keynote:lesson12
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| EULER定理 若曲线Γ：y = y(x)在等周条件（48）及边界条件（49）下使泛函（50）达到极值，则存在常数λ，使y(x)满足泛函\\ | EULER定理 若曲线Γ：y = y(x)在等周条件（48）及边界条件（49）下使泛函（50）达到极值，则存在常数λ，使y(x)满足泛函\\ | ||
| + | <note>note</note> | ||
| - | --- //[[1@1|徐斌]] 2010/06/22 16:30// | ||
| - | ==== 5 变分问题中的直接法 ===== | + | ===== 5 变分问题中的直接法 ===== |
| + | |||
| + | =====5 变分问题中的直接法 ===== | ||
| ==== 5.1 里兹法 ==== | ==== 5.1 里兹法 ==== | ||
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| *选取Y的基函数 | *选取Y的基函数 | ||
| - | $\varphi_1(x),\varphi_2(x),\cdots,\varphi+n(x),\cdots$ (57)\\ | + | $\varphi_1(x),\varphi_2(x),\cdots,\varphi_n(x),\cdots$ (57)\\ |
| 对任意$ y(x) \in Y,y(x) $ 都可以表示为$\{\varphi_i(x)\}$的有限维或无限维线性组合。特别的，对极值函数f(x),也有 | 对任意$ y(x) \in Y,y(x) $ 都可以表示为$\{\varphi_i(x)\}$的有限维或无限维线性组合。特别的，对极值函数f(x),也有 | ||
| Line 118: | Line 120: | ||
| *对每个n,考虑由$\varphi_1,\varphi_2,\cdots,\varphi_n,生成的线性子空间Yn，设\\ | *对每个n,考虑由$\varphi_1,\varphi_2,\cdots,\varphi_n,生成的线性子空间Yn，设\\ | ||
| $y_n(x) = sum_{i=1}^n \alpha \varphi_i(x) \in Y_n$\\ | $y_n(x) = sum_{i=1}^n \alpha \varphi_i(x) \in Y_n$\\ | ||
| - | 由泛函J[y]就确定了n元函数，\\ | + | 由泛函J[y]就确定了n元函数\\ |
| + | $J[a_1,\cdots,a_n] = J[y_n]=\int_{x_0}^{x_1}F[x,sum_{i=1}^n \alpha_i\varphi_i(x)，sum_{i=1}^n \alpha_i\varphi_i(x)^{'}]dx$\\ | ||
| + | *对每个n，选取$a_{1}^(n),a_{2}^(n),\cdots,a_{n}^(n)$使J[yn]、取极值，也就是由方程组\\ | ||
| + | $\frac{\partial}{\partial a_i}J[y_n] = 0, i = 1,2, \cdots,n,$\\ | ||
| + | 来确定$a_{1}^(n),a_{2}^(n),\cdots,a_{n}^(n)$，然后用得到的函数\\ | ||
| + | $f_n = sum_{i=1}^n \alpha_i^(n)\varphi_i(x)$\\ | ||
| + | 作为变分问题的近似解\\ | ||
| + | 几点说明：\\ | ||
| + | *上面求得的fn是序列（57）中前n个函数所有可能的线性组合中使泛函J[yn]达到极值的函数。这样得到的序列f1,f2,...,fn,...称为J[yn]的极小化序列。\\ | ||
| + | *因为 | ||
| + | $Y_1 \subset Y_2\subset \cdots \subset Y_n \cdots$\\ | ||
| + | 所以 | ||
| + | $J[f_1] \geq J[f_2] \geq \cdots \geq J[f_n]\cdots $\\ | ||
| + | 并且 | ||
| + | $\lim_{n \to \infty} J[f_n] = J[f]$ | ||
| + | ' | ||
| + | *但是，并不能保证fn极限存在，及时存在也不一定收敛于J[y]的极值函数f。但对于常用函数来说，fn不仅逐点收敛，而且一致收敛于f。因此，如果只限于讨论前n项和$f_n = sum_{i=1}^n \alpha_i^(n)\varphi_i(x)$，那么，fn就是变分问题的近似解。\\ | ||
| + | 基函数通常选取下列三个函数系之一： | ||
| + | * $\varphi_n(x) = (x-x_0)^n(x_1-x),n= 1,2,...$\\ | ||
| + | * $\varphi_n(x) = (x_1-x)^n(x-x_0),n= 1,2,...$\\ | ||
| + | * $\varphi_n(x) = sin\frac{n \pi (x-x_0)}{x_1-x_0},n= 1,2,...$\\ | ||
| - | <未完待续 by 徐斌>\\ | ||
| + | --- //[[1@1|徐斌]] 2010/06/22 17:35// | ||
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