費(fèi)琪

(1.西北大學(xué)數(shù)學(xué)系,陜西 西安 710127;2.西北大學(xué)非線性中心,陜西 西安 710069)

廣義 Riccati展開法及其在foam drainage方程中的應(yīng)用

費(fèi)琪1,2

(1.西北大學(xué)數(shù)學(xué)系,陜西 西安 710127;2.西北大學(xué)非線性中心,陜西 西安 710069)

應(yīng)用雙曲函數(shù)法結(jié)合Riccati方程,求得foam drainage方程的精確解.通過這種方法可以得到此方程的新的孤立波解與周期解,并且此方法可以用來求解其它許多的非線性演化方程.

雙曲函數(shù)法結(jié)合Riccati方程;行波解;foam drainage方程

1 引言

非線性現(xiàn)象在各科學(xué)領(lǐng)域與工程領(lǐng)域普遍存在,近些年來,解方程的方法層出不窮,如逆射法[1,2],雙曲函數(shù)法[3],sine-cosine法[4],tanh-sech法[5],F展開法[6],G′/G展開法[7]等. Mal fi et首先用tanh方法解非線性偏微分方程,后來此方法被廣泛應(yīng)用,Wazwas把這種方法推廣為tanh-coth方法.本文運(yùn)用tanh-coth法結(jié)合Riccati方程求解foam drainage方程新的行波解,同時體現(xiàn)出這種方法解非線性方程非常直觀有效.

泡沫出現(xiàn)在日常生活和工業(yè)中,foam drainage方程描述了在重力的作用下,泡沫垂直密度的變化情況.這個模型由Verbist和Weaire在1996年提出,它的解在食品生產(chǎn)、化學(xué)工業(yè)、消防、材料物理等領(lǐng)域中有很重要的作用,因此研究foam drainage方程在理論和實(shí)踐中有著

很重要的意義.

2 雙曲函數(shù)法結(jié)合Riccati方程基本思想及求解步驟

3 foam drainage方程

4 結(jié)論

本文在Maple軟件運(yùn)算功能的幫助下,運(yùn)用了雙曲函數(shù)法結(jié)合廣義Riccati方程成功求得了foam drainage方程新的行波解,這樣豐富了方程解的結(jié)果,有助于對方程所描述的物理現(xiàn)象有進(jìn)一步的了解和研究.

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[3]Fan Engui．Extended tanh-function method and its applications to nonlinear equations[J]．Physics Letters A,2000,277:212-218．

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Generalized Riccati equation expansion method and its application to the foam drainage equation

Fei Qi
(1.Department of Mathematics,Northwest University,Xi′an 710127,China;
2.Center for Nonlinear Studies,Northwest University,Xi′an 710069,China)

In this paper,the tanh-coth method combined with the Riccati equation is applied to the analysis of the foam drainage equation.We can gain new solitary wave solutions and periodic solutions.It is also a promising method to solve other nonlinear evolution equations.

the tanh-coth method combined with the Riccati equation,travelling wave, the foam drainage equation

O175.2

A

1008-5513(2012)01-0109-04

2011-02-12.

國家自然科學(xué)基金(10671156).

費(fèi)琪(1985-),碩士生,研究方向：偏微分方程.

2010 MSC:35Q58