| Citation: |
Jing Shao, Boling Guo, Lingling Duan. ANALYTICAL STUDY OF THE TEO-DIMENSIONAL TIME-FRTACTIONAL NAVIER-STOKES EQUATIONS[J]. Journal of Applied Analysis & Computation, 2019, 9(5): 1999-2022. doi: 10.11948/20190065
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ANALYTICAL STUDY OF THE TEO-DIMENSIONAL TIME-FRTACTIONAL NAVIER-STOKES EQUATIONS
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Abstract
In this paper, the two-dimensional (2D) Holf-Cole transformation with mass conservation in the frame of conformable derivative is developed, and then by introducing some exact solutions that satisfy linear differential equations and using the symbolic computation method, four exact solutions of 2D-nonlinear Navier-Stokes equations (NSEs) with the conformable time-fractional derivative are established. Some physical properties of the exact solutions are described preliminarily. Our results are the first ones on analytical study for the 2D time-fractional NSEs. -
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References
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Jing Shao, Boling Guo, Lingling Duan. ANALYTICAL STUDY OF THE TEO-DIMENSIONAL TIME-FRTACTIONAL NAVIER-STOKES EQUATIONS[J]. Journal of Applied Analysis & Computation, 2019, 9(5): 1999-2022. doi: 10.11948/20190065
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Figure 1. Case 1. The parameters are chosen as:
$ \nu=0.1 $ ,$ l=1 $ ,$ t=10 $ ,$ C=-1 $ ,$ C^{*}=-2 $ and$ \alpha=\frac{1}{2} $ -
Figure 2. Case 1. The parameters are chosen as:
$ \nu=0.1 $ ,$ l=1 $ ,$ C=-1 $ ,$ C^{*}=-2 $ and$ \alpha=\frac{1}{2} $ -
Figure 3. Case 1. The parameters are chosen as:
$ \nu=0.1 $ ,$ l=1 $ ,$ C=-1 $ ,$ C^{*}=-2 $ and$ \alpha=\frac{1}{2} $ -
Figure 4. Case 2. The parameters are chosen as:
$ \nu=0.1 $ ,$ t=1 $ ,$ x\times y\in[-5, 5]\times[-5, 5] $ ,$ a_{0}=\exp(1) $ and$ \alpha=\frac{1}{2} $ -
Figure 5. Case 2. The parameters are chosen as:
$ \nu=0.1 $ ,$ t=1 $ ,$ x\times y\in[-10, 10]\times[-10, 10] $ ,$ a_{0}=\exp(1) $ and$ \alpha=\frac{1}{2} $ -
Figure 6. Case 2. The parameters are chosen as:
$ \nu=0.1 $ ,$ t=10000 $ ,$ a_{0}=\exp(1) $ and$ \alpha=\frac{1}{2} $ -
Figure 7. Case 2. The parameters are chosen as:
$ \nu=0.1 $ ,$ t=10000 $ ,$ a_{0}=\exp(1) $ and$ \alpha=\frac{1}{2} $ -
Figure 8. Case 2. The parameters are chosen as:
$ \nu=0.1 $ ,$ t=10000 $ ,$ a_{0}=\exp(1) $ and$ \alpha=\frac{1}{2} $ -
Figure 9. Case 3. The parameters are chosen as:
$ \nu=0.1 $ ,$ C_{0}=10.0 $ ,$ C_{1}=0.005 $ ,$ C_{2}=500.0 $ , and$ \alpha=\frac{1}{4} $ -
Figure 10. Case 4. The parameters are chosen as:
$ \nu=0.01 $ ,$ t=10 $ ,$ a_{0}=\exp(3) $ ,$ X\times Y\in[-5, 5]\times[-5, 5] $ and$ \alpha=\frac{1}{4} $ -
Figure 11. Case 4. The parameters are chosen as:
$ \nu=0.01 $ ,$ t=10 $ ,$ a_{0}=\exp(3) $ ,$ X\times Y\in[-20, 20]\times[-20, 20] $ and$ \alpha=\frac{1}{4} $