In this paper, Lie point symmetry group of the Harry-Dym type equation with Riemann-Liouville fractional derivative is constructed. Then complete subgroup classification is obtained by means of the optimal system meth...In this paper, Lie point symmetry group of the Harry-Dym type equation with Riemann-Liouville fractional derivative is constructed. Then complete subgroup classification is obtained by means of the optimal system method. Finally, corresponding group-invariant solutions with reduced fractional ordinary differential equations are presented via similarity reductions.展开更多
In this paper, the authors consider the Harry-Dym equation on the line with decaying initial value. They construct the solution of the Harry-Dym equation via the solution of a 2 × 2 matrix Riemann-Hilbert problem...In this paper, the authors consider the Harry-Dym equation on the line with decaying initial value. They construct the solution of the Harry-Dym equation via the solution of a 2 × 2 matrix Riemann-Hilbert problem in the complex plane. Further, onecusp soliton solution is expressed in terms of the Riemann-Hilbert problem.展开更多
Three kinds of initial-value problems of the Harry-Dym equation are considered. Theexistence and regularity of solution are proved. Some special solutions are given. Severalexamples show that different initial-value p...Three kinds of initial-value problems of the Harry-Dym equation are considered. Theexistence and regularity of solution are proved. Some special solutions are given. Severalexamples show that different initial-value problems are related to K. D. V. equation onthe whole line, the half line or the finite interval respectively.展开更多
In this paper, we derive a general formula of the flow equtions for the Harry-Dym hierarchy. And three applications in n-reduction, （2 ＋ 1）-dimensional generalization, and Kupershmidt reduction, are considered.
In this paper, the homotopy analysis method (HAM) has been employed to obtain the approximate analytical solution of the nonlinear Harry-Dym (HD) equation, which is one of the most important soliton equations. Utilizi...In this paper, the homotopy analysis method (HAM) has been employed to obtain the approximate analytical solution of the nonlinear Harry-Dym (HD) equation, which is one of the most important soliton equations. Utilizing the HAM, thereby employing the initial approximation, variations of the 7th-order approximation of the Harry-Dym equation is obtained. It is found that effect of the nonzero auxiliary parameter on convergence rate of the series solution is undeniable. It is also shown that, to some extent, order of the fractional derivative plays a fundamental role in the prediction of convergence. The final results reported by the HAM have been compared with the exact solution as well as those obtained through the other methods.展开更多
The aim of this paper is to generate exact travelling wave solutions of the Harry-Dym equation through the methods of Adomian decomposition, He＇s variational iteration, direct integration, and power series. We show t...The aim of this paper is to generate exact travelling wave solutions of the Harry-Dym equation through the methods of Adomian decomposition, He＇s variational iteration, direct integration, and power series. We show that the two later methods are more successful than the two former to obtain more solutions of the equation.展开更多
基金Supported by the National Natural Science Foundations of China(Grant No.11201371,11371293,11371323)the National Natural Science Foundation of Shaanxi Province(Grant No.2012JQ1013,2015JM1037)the Foundation of Department of Education of Zhejiang Province(Grant No.Y201432097)
文摘In this paper, Lie point symmetry group of the Harry-Dym type equation with Riemann-Liouville fractional derivative is constructed. Then complete subgroup classification is obtained by means of the optimal system method. Finally, corresponding group-invariant solutions with reduced fractional ordinary differential equations are presented via similarity reductions.
基金supported by the National Natural Science Foundation of China(No.11271079)
文摘In this paper, the authors consider the Harry-Dym equation on the line with decaying initial value. They construct the solution of the Harry-Dym equation via the solution of a 2 × 2 matrix Riemann-Hilbert problem in the complex plane. Further, onecusp soliton solution is expressed in terms of the Riemann-Hilbert problem.
文摘Three kinds of initial-value problems of the Harry-Dym equation are considered. Theexistence and regularity of solution are proved. Some special solutions are given. Severalexamples show that different initial-value problems are related to K. D. V. equation onthe whole line, the half line or the finite interval respectively.
基金Supported by the Natural Science Foundation of China under Grant Nos.10671187 and 10971 109Supported by Program for New Century Excellent Talents in Universities under Grant No.NECT-08-0515
文摘In this paper, we derive a general formula of the flow equtions for the Harry-Dym hierarchy. And three applications in n-reduction, （2 ＋ 1）-dimensional generalization, and Kupershmidt reduction, are considered.
文摘In this paper, the homotopy analysis method (HAM) has been employed to obtain the approximate analytical solution of the nonlinear Harry-Dym (HD) equation, which is one of the most important soliton equations. Utilizing the HAM, thereby employing the initial approximation, variations of the 7th-order approximation of the Harry-Dym equation is obtained. It is found that effect of the nonzero auxiliary parameter on convergence rate of the series solution is undeniable. It is also shown that, to some extent, order of the fractional derivative plays a fundamental role in the prediction of convergence. The final results reported by the HAM have been compared with the exact solution as well as those obtained through the other methods.
文摘The aim of this paper is to generate exact travelling wave solutions of the Harry-Dym equation through the methods of Adomian decomposition, He＇s variational iteration, direct integration, and power series. We show that the two later methods are more successful than the two former to obtain more solutions of the equation.
