三角形区域上复合边值问题探讨
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摘要
针对三角形区域上的复合边值问题,采用问题转化求解的办法,研究该问题的解析解。利用SchwarzChristoffel公式给出圆域到三角形区域的共形映射,依据椭圆函数的相关理论推导出三角形区域到圆域的逆映射;将三角形区域上的复合边值问题转化为圆域上的复合边值问题,进一步通过消去法把复合边值问题化为带间断系数的Hilbert边值问题,通过解析延拓化为带间断系数的联结问题,得出三角区域上复合边值问题的解析解。
In view of the compound boundary value problem on the triangle region, the solution of the problem is solved by using the method of problem solving. Schwarz-Christoffel formula is used to give the conformal mapping from circular domain to triangle region, further by using elimination to change the composite boundary value problem into the Hilbert boundary value with discontinuous coefficients. By analytic continuation of the problem with discontinuous coefficients, the analytical solution of the composite boundary value problem is obtained.
In view of the compound boundary value problem on the triangle region, the solution of the problem is solved by using the method of problem solving. Schwarz-Christoffel formula is used to give the conformal mapping from circular domain to triangle region, further by using elimination to change the composite boundary value problem into the Hilbert boundary value with discontinuous coefficients. By analytic continuation of the problem with discontinuous coefficients, the analytical solution of the composite boundary value problem is obtained.
引文
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[2]LU J K.On compound boundary problem[J].Science Sinica,1965,14(11):1545-1555.
[3]张霞,李星.非正则型复合边值问题[J].宁夏大学学报:自然科学版,2002,23(3):193-197.
[4]GUO D J.Multiple positive solutions for first order impulsive singular integro-differential equation on the half line[J].Acta Mathematica Scientia:Series B,2012,32(6):2176-2189.
[5]路见可.平面弹性周期问题概论[M].武汉:武汉大学出版,2008.
[6]李星,路见可.双周期弹性断裂理论[M].北京:科学出版社,2015.
[7]李星.一类周期裂纹问题的数值解[J].宁夏大学学报:自然科学版,1998,19(1):45-46.
[8]LI X.Complex analytic methods in mechanics[J].Joumal of Ningxia University:Natural Science Edition,2001,22(3):272-273.
[9]李星,霍华颂,时朋朋.一维六方压电准晶对称条形体中共线双半无限快速传播裂纹的解析解[J].固体力学学报,2014,35(2):135-136.
[10]BEGEHR H,WEN G C.Nonlinear Elliptic Boundary Value Problems and Their Applications[M].Harlow:Longman,1996.
[11]LIN J,DU J Y.Stability of displacement of the second fundamental problem in plane elasticity[J].Acta Mathematica Scientia:Series B,2014,34(1):125-140.
[12]DUAN P,DU J Y.Riemann-Hilbert characterization for main bessel polynomials with varying large negative parameters[J].Acta Mathematica Scientia:Series B,2014,34(2):557-567.
[13]闻国椿.共性映射与边值问题[M].北京:高等教育出版社,1985.
[14]普利瓦洛夫.复变函数引论[M].闵嗣鹤,等译.哈尔滨:哈尔滨工业大学出版社,2013.