The main purpose of this paper is to obtain the wave solutions of conformable time fractional Boussinesq–Whitham–Broer–Kaup equation arising as a model of shallow water waves. For this aim, the authors employed aux...The main purpose of this paper is to obtain the wave solutions of conformable time fractional Boussinesq–Whitham–Broer–Kaup equation arising as a model of shallow water waves. For this aim, the authors employed auxiliary equation method which is based on a nonlinear ordinary differential equation. By using conformable wave transform and chain rule, a nonlinear fractional partial differential equation is converted to a nonlinear ordinary differential equation. This is a significant impact because neither Caputo definition nor Riemann–Liouville definition satisfies the chain rule. While the exact solutions of the fractional partial derivatives cannot be obtained due to the existing drawbacks of Caputo or Riemann–Liouville definitions, the reliable solutions can be achieved for the equations defined by conformable fractional derivatives.展开更多
使用G′/G展开方法对(1+1)维修正Broer-Kaup-Kupershmidt方程进行研究.对该方程进行行波变换,将非线性微分方程转变成常微分方程,并假设具有u(ξ)=∑n i=0 a i(G′/G)i形式的解,通过平衡线性最高阶导数项与最高阶非线性项的幂次来确定...使用G′/G展开方法对(1+1)维修正Broer-Kaup-Kupershmidt方程进行研究.对该方程进行行波变换,将非线性微分方程转变成常微分方程,并假设具有u(ξ)=∑n i=0 a i(G′/G)i形式的解,通过平衡线性最高阶导数项与最高阶非线性项的幂次来确定正整数n,将确定n的拟设形式的解代入方程中,令同次幂项的系数为零,得到一个代数方程组并求解,最终得到非线性微分方程的拟设形式的精确解.展开更多
Based on computerized symbolic computation,a new method and its algorithm are proposed for searching for exact travelling wave solutions of the nonlinear partial differential equations.Making use of our approach,we in...Based on computerized symbolic computation,a new method and its algorithm are proposed for searching for exact travelling wave solutions of the nonlinear partial differential equations.Making use of our approach,we investigate the Whitham-Broer-Kaup equation in shallow water and obtain new families of exact solutions,which include soliton-like solutions and periodic solutions.As its special cases,the solutions of classical long wave equations and modified Boussinesq equations can also be found.展开更多
Through a variable transformation, the Whitham-Broer-Kaup system is transformed into a parameter Levi system. Based on the Lax pair of the parameter Levi system, the N-fold Darboux transformation with multi-parameters...Through a variable transformation, the Whitham-Broer-Kaup system is transformed into a parameter Levi system. Based on the Lax pair of the parameter Levi system, the N-fold Darboux transformation with multi-parameters is constructed. Then some new explicit solutions for the Whitham-Broer-Kaup system are obtained via the given Darboux transformation.展开更多
文摘The main purpose of this paper is to obtain the wave solutions of conformable time fractional Boussinesq–Whitham–Broer–Kaup equation arising as a model of shallow water waves. For this aim, the authors employed auxiliary equation method which is based on a nonlinear ordinary differential equation. By using conformable wave transform and chain rule, a nonlinear fractional partial differential equation is converted to a nonlinear ordinary differential equation. This is a significant impact because neither Caputo definition nor Riemann–Liouville definition satisfies the chain rule. While the exact solutions of the fractional partial derivatives cannot be obtained due to the existing drawbacks of Caputo or Riemann–Liouville definitions, the reliable solutions can be achieved for the equations defined by conformable fractional derivatives.
文摘使用G′/G展开方法对(1+1)维修正Broer-Kaup-Kupershmidt方程进行研究.对该方程进行行波变换,将非线性微分方程转变成常微分方程,并假设具有u(ξ)=∑n i=0 a i(G′/G)i形式的解,通过平衡线性最高阶导数项与最高阶非线性项的幂次来确定正整数n,将确定n的拟设形式的解代入方程中,令同次幂项的系数为零,得到一个代数方程组并求解,最终得到非线性微分方程的拟设形式的精确解.
基金Supported by the National Natural Science Foundation of China(10771072)the Natural Science Foundation of Inner Mongolia(2009 MS0108)+1 种基金the High Education Science Research Programof Inner Mongolia(NJ10045)the Initial Funding of Scientific Research Project for Ph.D.of Inner Mongolia Normal University and the Natural Science Foundation of Inner Mongolia Normal University(ZRYB08017)
基金Project supported by Science Research Foundation of the Returned Overseas Chinese Scholar,SEM,the NSF of Zhejiang Prov-ince(LY13A010020)Program for HNU(HNUEYT2013)
文摘Based on computerized symbolic computation,a new method and its algorithm are proposed for searching for exact travelling wave solutions of the nonlinear partial differential equations.Making use of our approach,we investigate the Whitham-Broer-Kaup equation in shallow water and obtain new families of exact solutions,which include soliton-like solutions and periodic solutions.As its special cases,the solutions of classical long wave equations and modified Boussinesq equations can also be found.
文摘Through a variable transformation, the Whitham-Broer-Kaup system is transformed into a parameter Levi system. Based on the Lax pair of the parameter Levi system, the N-fold Darboux transformation with multi-parameters is constructed. Then some new explicit solutions for the Whitham-Broer-Kaup system are obtained via the given Darboux transformation.
基金Project supported by the National Natural Science Foundation of China(11501076)General Scientific Research Project of Liaoning Province(L2014279)+1 种基金Natural Science Foundation of Liaoning Province(20170540103)Foundation of Dalian Ocean University(HDYJ201409)
