观察方程 \Delta u=0 可以发现, 方程具有旋转不变性, 即 v(y)=0=u(x), \text{其中}y=O(x), O: \mathbb{R}^n\rightarrow\mathbb{R}^n 是一个等距变换. 于是假设 u 具有对称不变性, 从而做出 …

We now turn to studying Laplace’s equation ∆u = 0 and its inhomogeneous version, Poisson’s equation, ¡∆u = f: We say a function u satisfying Laplace’s equation is a harmonic function. …

ut ¡ux = 0: This equation models simple fluid motion, where u(x;t) is the height of the wave at time t and position x. Clearly, u(x;t) = 0 satisfies this equation, but so does u(x;t) = c for any …

2018年8月28日 · u(x, t) = √ 2 4t + 1e − x2 4t + 1 = 2 √8t + 2e − x2 4t + 1. the last step is totally not mandatory. This solution requires you to know the Green's function on the standard heat …

u(x,0) = (x if 0 < x < 1, 2 ¡ x if 1 < x < 2, u(0,t) = ux(2,t) = 0. (4.21) In this problem, we have a mixture of both fixed and no flux boundary conditions. Again, assuming separable solutions …

2024年4月10日 · We can see that Laplace’s equation would correspond to finding the equilibrium solution (i.e. time independent solution) if there were not sources. So, this is an equation that …

We will express u(x,t) as the sum of the steady-state solution and another solution w(x,t) that depends on time: u(x,t) = v(x)+w(x,t). As shown in the discussion in 10.6 (p. 614), w(x,t) also …

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答案： u=\varphi_1(x+at)+\varphi_2(x-at) 。 （在熟悉不过了，且注意映射名用哪个都可以，不影响） （因为得出 u_{\xi \eta}=0 后，说明 u 的交叉项为0，所以先对 \eta 积分得到一个只与 …

其实是解一个微分方程，令P = 4x^3y, Q = xf (x)，显然P,Q有连续的偏dao导数，又存在可微函数u，满足上面条件，那么Q对x的偏导 = P对y的偏导，得到一个微分方程，f (1) = 2，实际就是 …