辽宁石油化工大学学报 ›› 2012, Vol. 32 ›› Issue (3): 88-90.

• 数学与经济管理 • 上一篇    下一篇

一类非线性二阶常微分方程边值问题的解析解

刘国志１，刘倩囡２   

  1. １．辽宁石油化工大学理学院，辽宁抚顺１１３００１；
    ２．华北电力大学（北京）能源动力与机械工程学院，北京１０２２０６
  • 收稿日期:2011-12-21 出版日期:2012-09-20 发布日期:2017-07-06
  • 作者简介:刘国志（１９６２－），男，吉林长春市，教授，硕士。

AnalyticalSolutionofBoundaryValueProblemsforaClassof SecondOrderNonlinearOrdinaryDifferentialEqations

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  1. １． Ｃ ｏｌｌｅ ｇ ｅ 　 ｏ ｆ　 Ｓ ｃｉｅｎｃｅｓ Ｌ ｉａｏｎｉｎ ｇ　 Ｓ ｈｉｈｕａ　 Ｕ ｎｉｖｅｒｓｉｔ ｙ Ｆ ｕｓｈｕｎ　 Ｌ ｉａｏｎｉｎ ｇ１ １３００１ Ｐ． Ｒ． Ｃ ｈｉｎａ ２． Ｓ ｃｈｏｏｌ 　 ｏ ｆ
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    Ｒ ｅｃｅｉｖｅｄ２ １ 　 Ｄ ｅｃｅｍｂｅｒ 　 ２ ０１１ ｒ ｅｖｉｓｅｄ２ ３ 　 Ｆ ｅｂｒｕａｒ ｙ２ ０１２ ａ ｃｃｅ ｐ ｔｅｄ２ ５ 　 Ｍａｒｃｈ２ ０１２
  • Received:2011-12-21 Published:2012-09-20 Online:2017-07-06

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摘要：         在计算胶粒间双电层相互作用能时，需要计算胶粒间的电位分布，而两个胶粒间的电位分布满足Ｐｏｉｓｓｏｎ－Ｂｏｌｔｚｍａｎｎ方程ｄ２ｙ／ｄｔ２＝ｓｉｎｈｙ，这是一个二阶非线性常微分方程，无法求出其解析解。但是当ｙ>>１时，ｓｉｎｈｙ≈ｅｙ／２，故求解Ｐｏｉｓｓｏｎ－Ｂｏｌｔｚｍａｎｎ方程近似解问题转化为求ｄ２ｙ／ｄｔ２＝ｅｙ／２解析解。因此，寻找常微分方程解解的问题是工程实际的需要。通过对二阶非线性常微分方程边值问题的研究，给出了一类二阶非线性常微分方程ｄ２ｙ／ｄｔ２＝ｅｙ／ａ在等边值条件下的解析解表达式。

关键词: 　微分方程, 　Ｐｏｉｓｓｏｎ－Ｂｏｌｔｚｍａｎｎ方程, 　解析解

Abstract: Ascalculatingthedoublelayerinteraction betweencolloidalparticles,itisnecessarytocalculatepotential distributionbetweencolloidalparticles.Then potentialdistribution betweentwocolloidalparticles meetthe Poisson- Boltzmannequationd2y/dt2=sinhy.Duetoitsnonlinearnature,noanalyticalsolutionoftheequationhasbeenfound,however wheny≫1,sinhy ≈ey/2,theapproximatesolutionofPoisson-Boltzmannequationistransformedintosolvinganalytical solutionofd2y/dt2 =ey/2.Hence,findinganalyticalsolutionofnonlinearordinarydifferentialequationistheneedof engineering.Thereforethroughresearchingboundaryvalueproblemofsecondordernonlinearordinarydifferentialequation, analyticalsolutionofaclassofsecondordernonlinearordinarydifferentialequationd2y/dt2 =ey/a wasobtainedunderthe conditionsofequalboundaryvalue.

Key words: 　Poisson-Boltzmannequation , Analysissolution

引用本文

刘国志，刘倩囡. 一类非线性二阶常微分方程边值问题的解析解[J]. 辽宁石油化工大学学报, 2012, 32(3): 88-90.

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