• Previous Article
    Numerical analysis of coupled fractional differential equations with Atangana-Baleanu fractional derivative
  • DCDS-S Home
  • This Issue
  • Next Article
    On the existence and uniqueness of the solution of an optimal control problem for Schrödinger equation
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. AgueroM. Najera and M. Carrillo, Non classic solitonic structures in dna vibrational dynamics, Int. J. Modern Phys. B., 22 (2008), 2571-2582.

[4]

W. AlkaA. 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. GongJ. 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. JawadM. 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. KongS. 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. WangX. 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. AgueroM. Najera and M. Carrillo, Non classic solitonic structures in dna vibrational dynamics, Int. J. Modern Phys. B., 22 (2008), 2571-2582.

[4]

W. AlkaA. 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. GongJ. 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. JawadM. 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. KongS. 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. WangX. 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 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) $\sigma = 0.5$ , $\gamma = 0.5$
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) $\sigma = 0.5$ , $\gamma = 0.5$
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) $\sigma = 0.5$ , $\gamma = 0.5$
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) $\sigma = 0.5$ , $\gamma = 0.5$
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) $\sigma = 0.5$ , $\gamma = 0.5$
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) $\sigma = 0.5$ , $\gamma = 0.5$
[1]

A. Ducrot. Travelling wave solutions for a scalar age-structured equation. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 251-273. doi: 10.3934/dcdsb.2007.7.251

[2]

Christopher K. R. T. Jones, Robert Marangell. The spectrum of travelling wave solutions to the Sine-Gordon equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 925-937. doi: 10.3934/dcdss.2012.5.925

[3]

Arnaud Ducrot, Michel Langlais, Pierre Magal. Qualitative analysis and travelling wave solutions for the SI model with vertical transmission. Communications on Pure & Applied Analysis, 2012, 11 (1) : 97-113. doi: 10.3934/cpaa.2012.11.97

[4]

Claude-Michael Brauner, Josephus Hulshof, J.-F. Ripoll. Existence of travelling wave solutions in a combustion-radiation model. Discrete & Continuous Dynamical Systems - B, 2001, 1 (2) : 193-208. doi: 10.3934/dcdsb.2001.1.193

[5]

Tong Li, Kun Zhao. Global existence and long-time behavior of entropy weak solutions to a quasilinear hyperbolic blood flow model. Networks & Heterogeneous Media, 2011, 6 (4) : 625-646. doi: 10.3934/nhm.2011.6.625

[6]

Jibin Li, Fengjuan Chen. Exact travelling wave solutions and their dynamical behavior for a class coupled nonlinear wave equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 163-172. doi: 10.3934/dcdsb.2013.18.163

[7]

Jibin Li, Weigou Rui, Yao Long, Bin He. Travelling wave solutions for higher-order wave equations of KDV type (III). Mathematical Biosciences & Engineering, 2006, 3 (1) : 125-135. doi: 10.3934/mbe.2006.3.125

[8]

Tong Li, Sunčica Čanić. Critical thresholds in a quasilinear hyperbolic model of blood flow. Networks & Heterogeneous Media, 2009, 4 (3) : 527-536. doi: 10.3934/nhm.2009.4.527

[9]

Tong Li, Kun Zhao. On a quasilinear hyperbolic system in blood flow modeling. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 333-344. doi: 10.3934/dcdsb.2011.16.333

[10]

Chun Zong, Gen Qi Xu. Observability and controllability analysis of blood flow network. Mathematical Control & Related Fields, 2014, 4 (4) : 521-554. doi: 10.3934/mcrf.2014.4.521

[11]

Mudassar Imran, Youssef Raffoul, Muhammad Usman, Chi Zhang. A study of bifurcation parameters in travelling wave solutions of a damped forced Korteweg de Vries-Kuramoto Sivashinsky type equation. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 691-705. doi: 10.3934/dcdss.2018043

[12]

Min Li, Zhaoyang Yin. Blow-up phenomena and travelling wave solutions to the periodic integrable dispersive Hunter-Saxton equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6471-6485. doi: 10.3934/dcds.2017280

[13]

Jibin Li. Bifurcations and exact travelling wave solutions of the generalized two-component Hunter-Saxton system. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1719-1729. doi: 10.3934/dcdsb.2014.19.1719

[14]

Chueh-Hsin Chang, Chiun-Chuan Chen. Travelling wave solutions of a free boundary problem for a two-species competitive model. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1065-1074. doi: 10.3934/cpaa.2013.12.1065

[15]

Borys Alvarez-Samaniego, Pascal Azerad. Existence of travelling-wave solutions and local well-posedness of the Fowler equation. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 671-692. doi: 10.3934/dcdsb.2009.12.671

[16]

Tiberiu Harko, Man Kwong Mak. Travelling wave solutions of the reaction-diffusion mathematical model of glioblastoma growth: An Abel equation based approach. Mathematical Biosciences & Engineering, 2015, 12 (1) : 41-69. doi: 10.3934/mbe.2015.12.41

[17]

Chiun-Chuan Chen, Li-Chang Hung, Masayasu Mimura, Daishin Ueyama. Exact travelling wave solutions of three-species competition--diffusion systems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (8) : 2653-2669. doi: 10.3934/dcdsb.2012.17.2653

[18]

Wei Wang, Wanbiao Ma. Global dynamics and travelling wave solutions for a class of non-cooperative reaction-diffusion systems with nonlocal infections. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3213-3235. doi: 10.3934/dcdsb.2018242

[19]

Martin Bauer, Martins Bruveris, Philipp Harms, Peter W. Michor. Soliton solutions for the elastic metric on spaces of curves. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1161-1185. doi: 10.3934/dcds.2018049

[20]

Bendong Lou. Traveling wave solutions of a generalized curvature flow equation in the plane. Conference Publications, 2007, 2007 (Special) : 687-693. doi: 10.3934/proc.2007.2007.687

2017 Impact Factor: 0.561

Article outline

Figures and Tables

[Back to Top]