June 2019, 12(3): 447-454. doi: 10.3934/dcdss.2019029

New exact solutions for some fractional order differential equations via improved sub-equation method

Department of Mathematic Education, Adıyaman University, Adıyaman, Turkey

Received  June 2017 Published  September 2018

In this paper, improved sub-equation method is proposed to obtain new exact analytical solutions for some nonlinear fractional differential equations by means of modified Riemann Liouville derivative. The method is applied to time-fractional biological population model and space-time fractional Fisher equation successfully. Finally, simulations of new exact analytical solutions are presented graphically.

Citation: Berat Karaagac. New exact solutions for some fractional order differential equations via improved sub-equation method. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 447-454. doi: 10.3934/dcdss.2019029
References:
[1]

E. A. B. Abdel-SalamaE. A. Yousif and G. F. Hassanc, Solution of nonlinear space time fractional differential equations via the fractional projective Riccati expansion method, Physics, 2013 (2015), 657-675.

[2]

J. F. Alzaidy, Fractional sub-equation method and its applications to the space-time Fractional differential equations in Mathematical Physics, British journal of Math. and Comput. Sci., 3 (2013), 153-163. doi: 10.9734/BJMCS/2013/2908.

[3]

A. Atangana and B. Dumitru, New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769.

[4]

A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana aleanu derivatives with fractional order, Solitons and Fractals, 89 (2016), 447-454. doi: 10.1016/j.chaos.2016.02.012.

[5]

D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, World Scientific Publishing, Singapore, 2012. doi: 10.1142/9789814355216.

[6]

H. M. Baskonus and H. Bulut, Regarding on the prototype solutions for the nonlinear fractional-order biological population model, AIP Conference Proceedings, 1738 (2016), 290004. doi: 10.1063/1.4952076.

[7]

A. Bekir and O. Guner, Exact solutions of nonlinear Fractional differential equations by $ (G^{^{\prime }}/G)$−expansion method, Chin Phys. B., 22 (2013), 110202.

[8]

Z. Bin, $ (G^{^{\prime }}/G)$-expansion method for solving Fractional partial differential equations in the theory mathematical physics, Commun. in Theo. Physics., 58 (2012), 623-628.

[9]

M. Cui, Compact finite difference method for the Fractional diffusion equation, Jounal of Computational Physics, 228 (2009), 7792-7804. doi: 10.1016/j.jcp.2009.07.021.

[10]

A. M. A. El-SayedS. Z. Rida and A. A. M. Arafa, Exact solution of fractional order biological population model, Commun. in Theo. Phys., 52 (2009), 992-996. doi: 10.1088/0253-6102/52/6/04.

[11]

A. EsenY. UcarO. Tasbozan and N. M. Yagmurlu, A galerkin finite element method to solve fractional diffusion and fractional Diffusion-Wave equations, Mathematical Modelling and Analysis, 18 (2013), 260-273. doi: 10.3846/13926292.2013.783884.

[12]

R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 355-369. doi: 10.1111/j.1469-1809.1937.tb02153.x.

[13]

O. Guner and A. Bekir, On the concept of exact solution for nonlinear differential equations of fractional-order, Math. Meth. Appl. Sci., 39 (2016), 4035-4043. doi: 10.1002/mma.3845.

[14]

Q. HuangG. Huang and H. Zhang, A Finite element solution for the fractional advection-dispersion equation, Advances in Water Resorces, 131 (2008), 1578-1589. doi: 10.1016/j.advwatres.2008.07.002.

[15]

H. Jafari, H. Tajadedi, N. Kadkhoda and D. Baleanu, Fractional sub-equation method for Cahn-Hilliard and Klein-Gordon equations, Abstract and Applied Analysis, 2013 (2013), Art. ID 587179, 5 pp.

[16]

Y. Liu, Study on space-time fractional nonlinear biological equation in Radial Symmetry, Mathematical Problems in Engineering, 2013 (2013), Art. ID 654759, 6 pp.

[17]

Y. LiuZ. Li and Y. Zhang, Homotophy Perturbation Method to fractional biological population equation, Fractional Differential Calculus, 1 (2011), 117-124. doi: 10.7153/fdc-01-07.

[18]

Z. Odibat and S. Momoni, Application of Variational Iteration Method to nonlinear differential equations of fractional order, International Journal of Science and Numerical Simulation, 7 (2006), 27-34. doi: 10.1515/IJNSNS.2006.7.1.27.

[19]

I. Podlubny, An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, San Diego-Boston-New York-London-Tokyo-Toronto, 1999.

[20]

B. S. T. Alkahtani, Chua's circuit model with Atangana aleanu derivative with fractional order Chaos, Solitons and Fractals, 89 (2016), 539-546. doi: 10.1016/j.chaos.2016.03.012.

[21]

M. SafariD. D. Ganji and M. Moslemi, Application of He's Variational Iteration method and Adomian's decomposition method to the fractional KdV-Burgers-Kuramoto equation, Computer and Mathematics with Applications., 58 (2009), 2091-2097. doi: 10.1016/j.camwa.2009.03.043.

[22]

S. Sahoo and S. S. Ray, Improved fractional sub-equation method for (3+1)-dimensional generalized fractional KdV- Zakharov-Kuznetsov equations, Computers and Mathematics with Applications, 70 (2015), 158-166. doi: 10.1016/j.camwa.2015.05.002.

[23]

G. W. Wang and T. Z. Xu, The Improved fractional sub-equation method and its applications to nonlinear fractional equations, Theoretical and Mathematical Physics, 66 (2014), 595-602.

[24]

G.-ch. Wu and E. W. M. Lee, Fractional Variational Iteration Method and its application, Physics Letter A, 374 (2010), 2506-2509. doi: 10.1016/j.physleta.2010.04.034.

[25]

Y. Zhang, A finite difference method for fractional partial differential equation, Appl. Math. and Comput., 2015 (2009), 524-529. doi: 10.1016/j.amc.2009.05.018.

[26]

S. ZhangQ. A. ZongD. Liu and Q. Gao, A generalized exp-function method for fractional Riccati differential equations, Comm. Fract. Calc., 1 (2010), 48-52.

[27]

S. Zhang and H. Q. Zhang, Fractional sub-equation method and its applications to nonlinear fractional PDEs, Phys. Lett. A,, 375 (2011), 1069-1073. doi: 10.1016/j.physleta.2011.01.029.

show all references

References:
[1]

E. A. B. Abdel-SalamaE. A. Yousif and G. F. Hassanc, Solution of nonlinear space time fractional differential equations via the fractional projective Riccati expansion method, Physics, 2013 (2015), 657-675.

[2]

J. F. Alzaidy, Fractional sub-equation method and its applications to the space-time Fractional differential equations in Mathematical Physics, British journal of Math. and Comput. Sci., 3 (2013), 153-163. doi: 10.9734/BJMCS/2013/2908.

[3]

A. Atangana and B. Dumitru, New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769.

[4]

A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana aleanu derivatives with fractional order, Solitons and Fractals, 89 (2016), 447-454. doi: 10.1016/j.chaos.2016.02.012.

[5]

D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, World Scientific Publishing, Singapore, 2012. doi: 10.1142/9789814355216.

[6]

H. M. Baskonus and H. Bulut, Regarding on the prototype solutions for the nonlinear fractional-order biological population model, AIP Conference Proceedings, 1738 (2016), 290004. doi: 10.1063/1.4952076.

[7]

A. Bekir and O. Guner, Exact solutions of nonlinear Fractional differential equations by $ (G^{^{\prime }}/G)$−expansion method, Chin Phys. B., 22 (2013), 110202.

[8]

Z. Bin, $ (G^{^{\prime }}/G)$-expansion method for solving Fractional partial differential equations in the theory mathematical physics, Commun. in Theo. Physics., 58 (2012), 623-628.

[9]

M. Cui, Compact finite difference method for the Fractional diffusion equation, Jounal of Computational Physics, 228 (2009), 7792-7804. doi: 10.1016/j.jcp.2009.07.021.

[10]

A. M. A. El-SayedS. Z. Rida and A. A. M. Arafa, Exact solution of fractional order biological population model, Commun. in Theo. Phys., 52 (2009), 992-996. doi: 10.1088/0253-6102/52/6/04.

[11]

A. EsenY. UcarO. Tasbozan and N. M. Yagmurlu, A galerkin finite element method to solve fractional diffusion and fractional Diffusion-Wave equations, Mathematical Modelling and Analysis, 18 (2013), 260-273. doi: 10.3846/13926292.2013.783884.

[12]

R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 355-369. doi: 10.1111/j.1469-1809.1937.tb02153.x.

[13]

O. Guner and A. Bekir, On the concept of exact solution for nonlinear differential equations of fractional-order, Math. Meth. Appl. Sci., 39 (2016), 4035-4043. doi: 10.1002/mma.3845.

[14]

Q. HuangG. Huang and H. Zhang, A Finite element solution for the fractional advection-dispersion equation, Advances in Water Resorces, 131 (2008), 1578-1589. doi: 10.1016/j.advwatres.2008.07.002.

[15]

H. Jafari, H. Tajadedi, N. Kadkhoda and D. Baleanu, Fractional sub-equation method for Cahn-Hilliard and Klein-Gordon equations, Abstract and Applied Analysis, 2013 (2013), Art. ID 587179, 5 pp.

[16]

Y. Liu, Study on space-time fractional nonlinear biological equation in Radial Symmetry, Mathematical Problems in Engineering, 2013 (2013), Art. ID 654759, 6 pp.

[17]

Y. LiuZ. Li and Y. Zhang, Homotophy Perturbation Method to fractional biological population equation, Fractional Differential Calculus, 1 (2011), 117-124. doi: 10.7153/fdc-01-07.

[18]

Z. Odibat and S. Momoni, Application of Variational Iteration Method to nonlinear differential equations of fractional order, International Journal of Science and Numerical Simulation, 7 (2006), 27-34. doi: 10.1515/IJNSNS.2006.7.1.27.

[19]

I. Podlubny, An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, San Diego-Boston-New York-London-Tokyo-Toronto, 1999.

[20]

B. S. T. Alkahtani, Chua's circuit model with Atangana aleanu derivative with fractional order Chaos, Solitons and Fractals, 89 (2016), 539-546. doi: 10.1016/j.chaos.2016.03.012.

[21]

M. SafariD. D. Ganji and M. Moslemi, Application of He's Variational Iteration method and Adomian's decomposition method to the fractional KdV-Burgers-Kuramoto equation, Computer and Mathematics with Applications., 58 (2009), 2091-2097. doi: 10.1016/j.camwa.2009.03.043.

[22]

S. Sahoo and S. S. Ray, Improved fractional sub-equation method for (3+1)-dimensional generalized fractional KdV- Zakharov-Kuznetsov equations, Computers and Mathematics with Applications, 70 (2015), 158-166. doi: 10.1016/j.camwa.2015.05.002.

[23]

G. W. Wang and T. Z. Xu, The Improved fractional sub-equation method and its applications to nonlinear fractional equations, Theoretical and Mathematical Physics, 66 (2014), 595-602.

[24]

G.-ch. Wu and E. W. M. Lee, Fractional Variational Iteration Method and its application, Physics Letter A, 374 (2010), 2506-2509. doi: 10.1016/j.physleta.2010.04.034.

[25]

Y. Zhang, A finite difference method for fractional partial differential equation, Appl. Math. and Comput., 2015 (2009), 524-529. doi: 10.1016/j.amc.2009.05.018.

[26]

S. ZhangQ. A. ZongD. Liu and Q. Gao, A generalized exp-function method for fractional Riccati differential equations, Comm. Fract. Calc., 1 (2010), 48-52.

[27]

S. Zhang and H. Q. Zhang, Fractional sub-equation method and its applications to nonlinear fractional PDEs, Phys. Lett. A,, 375 (2011), 1069-1073. doi: 10.1016/j.physleta.2011.01.029.

Figure 1.  The numerical simulations of real and imaginal part for fractional Biological Population Model for $u^{3}_{1}$, respectively
Figure 2.  The numerical simulations of fractional Fisher equation for $u^{1}_{1}$, $u^{1}_{2}$, $u^{3}_{1}$ and $u^{4}_{1}$, respectively
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