数学物理方程与特殊函数-第三版-课后答案-王元明

2222222222220,0;0,0;0,0;0,0;22222222uuuuuuuuaxltaxltaxltaxlttxtxtxtx⎧⎧⎧⎧∂∂∂∂∂∂∂∂====⎪⎪⎪⎪⎪⎪⎪⎪∂∂∂∂∂∂∂∂⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎨⎨⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩⎩⎩⎩,,,,22222222llllllll⎛⎞⎛⎞⎛⎞⎛⎞⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎠⎝⎠llll0000例设弦的两端固定于x=x=x=x=0000和x=lx=lx=lx=l，弦的初始位移如下图，初速度为零，求弦满足的定解条件。解：,0,0,0,02222,0,0,0,000000000,,,,2222llllxxxxxxxxuuuuuuuuttttltltltltttttlxxllxxllxxllxxl⎧⎧⎧⎧≤≤≤≤⎪⎪⎪⎪∂∂∂∂⎪⎪⎪⎪========⎨⎨⎨⎨====∂∂∂∂⎪⎪⎪⎪====−≤−≤−≤−≤⎪⎪⎪⎪⎩⎩⎩⎩0;0;0;0;0000uuuuuuuuxxlxxlxxlxxl================练习：求下列定解问题的解(((())))(((())))(((())))(((())))(((())))2222,0,0,0,0,0,0,0,00,,0,00,,0,00,,0,00,,0,0,0,0,0,0,0,0,0,0txxtxxtxxtxxxxxxxxxxuauxltuauxltuauxltuauxltutulttutulttutulttutulttuxxxluxxxluxxxluxxxlϕϕϕϕ⎧⎧⎧⎧====⎪⎪⎪⎪⎪⎪⎪⎪========⎨⎨⎨⎨⎪⎪⎪⎪=≤≤=≤≤=≤≤=≤≤⎪⎪⎪⎪⎩⎩⎩⎩(((())))222211110000:,cos:,cos:,cos:,cos11112222nanananattttllllnnnnnnnnaaaannnnuxtaexuxtaexuxtaexuxtaexllllππππππππ⎛⎞⎛⎞⎛⎞⎛⎞∞∞∞∞−−−−⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎠⎝⎠========++++∑∑∑∑解为(((())))00002222cos0,1,2,...cos0,1,2,...cos0,1,2,...cos0,1,2,...llllnnnnnnnnaxxdxnaxxdxnaxxdxnaxxdxnllllllllππππϕϕϕϕ========∫∫∫∫，其中000000000(0,0)0(0,0)0(0,0)0(0,0)0,();0,();0,();0,();0,0.0,0.0,0.0,0.xxyyxxyyxxyyxxyyxxaxxaxxaxxayyyyyyyyyybyybyybyybuuxaybuuxaybuuxaybuuxaybuufyuufyuufyuufyuuuuuuuu================⎧⎧⎧⎧+=+=+=+=⎪⎪⎪⎪⎪⎪⎪⎪========⎨⎨⎨⎨⎪⎪⎪⎪========⎪⎪⎪⎪⎩⎩⎩⎩(0,0)(0,0)(0,0)(0,0)::::xaybxaybxaybxayb≤≤≤≤≤≤≤≤≤≤≤≤≤≤≤≤：在矩形中求解拉普拉斯方程的练定解问题习提示：将第三式看作边界条件。(,)(,)(,)(,)P54(1P54(1P54(1P54(1(-),(-),(-),(-),3)3)3)3)0000(,).(,).(,).(,).aaaauaTuaTuaTuaTuuuuθθπθθθπθθθπθθθπθρθρθρθρθ====一半径为的半圆形平板,其圆周边界上的温度保持而直径边界上的温度保持为度,板的侧面绝缘,试求稳恒状态下的温度分布规律例::::提示定解问题(((())))22222222222222222222111111110,0,0,0,0,0,0,0,(,)(),(,)(),(,)(),(,)(),(,0)(,)(,0)(,)(,0)(,)(,0)(,)0000,,,,,,,,00000,00,00,00,0uuuuuuuuuuuuaaaauaTuaTuaTuaTuuauuauuauuaρρρρρρρρρρρρθπθπθπθπθθθθρρθρρθρρθρρθθθπθθθπθθθπθθθπθρρρρρρρρπππππρπρπρπρ⎧⎧⎧⎧∂∂∂∂∂∂∂∂∂∂∂∂++=++=++=++=⎪⎪⎪⎪∂∂∂∂∂∂∂∂∂∂∂∂⎪⎪⎪⎪⎪⎪⎪⎪=−=−=−=−⎨⎨⎨⎨⎪⎪⎪⎪==≤≤==≤≤==≤≤==≤≤⎩⎩⎩⎩⎪⎪⎪⎪⎪⎪⎪⎪|(0,)||(0,)||(0,)||(0,)|....,,,,0000uuuuθθθθθπθπθπθπ++++≤≤≤≤≤≤≤≤∞∞∞∞(,)()(),(,)()(),(,)()(),(,)()(),uRuRuRuRρθρθρθρθρθρθρθρθ=Φ=Φ=Φ=Φ令经过分离变量后得(((())))(((())))0,0,0,0,0,0,0,0,00,00,00,00,λθπλθπλθπλθπππππ′′′′′′′′Φ+Φ=Φ+Φ=Φ+Φ=Φ+Φ=⎧⎧⎧⎧⎨⎨⎨⎨Φ=Φ=Φ=Φ=Φ=Φ=Φ=Φ=⎩⎩⎩⎩(((())))22220,0,0,0,0.0.0.0.RRRRRRRRRRRRRRRRρρλρρλρρλρρλ′′′′′′′′′′′′⎧⎧⎧⎧+−=+−=+−=+−=⎪⎪⎪⎪⎨⎨⎨⎨+∞+∞+∞+∞⎪⎪⎪⎪⎩⎩⎩⎩特征值2222,1,2,...,1,2,...,1,2,...,1,2,...nnnnnnnnnnnnλλλλ========特征函数(((())))sin,1,2,...sin,1,2,...sin,1,2,...sin,1,2,...nnnnnnnnbnnbnnbnnbnnθθθθθθθθΦ==Φ==Φ==Φ==(),1,2,...(),1,2,...(),1,2,...(),1,2,...nnnnnnnnnnnnRcnRcnRcnRcnρρρρρρρρ========(((())))1111,sin,,sin,,sin,,sin,nnnnnnnnnnnnudnudnudnudnρθρθρθρθρθρθρθρθ∞∞∞∞========∑∑∑∑(,)(),(,)(),(,)(),(,)(),uaTuaTuaTuaTθθπθθθπθθθπθθθπθ=−=−=−=−将代入上式1111sin()sin()sin()sin()nnnnnnnnnnnndanTdanTdanTdanTθθπθθθπθθθπθθθπθ∞∞∞∞=====−=−=−=−∑∑∑∑222200000000sind()sindsind()sindsind()sindsind()sindnnnnnnnndanTndanTndanTndanTnππππππππθθθπθθθθθθπθθθθθθπθθθθθθπθθθ=−=−=−=−∫∫∫∫∫∫∫∫33334444(1cos),(1cos),(1cos),(1cos),nnnnnnnnTTTTdndndndnananananππππππππ=−=−=−=−(((())))333311114444,[1(1)]sin.,[1(1)]sin.,[1(1)]sin.,[1(1)]sin.nnnnnnnnnnnnnnnnTTTTununununananananρρρρρθθρθθρθθρθθππππ∞∞∞∞=====−−=−−=−−=−−∑∑∑∑sinsinsinsincos,cos,cos,cos,xxxxxxxxρθθρθθρθθρθθθθθθρρρρ∂∂−∂∂−∂∂−∂∂−========∂∂∂∂∂∂∂∂coscoscoscossin,sin,sin,sin,yyyyyyyyρθθρθθρθθρθθθθθθρρρρ∂∂∂∂∂∂∂∂========∂∂∂∂∂∂∂∂同理coscoscoscossinsinsinsinxxxxyyyyρθρθρθρθρθρθρθρθ====⎧⎧⎧⎧⎨⎨⎨⎨====⎩⎩⎩⎩1cossin1cossin1cossin1cossin0sincos0sincos0sincos0sincosxxxxxxxxxxxxxxxxρθρθρθρθθρθθρθθρθθρθρθρθρθρθθρθθρθθρθθρθ∂∂∂∂∂∂∂∂⎧⎧⎧⎧=⋅−⋅=⋅−⋅=⋅−⋅=⋅−⋅⎪⎪⎪⎪⎪⎪⎪⎪∂∂∂∂∂∂∂∂⎨⎨⎨⎨∂∂∂∂∂∂∂∂⎪⎪⎪⎪=⋅+=⋅+=⋅+=⋅+⎪⎪⎪⎪∂∂∂∂∂∂∂∂⎩⎩⎩⎩(,)(,)(,)(,)uuuuρθρθρθρθ22222222222222220000uuuuuuuuxyxyxyxy∂∂∂∂∂∂∂∂+=+=+=+=∂∂∂∂∂∂∂∂已知sinsinsinsincoscoscoscosxxxxuuuuuuuuuuuuuuuuxxxxxxxxuuuuθθθθθθθθρθρθρθρθρθρθρθρθρθρθρθρθρρρρ∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂=⋅+⋅=−⋅=⋅+⋅=−⋅=⋅+⋅=−⋅=⋅+⋅=−⋅∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂极坐标系下的拉普拉斯方程的表达式222222222222222222222222222222222222cos(sin)cos(sin)cos(sin)cos(sin)sinsinsinsin1sin1sin1sin1sincoscoscoscosuuuuuuuuuuuuuuuuxxxxuuuuxxxxxxxxxxxxxxxxxxxxxxxxxxxxuuuuuuuuθθθθθθθθρρθρρρθρρρθρρρθρθθθθρθθρρθθρρθθρρθθρθθθθθθθθρθθρθθρθθρθθρθρθρθρθθθθθρρρρρρρρθθθθρρρρ⎛⎞⎛⎞⎛⎞⎛⎞∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂=⋅+⋅+⋅−⋅=⋅+⋅+⋅−⋅=⋅+⋅+⋅−⋅=⋅+⋅+⋅−⋅⎜⎟⎜⎟⎜⎟⎜⎟∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂⎝⎠⎝⎠⎝⎠⎝⎠⎛⎞⎛⎞⎛⎞⎛⎞∂∂∂∂∂∂∂∂−⋅+⋅⋅−⋅+⋅⋅−⋅+⋅⋅−⋅+⋅⋅⎜⎟⎜⎟⎜⎟⎜⎟∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂⎝⎠⎝⎠⎝⎠⎝⎠∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂⎛⎞⎛⎞⎛⎞⎛⎞∂∂∂∂−⋅−⋅−⋅−⋅⎜⎟⎜⎟⎜⎟⎜⎟∂∂∂∂⎝⎝⎝⎝−⋅−⋅−⋅−⋅∂∂∂∂⎠⎠⎠⎠∂∂∂∂222222222222222222222222.......,0.......,0.......,0.......,0uuuuuuuuuuuuyxyyxyyxyyxy∂∂∂∂∂∂∂∂∂∂∂∂=+==+==+==+=∂∂∂∂∂∂∂∂∂∂∂∂同理代入22222222222222222222111111110000uuuuuuuuuuuuρρρρθρρρρθρρρρθρρρρθ∂∂∂∂∂∂∂∂∂∂∂∂++=++=++=++=∂∂∂∂∂∂∂∂∂∂∂∂sinsinsinsincoscoscoscosuuuuuuuuθθθθθθθθρθρρθρρθρρθρ∂∂∂∂∂∂∂∂−⋅−⋅−⋅−⋅∂∂∂∂∂∂∂∂练习求解定解问题2222222222222222222200000000000022222222sincos,0,0;sincos,0,0;sincos,0,0;sincos,0,0;|3,|6;0|3,|6;0|3,|6;0|3,|6;04444|31,|sin,0.|31,|sin,0.|31,|sin,0.|31,|sin,0.xxlxxlxxlxxlttttttttuuuuuuuuaxxxltaxxxltaxxxltaxxxlttxlltxlltxlltxlluutuutuutuutxuxuxuxuuxxluxxluxxluxxlltlltlltlltlππππππππππππ================⎧⎧⎧⎧∂