在矩形域[tex=12.143x1.357]YZEGTSaSXxmSbEE+7iMM6mHFpUxqr132eMIbRwiLSg8=[/tex]上求解[tex=3.5x1.143]krDGf5jEC9nNIfJ1Hz/11Q==[/tex],且u在边界上的值为零.
解:由题意得定解问题为[tex=17.786x7.071]fnpmC2J6JmQBLyo5NmGAz0wgJNA+maxgt0UPCsAeQ1iLca8pQuCxk8lvORRiWHHjFCyKhdei2oRSmaDacelAKMfcTojl9ocqVlvqwf16ki+7luOtTby0IoSQnlB0BqnzhK5f7/rPF475ccZb2hHgmHOe2PpVYvIvyK+5p0fNl3cwt3qQoImTY1KkTQVhJEgDnvPtMes77lXtPnLjnNr7F2FcPix+lDmE3NP5WzwMuoApO9i0GzbrYce3m1f37SQMaaZ67tZvOVcTGVKkSo2n/ExAvxmaRCLAlRzjhqLgLoE=[/tex]取试探解[tex=6.571x1.357]hkeK84z+uWnVIiIINmio5w==[/tex]，使其满足[tex=8.5x3.357]fnpmC2J6JmQBLyo5NmGAz0wgJNA+maxgt0UPCsAeQ1j4qigJ95bSJPDd2Q76RhaLPmt5mc8PgfRmiHPzI4NM+MbKpfLr/VnXgB2lsBBwiItg/fJaZgav3nnfWDl/AmZvzbJNigAZN6mLpmifZ46AZQ==[/tex]得到[tex=8.429x3.929]fnpmC2J6JmQBLyo5NmGAzwn7jrExIota0O8D1Spct4vg7VgAuvfF/1k2W4BoqD35ciqhSnJQsiK5aDqj+KChGexM3XmYHMAFA0xFdWCOmZ8=[/tex]，解得[tex=5.714x5.643]Ck4j1YFlvVH5wCAykOEMi3M+mMOVqAPiPbi16TSRdIioyvsnsSVUHdqrKkRYuyLM2F2ZoB6ba8IprPeine+KM70L89M0y4YoCNLB8EKQmcYbebim8xG6/bXS3bt0+0yZEX/yWDXXbdl1kDkxdv84aw==[/tex]，令[tex=10.286x1.5]/ZNUZQSyD5Fw3aldBET78+zHzTrt5Bf59mu/KLdbwso=[/tex]，则定解问题转化为求解w(x,y)，[tex=12.0x4.643]fnpmC2J6JmQBLyo5NmGAz7LibI7ay13F4nv3CfCjlU8YsEICLEAv+F/cay/aJ4IP25HN5J0mcQotHD4OBG75TkZbyI+Ojuw85Zud8UhoO6/AzUGIzqsSG6dfe5naw1/kKCEiLtyT1iGfzkxQySQScuk3K0nLur2pUAZNCwOviJVjxReNA7t+ibcnV3HbQl9lgHItC7KrXM8+q7Epg8y8EQ+S3kbOBdqipqvW6e7MuNw=[/tex]。由x的边界条件为第一类边界条件，可知其本征函数为[tex=3.571x2.143]tm6VRRbM/Df8Qog4Yg0Ze210ieKINURp+N033vrD2Ps=[/tex]，则可设[tex=21.786x3.286]Xsz2mSzELwcOz8I+3cWFkdrX/08TzKR6DGGK2wZYhfdads1idkh9IPssedAe/1PViL2lNUMm7u3kDE7CTb+GfGD987IkZmgTAqFTvW75M4gFuCorEvOBPiZdtNFkaklrGRAkalWeogGrlP4pB+kedfChXenr9zacwnOwXEZzRkI=[/tex]，代入y的边界条件可得[tex=23.5x12.786]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[/tex][tex=15.429x5.5]Ck4j1YFlvVH5wCAykOEMixZKqdq1Jwy3Dt+NGj9/8XZU3a89SukBJR631gUO4kmDZviYx7BEhR8gkUzg8ReTT8r8DAXx0q3XQ1F0w56FfCmklO0ug15Zx/IqEV6JjY36/QfkSeBiDfFaR0qENy7vzilCo5OmAab0g7XcDd2up1aSrlg//FvMbIbufa8/IY3X8fLpdexZv4wer8Jre1YEMaIFHO2L78KE/C4kklCQVyY=[/tex][tex=36.071x11.786]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[/tex][tex=26.286x11.643]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[/tex][tex=24.714x3.571]gTUnBXH4tCfGAxrGs7EZKUcvMLt6BwjEmbfDEQNlaLJNqctxIJfz+LSS3M1KbNFFeU9o0eZCmC6vPCNBwufzi1d5PjqD1x4bl+X887CFsAVLLgejAHuq0YeJjJYevi00WnbFUYJkj4qpEksRkVxeVKNIAMlS3+RjgAJjjV4ChY81qJzS/al4v9QdVaWQ6Qi71oKdm4CIR7NuCEtvTrEaLQ==[/tex]所以[tex=27.357x3.571]/ZNUZQSyD5Fw3aldBET7817yhVyb+yG8f42qeFU3U28Zft+rK+iUs/oqcD0xrxcK+ss3HHDsKOni6zrNrTXN+8iNdoesDWQxjEVZoCV7HmxITOtCuVhwOo/Qu2ozO3+W5+TuwU1hmA4VayY0RXORcG+12D7YhXALQ4CGfs5Di6GWY9LKU+EZ8VsIWajwiid+ZKTs4Rg0Jp/f+uqu9Jg1qm9kHMP7PMo04lOf+cN+lJo=[/tex]
举一反三
- 在矩形域[tex=12.143x1.357]u7dB75MB04hJwL0Is5oscQxK5Nf4GUiqovCWAfDqfuk=[/tex]上求解[tex=4.429x1.429]gVMDFPnXKt+7e/x3WSBw+74Y3LVhMo5JC2ws5SZvETE=[/tex]，且[tex=0.643x0.786]cnVwa8IjZzNSEmAUXJ8VCQ==[/tex]在边界上的值为零。
- 在矩形区域[tex=8.857x1.214]8uiQoSOJkMmOQsq0h09plA==[/tex]上求解拉普拉斯方程[tex=2.714x1.0]XZxNuDBDKDChdvMyVUROEg==[/tex]，使满足如下边界条件，其中A，B为常数[tex=6.857x1.357]WfzXYNdOb3O1ndzxlvYEHnMpo5FsSbgx3aFZTr7uEps=[/tex]在矩形区域[tex=8.857x1.214]8uiQoSOJkMmOQsq0h09plA==[/tex]上求解拉普拉斯方程[tex=2.714x1.0]XZxNuDBDKDChdvMyVUROEg==[/tex]，使满足如下边界条件，其中A，B为常数[tex=3.643x1.357]WfzXYNdOb3O1ndzxlvYEHsKiAdLL4XLKHB0nve3QZbY=[/tex]在矩形区域[tex=8.857x1.214]8uiQoSOJkMmOQsq0h09plA==[/tex]上求解拉普拉斯方程[tex=2.714x1.0]XZxNuDBDKDChdvMyVUROEg==[/tex]，使满足如下边界条件，其中A，B为常数[tex=6.786x2.143]WfzXYNdOb3O1ndzxlvYEHiABdUA3ykZtcmOVGU4uAjeseETAXJruXfdgrNdSjIM/[/tex][br][/br]在矩形区域[tex=8.857x1.214]8uiQoSOJkMmOQsq0h09plA==[/tex]上求解拉普拉斯方程[tex=2.714x1.0]XZxNuDBDKDChdvMyVUROEg==[/tex]，使满足如下边界条件，其中A，B为常数[tex=3.5x1.5]WfzXYNdOb3O1ndzxlvYEHsMfkLaQGw4GGtDXzCjdZk0=[/tex]
- 设 [tex=2.857x1.286]tj1rvgP4AHIdbrLux0kAEQ==[/tex] 定义于闭矩形域 [tex=6.929x1.286]JyGyRy+hyV7lwTESv8XFUzY1qFL+aRyepgIRw7xfGt4=[/tex], 若 [tex=0.643x1.286]+RQz+inOZSqc5WvKyEpD0Q==[/tex] 对 [tex=0.571x1.286]Hz6y44ELFVLLNrLVhO3CQA==[/tex] 在 [tex=1.857x1.286]TaQDUPTPF82mJndYOqgzrA==[/tex] 上处处连续. 对 [tex=0.571x1.286]XubEW9+1+hkJqH7jXe5MrA==[/tex] 在 [tex=1.929x1.286]vPlUmwL8t1REs9r1XOy2kg==[/tex] 上(且关于 [tex=0.571x1.286]Hz6y44ELFVLLNrLVhO3CQA==[/tex])为一致连续, 证明 [tex=0.643x1.286]+RQz+inOZSqc5WvKyEpD0Q==[/tex] 在 [tex=0.714x1.286]yQZEV57S9rHjYvgfJydTyg==[/tex] 上处处连续.
- 设 [tex=1.857x1.357]bZ4KhrFbnCaidqbMGQZfww==[/tex] 是 [tex=1.071x1.214]NjrYEUtAtazIvmN6YwFSV0Vu3vRnVOZ4lf7jPz8aobQ=[/tex] 上的三次不可约多项式. 证明[tex=2.429x1.357]lrCiwS81ZLblJbuP1EmZ5A==[/tex] 在 [tex=1.071x1.214]NjrYEUtAtazIvmN6YwFSV0Vu3vRnVOZ4lf7jPz8aobQ=[/tex] 上的分裂域是 8 阶域.
- 向量[tex=5.643x1.286]UOUVlYY3Owd/9Y+4aGhD2Q==[/tex]在[tex=4.786x1.286]x/DRKltwGOjd6FFY9joZ6Q==[/tex]上的投影[tex=3.214x1.286]HwD6aHO6Qt0l6J++EPGgPBkdil9ILD3xu4YblbhvSoE=[/tex][input=type:blank,size:6][/input] ，[tex=0.5x1.286]PGyKeLDo0qv9T0n29ldi6w==[/tex]在[tex=0.571x1.286]mRKL/orzOudCEARA8qn3Kw==[/tex]上的投影[tex=3.143x1.286]HwD6aHO6Qt0l6J++EPGgPJ4STKvTqeKlzMVUIz66NNQ=[/tex][input=type:blank,size:6][/input] .
内容
- 0
在矩形区域[tex=8.857x1.214]8uiQoSOJkMmOQsq0h09plA==[/tex]上求解拉普拉斯方程[tex=2.714x1.0]XZxNuDBDKDChdvMyVUROEg==[/tex]，使满足如下边界条件，其中A，B为常数[tex=3.643x1.357]WfzXYNdOb3O1ndzxlvYEHsKiAdLL4XLKHB0nve3QZbY=[/tex]
- 1
在矩形区域[tex=8.857x1.214]8uiQoSOJkMmOQsq0h09plA==[/tex]上求解拉普拉斯方程[tex=2.714x1.0]XZxNuDBDKDChdvMyVUROEg==[/tex]，使满足如下边界条件，其中A，B为常数[tex=3.5x1.5]WfzXYNdOb3O1ndzxlvYEHsMfkLaQGw4GGtDXzCjdZk0=[/tex]
- 2
在矩形区域[tex=8.857x1.214]8uiQoSOJkMmOQsq0h09plA==[/tex]上求解拉普拉斯方程[tex=2.714x1.0]XZxNuDBDKDChdvMyVUROEg==[/tex]，使满足如下边界条件，其中A，B为常数[tex=6.786x2.143]WfzXYNdOb3O1ndzxlvYEHiABdUA3ykZtcmOVGU4uAjeseETAXJruXfdgrNdSjIM/[/tex]
- 3
在矩形区域[tex=8.857x1.214]8uiQoSOJkMmOQsq0h09plA==[/tex]上求解拉普拉斯方程[tex=2.714x1.0]XZxNuDBDKDChdvMyVUROEg==[/tex]，使满足如下边界条件，其中A，B为常数[tex=3.643x1.357]WfzXYNdOb3O1ndzxlvYEHsKiAdLL4XLKHB0nve3QZbY=[/tex]
- 4
设函数 f(x, y) 在矩形 [tex=6.929x1.357]uGPfxrMJGcQ2XAB2RRDcZcFeR/9W2pItaqM4v+BKP2g=[/tex] 上有界，而且除了曲线段 [tex=8.214x1.214]YF1aBQ4Go6V0tFM95u1zoz4002vODJ+cqewc3B5DDqn4WPxC0rz4N/2Bs6wpQ7lQ[/tex] 外， f(x, y) 在 D 上其他点连续。证明 f 在 D 上可积.