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1数学物理方法习题精选§8-31..2sin52,2)0;0(020202xxlulutautlxuautlxxxxxt, 2.2222222202000,(0)0,1315coscoscos023252xxxltttuuaxltxuulatxxxuxulll ,3.2222202000,(0)0,21315coscoscos,035xxxxltttuuaxltxuulxxxuxulll 222),(taxtxv),(),(),(txwtxvtxu0)0,(,5cos513cos31cos)0,(0),(,0),0(02xwlxlxlxxwtlwtwwawtxxxxtt4.xlAxllaAxuxutlaAuuuautlxxxxttsin2sin2)0,(,0)0,(2sin,00025.xlAaxuxuatlAuuuautlxxxxxttsin)0,(,0)0,(sin,00022§8-41.在圆域a上求解:04auu2.在圆域a上求解:0auxyu。3.在圆域0上求解:cuyxbauo)(222§9-11.在球坐标系中,拉普拉斯方程为.0sin1)(sinsin1)(12222222ururrurrr试将方程分离为三个常微分方程。2.在柱坐标系中,拉普拉斯方程为:2222211()0.uuuz试将方程分离为三个常微分方程。3.在球坐标系中,亥姆霍兹方程为:.0sin1)(sinsin1)(122222222ukururrurrr试将方程分离为三个常微分方程。4.在球坐标系中,亥姆霍兹方程为:01)(1222222VkzVVV试将方程分离为三个常微分方程。5.在球坐标系中,氢原子的定态问题薛定谔方程为2222222222111[()(sin)].8sinsinhuuuZeruEumrrrrrr其中Eezmh,,,,都是常数,试将方程分离为三个常微分方程。6.平面极坐标中二维波动方程为:022uautt其中,22222211uuuu,试将方程分离为三个常微分方程。37.平面极坐标中二维输运方程为:022uaut其中,22222211uuuu,试将方程分离为三个常微分方程。§10-11.求解球形区域内部的定解问题:20cos)(,00rrurru2.求解球形区域内部的定解问题:20sin)(,00rrurru3.求解球形区域外部的定解问题:0sin)(,0200rrruurru4.求解空心球区域内的定解问题:21021cos)(,021uuuurrrurrrr,0u、1u均这常数。5.求解球形区域内部的定解问题:0)(,cos00rrurrAru,A为常数。§10-31.求解球形区域内部的定解问题:sincossin)(,0200orruurru[提示:)12cos3(41)(cos,2sin23)(cos,sin3)(cos0212222PPP.]2.求解球形区域外部的定解问题:0)31sinsin()(,02200rorruururru[提示:)12cos3(41)(cos,2sin23)(cos,sin3)(cos0212222PPP.]43.求解球形区域内部的定解问题:)21sin(cossin4)(,0200rrurru[提示:)12cos3(41)(cos,2sin23)(cos,sin3)(cos0212222PPP.]4.求解球形区域外部的定解问题:0)21sin(cossin4)(,0200rrruurru[提示:)12cos3(41)(cos,2sin23)(cos,sin3)(cos0212222PPP.]5.求解球形区域内部的定解问题:31coscoscos)(,022200rrrurru[提示:)12cos3(41)(cos,2sin23)(cos,sin3)(cos0212222PPP.]6.求解球形区域内部的定解问题:0)(,cossin00rrurrAru,A为常数。§11-21.0),(,0000Lzzufuuuuo有限值2.)(,0,0000fuuuuuLzzo有限值3..0),(,00100Lzzufuuuuo有限值54.).(,0,00200fuuuuuLzzo有限值5..23sin)(0,0,0010002zLfuuuuuuautLzzzto有限值6..3cos)(0,0,000002zLfuuuuuuautLzzzzto有限值7..,0,00200Lzzuuuuuo有限值提示:1)1,0(2)1,0()1,0(1)1,0(202)(])(41[)(2nononnnxJxxJx8..0,,002000Lzzuuuuuo有限值提示:)(])[(421)2,0(12)2,0()2,0(020202ononnnxJxxJ9.0,/,0002002002tttttuuuuuuauo有限值;提示:在柱坐标系中,若V与、z无关,)(RV,则RRV1;1)1,0(2)1,0()1,0(1)1,0(202)(])(41[)(2nononnnxJxxJx610.0,00002uuuuuauott有限值,求本征振动。提示:u与z无关,亥姆霍兹方程分离变数:)()(),(RV0)(102222RkddRdRd11..0,0,0sin0002tttttuuuutAuauo有限值提示:在柱坐标系中,若V与z、无关,)(RV,则RRV1。§11-41..,,000000qzukqzukuuuLzzo有限值2..,),(00000qzukuuuzfuuLzzo有限值3..,,000000uuuuuqkuuLzzo有限值7部分试题解答§8-34.解:tlaxXtxv2sin)(),(),(),(),(txwtxvtxu代入泛定方程,得xlAxllaAxwxXlaxwtlaAtlwtlalXtwtlaXwawtlaXaXlatxxttsin2sin2)0,()(2,0)0,(2sin),(2sin)(,0),0(2sin)0(0)(2sin])2([222分为两个定解问题xlAxllaAxXlaxwxwtlwtwwawtxxttsin2sin2)(2)0,(,0)0,(0),(,0),0(02,AlXXXlX)(,0)0(0422求解)(xX:xlAxXAccxlcxlcxX2sin)(02sin2cos)(2121tlaxlAtxv2sin2sin),(xlAxwxwtlwtwwawtxxttsin)0,(,0)0,(0),(,0),0(02xlntlanDtlanCtxwnnnsin)sincos(),(11,0,0sinsin0sin111nDaAlDCxlAxlnlanDxlnCnnnnnn8xltlaaAltxwsin)sin),(xltlaaAlxltlaAtxusinsin2sin2sin),(5.解:atlxXtxvsin)(),(),(),(),(txwtxvtxu代入泛定方程,得xlAlxwalxXxwatlAtlwatllXtwatlXwawtlaXXltxxxttsin)0,()(,0)0,(sin),(sin)(,0),0(sin)0(0)(2sin)(2222AlXXXlX)(,0)0(022,xlAaalxXxwxsin)()0,(,0)0,(0,0002AlCAlClXCXxlCxlCxX22121,)(0)0(,sincos)(xlAlxXsin)(atlxlAltxvsinsin),(0)0,(,0)0,(0,0002xwx,0),(txwatlxlAltxusinsin),(§10-15.解:用§8-4的特殊处理法找特解,因为zrcos,简单的特解是cos101)(1013222ArzzyxA。令wucos1013Ar9cos101)(,03000Arwrrwrr)(cos),(0llllPrCrw)(cos101cos101)(cos1303000PArArPrCllll201101ArCcos101),(20rArrwcos)(101),(202rrrArucos101)(,03000Arwrrwrr§10-36.0)(,cossin00rrurrAru解:找特解,因为xrcossin,简单的特解是cossin101)(1013222ArxzyxA,cossinArAxcos)(cos101113PArwucos)(cos101)(,0113000PArwrrwrr0)(cos)sincos(),,(mmlmlmlmllPmBmArrwcos)(cos101)(cos)sincos(113000PArPmBmArmmlmlmlmll.0);1,1(,0;1012011mlmlBmlAArAcos)(cos)(101cos)(cos10111202
本文标题:数理方程8-11章习题精选(计算题)
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