Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: , multi-step hybrid methods for solving special second-order differential equations y″(t) = f(t,y(t)) are presented and studied. The new methods inherit the frameworks of RKN methods and linear multi-step methods and include two-step hybrid methods proposed by Coleman (IMA J. Numer. Anal. 23, 197–220, 8) as special cases. The order conditions of the methods re derived by using the SN-series defined on the set SNT of SN-trees. Based on the order conditions,  construct two explicit four-step hybrid methods, which are convergent of order six and seven, respectively. Numerical results show that our new methods are more efficient in comparison with the ll-known high quality methods proposed in the scientific literature.
['Jiyong Li', 'Xianfen Wang']
I am writing a research paper on a idea Multi-step hybrid methods for special second-order differential equations y″(t) = f(t,y(t))
[{"from": "human", "value": "I am writing a research paper on a idea Multi-step hybrid methods for special second-order differential equations y\u2033(t) = f(t,y(t))"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: , multi-step hybrid methods for solving special second-order differential equations y\u2033(t) = f(t,y(t)) are presented and studied. The new methods inherit the frameworks of RKN methods and linear multi-step methods and include two-step hybrid methods proposed by Coleman (IMA J. Numer. Anal. 23, 197\u2013220, 8) as special cases. The order conditions of the methods re derived by using the SN-series defined on the set SNT of SN-trees. Based on the order conditions,  construct two explicit four-step hybrid methods, which are convergent of order six and seven, respectively. Numerical results show that our new methods are more efficient in comparison with the ll-known high quality methods proposed in the scientific literature."}]