Numerical Solution of Partial Differential Equations
部分课后习题答案部分代码部分图片解析解图2.2解析解
\[ u(x,t) = \sum_{m=1}^{\infty} a_m\mbox{e}^{-(m\pi)^2t}\sin m\pi x, \] 其中 \[ a_m = (8/m^2\pi^2)\sin(0.5m\pi). \]
图2.4解析解
\[ u(x,t) = \sum_{m=1}^{\infty} a_m\mbox{e}^{-(m\pi)^2t}\sin m\pi x, \] 其中 \[ a_m = (4/m^3\pi^3)(1-\cos(m\pi)). \] 图2.10解析解(点此查看计算过程)
\[ u(x,t) = \sum_{m=1}^{\infty}a_m\mbox{e}^{-(0.5m\pi)^2t}\cos(0.5m\pi x), \] 其中 \[ a_m = \int_{0}^{2}f(x)\cos(0.5m\pi x) \mbox{d}x, \ f(x) = \left\{ \begin{array}{ll} 1-x^2, & 0\leq x \leq 1, \\ (x-2)^2-1, & 1 <x \leq 2. \\ \end{array}\right. \]
图2.11解析解
\[ u(x,t) = 2/3+a_m\mbox{e}^{-(m\pi)^2t}\cos(m\pi x), \] 其中 \[ a_m = \int_{-1}^{1} (1-x^2)\cos(m\pi x)\mathrm{d}x. \]
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