| Citation: |
Muhammad Zafarullah Baber, Tahir Shahzad, Muhammad Waqas Yasin, Tukur Abdulkadir Sulaiman, Mustafa Bayram, Taha Radwan, Karim K. Ahmed. WAVES PROPAGATION FOR THE CONFORMABLE TIME-FRACTIONAL BENJAMIN-BONA-MAHONY-BURGERS EQUATION ARISING ON THE SURFACE OF WATER[J]. Journal of Applied Analysis & Computation, 2027, 17(3): 1761-1791. doi: 10.11948/20260145
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WAVES PROPAGATION FOR THE CONFORMABLE TIME-FRACTIONAL BENJAMIN-BONA-MAHONY-BURGERS EQUATION ARISING ON THE SURFACE OF WATER
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Abstract
This study investigates solitary wave solutions of the conformable time-fractional Benjamin-Bona-Mahony-Burgers (TFBBMB) equation. This is a special type of Sobolev-type equation that can describe long waves on the surface of water in a channel with a minimal amplitude. Three techniques are employed: The generalized projective Riccati equation (GPRE) approach, the modified auxiliary equation (MAE) technique, and the generalized exponential rational function (GERF) approach. We obtain formally generated expressions that are symbolically verified by direct substitution into the original equation. The solutions are obtained in the form of exponential functions, rational functions, hyperbolic functions, and trigonometric forms. Real nonsingular solutions are identified and presented with appropriate parameter restrictions, while physically admissible solutions are highlighted for their relevance to wave propagation phenomena. Moreover, linear stability analysis is discussed for the classical BBMB equation. Unique physical problems (initial and boundary conditions) are constructed for real-valued, nonsingular solutions that can serve as benchmark problems for numerical methods.
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References
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Muhammad Zafarullah Baber, Tahir Shahzad, Muhammad Waqas Yasin, Tukur Abdulkadir Sulaiman, Mustafa Bayram, Taha Radwan, Karim K. Ahmed. WAVES PROPAGATION FOR THE CONFORMABLE TIME-FRACTIONAL BENJAMIN-BONA-MAHONY-BURGERS EQUATION ARISING ON THE SURFACE OF WATER[J]. Journal of Applied Analysis & Computation, 2027, 17(3): 1761-1791. doi: 10.11948/20260145
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Figure 1.
The physical representation for the solution
$ w_{1}(x, t) $ $ \gamma = 0.2, \beta = 0.1, \delta _2 = 0.2, A = 0.3, c = 2.1, \delta _1 = -1, $ $ \epsilon = 0.99. $ -
Figure 2.
The physical representation for the solution
$ w_{3}(x, t) $ $ \beta = 3.1, A = 0.3, \gamma = 0.3, c = .09, \delta _1 = .3, \delta _2 = 0.2, $ $ \epsilon = 0.99. $ -
Figure 3.
The physical representation for the solution
$ w_{6}(x, t) $ $ \alpha = 2.7, A = 0.3, \delta _2 = 0.2, \gamma = 5.2, c = 3.1, \delta _1 = -1, \beta = .1, $ $ \epsilon = 0.99. $ -
Figure 4.
The physical representation for the solution
$ w_{11}(x, t) $ $ \alpha = -1.5,\beta = -9.1, $ $ \epsilon = 0.9. $ -
Figure 5.
The physical representation for the solution
$ w_{12}(x, t) $ $ \alpha = 0.9,\beta = 1.1, $ $ \epsilon = 0.7. $ -
Figure 6.
The physical representation for the solution
$ w_{14}(x, t) $ $ \alpha = 0.1,\beta = 0.7,\gamma = 0.3,x = 1, $ $ \epsilon = .25. $ -
Figure 7.
The physical representation for the solution
$ w_{15}(x, t) $ $ \alpha = 4.1,a_1 = 0.3,\beta = 2.1,\gamma =3.1,c = -0.9, $ $ \epsilon = 0.7. $ -
Figure 8.
The physical representation for the solution
$ w_{18}(x, t) $ $ \alpha = 2.5,a_2 = 0.2,\beta = 3.1,\gamma = 1.4, $ $ \epsilon = 0.7. $ -
Figure 9.
The physical representation for the solution
$ w_{19}(x, t) $ $ \alpha = 0.5,a_1 = 2.1,a_2 = 3.2,\beta = 0.1,\gamma = 2.4, $ $ \epsilon = 0.2. $ -
Figure 10.
The physical representation for the solution
$ w_{26}(x, t) $ $ \alpha = 0.9,\beta = 2.1,\gamma = 2.5, $ $ \epsilon = 0.5. $ -
Figure 11.
The physical representation for the solution
$ w_{27}(x, t) $ $ \alpha = 0.7,\beta = 1.1,\beta _0 = 4.2,\beta _1= 2.2, A = 0.3,\gamma = 4.5,c = 0.1,\nu = 0.1, $ $ \epsilon =0.9. $ -
Figure 12.
The physical representation for the solution
$ w_{28}(x, t) $ $ \alpha = 0.7,\beta = 1.1,\beta _0 = 4.2,\beta _1 = 2.2, A = 0.3,\gamma = 4.5, c = 3.1,\nu = 2.1, $ $ \epsilon =0.3. $ -
Figure 13.
The physical representation for the solution
$ w_{29}(x, t) $ $ \alpha = 0.7,\beta = 1.1,\beta _0=-0.2, \beta _1 = 2.2,A = 0.3,\gamma = 1.5,c = 0.001,\nu = 2.1, $ $ \epsilon = 0.5. $ -
Figure 14.
The physical representation for the solution
$ w_{30}(x, t) $ $ \alpha = -0.7,\beta _0 = 0.1,\beta _1 = 2.2,\beta = 0.1,A = 0.3,\gamma = -3.5,c = 2.9,\nu = 3.4, $ $ \epsilon = 0.9. $ -
Figure 15.
The physical representation for the solution
$ w_{31}(x, t) $ $ \alpha = 0.9,\beta = 1.1,\beta _0 = 0.2,\beta _1 = 2.2,A = 0.3,\gamma = 6.5,c = 0.9,\nu = 2.1, $ $ \epsilon =0.1. $ -
Figure 16.
The 2-dimension behavior of IC and BCs for
$ w_3(x, t). $ -
Figure 17.
The 2-dimension behavior of IC and BCs for
$ w_4(x, t). $ -
Figure 18.
The 2-dimension behavior of IC and BCs for
$ w_5(x, t). $ -
Figure 19.
The 2-dimension behavior of IC and BCs for
$ w_{18}(x, t). $ -
Figure 20.
The 2-dimension behavior of IC and BCs for
$ w_{19}(x, t). $ -
Figure 21.
The 2-dimension behavior of IC and BCs for
$ w_{30}(x, t). $ -
Figure 22.
The 2-dimension behavior of IC and BCs for
$ w_{31}(x, t). $ -
Figure 23.
The 2-dimension behavior of IC and BCs for
$ w_{33}(x, t). $