In this paper, we apply the particle method to solve the numerical solution of a family of non-li-near Evolutionary Partial Differential Equations. It is called b-equation because of its bi-Hamiltonian structure. We i...In this paper, we apply the particle method to solve the numerical solution of a family of non-li-near Evolutionary Partial Differential Equations. It is called b-equation because of its bi-Hamiltonian structure. We introduce the particle method as an approximation of these equations in Lagrangian representation for simulating collisions between wave fronts. Several numerical examples will be set to illustrate the feasibility of the particle method.展开更多
In this paper, for b ∈ （-∞,∞） and b ≠ -1, -2, we investigate the explicit periodic wave solutions for the generalized b-equation ut ＋ 2kux - uxxt ＋ （1 ＋ b）u2ux =buxuxx ＋ uuxxx, which contains the generaliz...In this paper, for b ∈ （-∞,∞） and b ≠ -1, -2, we investigate the explicit periodic wave solutions for the generalized b-equation ut ＋ 2kux - uxxt ＋ （1 ＋ b）u2ux =buxuxx ＋ uuxxx, which contains the generalized Camassa-Holm equation and the generalized Degasperis-Procesi equation. Firstly, via the methods of dynamical system and elliptic integral we obtain two types of explicit periodic wave solutions with a parametric variable a. One of them is made of two elliptic smooth periodic wave solutions. The other is composed of four elliptic periodic blow-up solutions. Secondly we show that there exist four special values for a. When a tends to these special values, these above solutions have limits. From the limit forms we get other three types of nonlinear wave solutions, hyperbolic smooth solitary wave solution, hyperbolic single blow-up solution, trigonometric periodic blow-up solution. Some previous results are extended. For b = -1 or b = -2, we guess that the equation does not have any one of above solutions.展开更多
This paper is concerned with the optimal distributed control problem governed by b-equation. We firstly investigate the existence and uniqueness of weak solution for the controlled system with appropriate initial valu...This paper is concerned with the optimal distributed control problem governed by b-equation. We firstly investigate the existence and uniqueness of weak solution for the controlled system with appropriate initial value and boundary condition. By contrasting with our previous result, the proof without considering viscous coefficient is a big improvement. Secondly, based on the well-posedness result, we find a unique optimal control for the controlled system with the quadratic cost functional. Moreover, by means of the optimal control theory, we obtain the sufficient and necessary optimality condition of an optimal control, which is another major novelty of this paper. Finally, we also present the optimality conditions corresponding to two physical meaningful distributive observation cases.展开更多
文摘In this paper, we apply the particle method to solve the numerical solution of a family of non-li-near Evolutionary Partial Differential Equations. It is called b-equation because of its bi-Hamiltonian structure. We introduce the particle method as an approximation of these equations in Lagrangian representation for simulating collisions between wave fronts. Several numerical examples will be set to illustrate the feasibility of the particle method.
基金Supported by the National Natural Science Foundation of China(No.11401222)Natural Science Foundation of Guangdong Province(No.S2012040007959)+1 种基金The Fundamental Research Funds for the Central Universities(No.2014ZZ0064)Pearl River Science and Technology Nova Program of Guangzhou
文摘In this paper, for b ∈ （-∞,∞） and b ≠ -1, -2, we investigate the explicit periodic wave solutions for the generalized b-equation ut ＋ 2kux - uxxt ＋ （1 ＋ b）u2ux =buxuxx ＋ uuxxx, which contains the generalized Camassa-Holm equation and the generalized Degasperis-Procesi equation. Firstly, via the methods of dynamical system and elliptic integral we obtain two types of explicit periodic wave solutions with a parametric variable a. One of them is made of two elliptic smooth periodic wave solutions. The other is composed of four elliptic periodic blow-up solutions. Secondly we show that there exist four special values for a. When a tends to these special values, these above solutions have limits. From the limit forms we get other three types of nonlinear wave solutions, hyperbolic smooth solitary wave solution, hyperbolic single blow-up solution, trigonometric periodic blow-up solution. Some previous results are extended. For b = -1 or b = -2, we guess that the equation does not have any one of above solutions.
文摘This paper is concerned with the optimal distributed control problem governed by b-equation. We firstly investigate the existence and uniqueness of weak solution for the controlled system with appropriate initial value and boundary condition. By contrasting with our previous result, the proof without considering viscous coefficient is a big improvement. Secondly, based on the well-posedness result, we find a unique optimal control for the controlled system with the quadratic cost functional. Moreover, by means of the optimal control theory, we obtain the sufficient and necessary optimality condition of an optimal control, which is another major novelty of this paper. Finally, we also present the optimality conditions corresponding to two physical meaningful distributive observation cases.
