对于一维摄动Gelfand两点边值问题的注记  

A Note on One-dimensional Perturbed Gelfand Two-point Boundary Value Problem

作　　者：秦奋 秦惠增 QIN Fen;QIN Hui-zeng(The School of Mathematics and Statistics,Shandong University of Technology,Zibo 255049,China)

机构地区：[1]山东理工大学数学与统计学院,山东淄博255049

出　　处：《数学的实践与认识》2021年第24期320-328,共9页Mathematics in Practice and Theory

基　　金：国家自然科学基金(61379009,61771010)。

摘　　要：考虑下面两点边值问题的三重正解问题{u''(t)+λe^(-β/α+u)=0,-1<t<1,u(-1)=u(1)=0,其中λ,β,α>0.通过数值方法我们找到直线β=Kα,其中4.06867223427747<K<4.06867223427762,当(α,β)在这个直线的下方时,这个边值问题有唯一正解;当(α,β)在这个直线的上方时,这个边值问题存在三重正解.特别当β=α^(2),λ被换成λe^(α)时,这个边值问题出现在燃烧理论中,被称为著名一维摄动Gelfand两点边值问题.人们的一个研究焦点是寻找α0,使得当α>α0>0时,这个边值问题存在多重正解.目前最好的结果是α0≈4.069.然而由直线β=Kα以及β=α^(2),我们得到α0=K∈(4.06867223427747,4.06867223427762).In this article,we briefly introduce the triple positive solution problem of twopoint boundary value problem{u''(t)+λe^(-β/α+u)=0,-1<t<1,u(-1)=u(1)=0,whereλ>0,α>0.We find the straight lineβ=Kαthrough numerical method,where4.06867223427747<K<4.06867223427762.When(α,β)is below the line,the boundary value problem has a unique positive solution;When(α,β)is above the line,the boundary value problem has triple positive solutions.Especially whenβ=α^(2),λis replaced byλe^(α),this boundary value problem appears in combustion theory and is known as the famous onedimensional perturbation Gelfand two-point boundary value problem.The historical best result isα0≈4.069 so far,however fromβ=Kαandβ=α^(2),we found more accurateα0=K∈(4.06867223427747,4.06867223427762).

关 键 词：Gelfand问题 边值问题 正解 多解 高精度快速计算 

分 类 号：O175.8[理学—数学]