, USA, 1993.  Google Scholar

[29]

M. R. Miura, Backlund Transformation, Springer-Verlag, Berlin-New York, 1976.  Google Scholar

[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.  Google Scholar

[31]

K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

C. Rogers and W. F. Shadwick, Backlund Transformation, Academic Press, New York, NY, USA, 1982. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

S. Yomosa, Solitary waves in large blood vessels, J. Phys. Soc. Jpn., 56 (1987), 506-520.  doi: 10.1143/JPSJ.56.506.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, Pa, USA, 1981.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

N. Bourbaki, Commutative Algebra, Addison-Wesley, Paris, 1972. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

T. R. Ding and C. Z. Li, Ordinary Differential Equations, Peking University Press, Peking, 1996. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

X. Gong, J. Tian and J. Wang, First integral method for an oscillator system, Electronic Journal of Differential Equations, 96 (2013), 1-12.   Google Scholar

[15]

Y. Hashimuze, Nonlinear pressure waves in a fluid-filled elastic tube, J. Phys. Soc. Jpn., 54 (1985), 3305-3312.   Google Scholar

[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.  Google Scholar

[17]

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing, New Jersey, NJ, USA, 2000. doi: 10.1142/9789812817747.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[25]

Z. B. Li and J. H. He, Fractional complex transform for fractional differential equations, Mathematical and Computational Applications, 15 (2010), 970-973.   Google Scholar

[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.  Google Scholar

[27]

V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer, Berlin, Germany, 1991. doi: 10.1007/978-3-662-00922-2.  Google Scholar

[28]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, NY, USA, 1993.  Google Scholar

[29]

M. R. Miura, Backlund Transformation, Springer-Verlag, Berlin-New York, 1976.  Google Scholar

[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.  Google Scholar

[31]

K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

C. Rogers and W. F. Shadwick, Backlund Transformation, Academic Press, New York, NY, USA, 1982. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

S. Yomosa, Solitary waves in large blood vessels, J. Phys. Soc. Jpn., 56 (1987), 506-520.  doi: 10.1143/JPSJ.56.506.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

Figure 1.  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) $\sigma = 0.5$ , $\gamma = 0.5$
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Figure 2.  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) $\sigma = 0.5$ , $\gamma = 0.5$
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Figure 3.  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) $\sigma = 0.5$ , $\gamma = 0.5$
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Figure 4.  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) $\sigma = 0.5$ , $\gamma = 0.5$
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Figure 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) $\sigma = 0.5$ , $\gamma = 0.5$
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Figure 6.  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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