RESEARCH ON ANALYTICAL SOLUTION, DYNAMIC ANALYSIS AND PORT ENGINEERING APPLICATION OF THE SIMPLIFIED MODIFIED CAMASSA-HOLM EQUATION (SMCH) UNDER FRACTIONAL DERIVATIVES

Figure (a)-(c) are three-dimensional diagrams by $ \theta_{1.1}(x, t;\alpha) $. Figure (d) is the two-dimensional diagram when $ t=16 $. The $ E_1=2, \rho=-1, \omega=1.5 $ and (a): $ \alpha = 0.25 $; (b): $ \alpha = 0.50 $; (c): $ \alpha = 0.75 $.

Figure (a)-(c) are three-dimensional diagrams by $ \theta_{1.2}(x, t;\alpha) $. Figure (d) is the two-dimensional diagram when $ t=8 $. The $ E_1 = 3, \rho = -1.5, \omega = 2 $ and (a): $ \alpha = 0.35 $; (b): $ \alpha = 0.65 $; (c): $ \alpha = 0.85 $.

Figure (a)-(c) are three-dimensional diagrams by $ \theta_{2.3}(x, t;\alpha) $. Figure (d) is the two-dimensional diagram when $ t=20 $. The $ D_1 = 2, \rho=2, \omega=1.5 $ and (a): $ \alpha = 0.25 $; (b): $ \alpha = 0.55 $; (c): $ \alpha = 0.85 $.

Figure (a)-(c) are three-dimensional diagrams by $ \theta_{3.1}(x, t;\alpha) $. Figure (d) is the two-dimensional diagram when $ t=15 $. The $ D_1 = 3, \omega = 1, \rho = -1.5, E_1 = -4.5 $ and (a): $ \alpha = 0.25 $; (b): $ \alpha = 0.50 $; (c): $ \alpha = 0.75 $.

Figure (a)-(c) are three-dimensional diagrams by $ \theta_{4.1}(x, t;\alpha) $. Figure (d) is the two-dimensional diagram when $ t=15 $. The $ D_1 = 1.5, \omega = 1, \rho = -2.5, E_1 = 3.75 $ and (a): $ \alpha = 0.45 $; (b): $ \alpha = 0.65 $; (c): $ \alpha = 0.85 $.

The effect of the free parameter $ \omega $ on the solution profiles. Figure (a) The absolute value of the solution $ \theta_{1.1}(x, t; \alpha) $ with $ \alpha = 0.50 $, $ E_1 = 2 $, $ \rho = -1 $. Figure (b) The absolute value of the solution $ \theta_{1.2}(x, t; \alpha) $ with $ \alpha = 0.65 $, $ D_1 = 3 $, $ \rho = -1.5 $, $ E_1 = -4.5 $.

Comparison of the derived solutions $ \theta_{1.2}(x, t) $ and $ \theta_{4.1}(x, t) $ with MFD and MRLD of different fractional orders. Figure (a) Three-dimensional graphical representation of $ \theta_{1.2}(x, t) $. Figure (b) Two-dimensional cross-sectional comparison of $ \theta_{1.2}(x, t) $, with parameters: $ \alpha = 0.8 $, $ \beta = 0.5 $, $ y = 0 $, $ E_1 = 3 $, $ \rho = -1.5 $, $ \omega = 2 $, $ c = -2 $, $ k = 1 $, $ t = 5 $. Figure (c) Three-dimensional graphical representation of $ \theta_{4.1}(x, t) $. Figure (d) Two-dimensional cross-sectional comparison of $ \theta_{4.1}(x, t) $, with parameters: $ \alpha = 0.6 $, $ \beta = 0.8 $, $ y = 0 $, $ D_1 = 1.5 $, $ \omega = 1 $, $ \rho = -2.5 $, $ E_1 = 3.75 $, $ c = -1 $, $ k = 1 $, $ t = 15 $.

Figure (a)-(b) depict the phase diagrams and three-dimensional spatial diagrams corresponding to different initial conditions near $ A_{0}(0, 0) $, with the parameters $ \lambda=2 $, $ \mu=1 $ and $ \omega=1 $. Figure (c)-(d) depict the phase diagrams and three-dimensional spatial diagrams corresponding to different initial conditions near $ A_{0}(0, 0) $, with the parameters $ \lambda=1 $, $ \mu=1 $ and $ \omega=2 $.

Figure (a)-(b) depict the phase diagrams and three-dimensional spatial diagrams corresponding to different initial conditions near $ A_{1}(1, 0) $, with the parameters $ \lambda=1 $, $ \mu=1 $ and $ \omega=2 $. Figure (c)-(d) depict the phase diagrams and three-dimensional spatial diagrams corresponding to different initial conditions near $ A_{1}(1, 0) $, with the parameters $ \lambda=2 $, $ \mu=-1 $ and $ \omega=1 $.

LE diagram. Figure(a) is the LE for trajectory corresponding to initial values near the equilibrium point $ A_{0}(0, 0) $. Figure(b) is the LE diagram for trajectory corresponding to initial values near the equilibrium point $ A_{1}(1, 0) $. Where $ \lambda=1 $, $ \mu=1 $ and $ \omega=2 $.

Figures (a) and (b) show the absolute error range between the numerical and analytical solutions of $ \theta_{1.2} $ for the two methods, and Figures (c) and (d) present the corresponding polar coordinate diagrams.

Figures (a) and (b) show the numerical solutions of $ \hat{\theta}_{1.2} $ obtained using the RK44 and Huta86 methods, respectively.

Figure (a) shows the absolute error performance of the classical RK44 method when solving the equation, and (b) shows that of the Huta86 method.