In this work, we present travelling wave solutions for the Burgers, Burgers-Huxley and modified Burgers-KdV equations. The (G /G)-expansion method is used to determine travelling wave solutions of these sets of equations. The travelling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions. It is shown that the proposed method is direct, effective and can be used for many other nonlinear evolution equations in mathematical physics.
In this paper, a novel fractional‐integer integral type sliding mode technique for control and generalized function projective synchronization of different fractional‐order chaotic systems with different dimensions in the presence of disturbances is presented. When the upper bounds of the disturbances are known, a sliding mode control rule is proposed to insure the existence of the sliding motion in finite time. Furthermore, an adaptive sliding mode control is designed when the upper bounds of the disturbances are unknown. The stability analysis of sliding mode surface is given using the Lyapunov stability theory. Finally, the results performed for synchronization of three‐dimensional fractional‐order chaotic Hindmarsh‐Rose (HR) neuron model and two‐dimensional fractional‐order chaotic FitzHugh‐Nagumo (FHN) neuron model.
In this paper, an adaptive modified function projective synchronization (AMFPS) scheme of different dimensions fractional-order chaotic systems with fully unknown parameters is presented. On the basis of fractional Lyapunov stability theory and adaptive control law, a new fractional-order controller and suitable update rules for unknown parameters are designed to realize the AMFPS of different fractional-order chaotic systems with non-identical orders and different dimensions. Theoretical analysis and numerical simulations are given to verify the validity of the proposed method. Additionally, synchronization results are applied to secure communication via modified masking method. Due to the unpredictability of the scale function matrix and using of fractional-order systems with different dimensions and unequal orders, the proposed scheme has higher security. The security analysis demonstrate that the proposed algorithm has a large key space and high sensitivity to encryption keys and it is resistance to all kind of attacks.
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