Summary of Separation of Variables — Question 4

Question 4 – Solution

Strategy. Homogenizing boundary values creates a source involving the first or second time derivative of the lifting, depending on the PDE.

Step 1: Transform the PDE and every initial datum. Since $\ell_{xx}=0$, the zero-endpoint fields $v=H-\ell$ and $z=W-\ell$ obey $$v_t-v_{xx}=-\ell_t,\qquad z_{tt}-z_{xx}=-\ell_{tt}.$$ Their initial displacements are the respective original profiles minus $\ell(x,0)$; the wave initial velocity is the original velocity minus $\ell_t(x,0)$. Transforming only the endpoints would omit essential source and initial terms.

Step 2: Test the actual moving line. Here $\ell=t(1+x)$, so $\ell_t=1+x$ but $\ell_{tt}=\ell_{xx}=0$. It fails the heat PDE but solves the wave PDE. The wave’s transformed displacement and velocity are both zero, giving $$\boxed{W(x,t)=t(1+x).}$$ The heat correction has $v_t-v_{xx}=-(1+x)$ and $v(x,0)=0$.

Step 3: Solve the forced heat correction. Seek $P''=1+x$ with $P(0)=P(1)=0$. Integration gives $$P(x)=\frac{x^2}{2}+\frac{x^3}{6}-\frac{2x}{3} =\frac{x(x-1)(x+4)}6.$$ The sine coefficients of $1+x$ are $a_n=2[1-2(-1)^n]/(n\pi)$. Twice integrating $P''$ against a sine gives $p_n=-a_n/(n\pi)^2$. Thus $$\boxed{H=t(1+x)+P(x)+ \sum_{n\ge 1}\frac{2[1-2(-1)^n]}{(n\pi)^3} e^{-(n\pi)^2t}\sin(n\pi x).}$$

Step 4: Verify and interpret the offset. At both ends, $P$ and every sine vanish. At $t=0$ the summable sine coefficients reconstruct $-P$ uniformly, giving zero initial temperature. For $t>0$, Gaussian damping permits differentiated convergence; substitution verifies $H_t=H_{xx}$. The initial flat field has $H_{xx}=0$ while the boundary time derivatives are one and two, so joint classical corner smoothness is not asserted. Finally $H-\ell\to P<0$ on $0<x<1$, uniformly on the closed interval, whereas $W-\ell=0$ at all times. An identical boundary lifting can have different physical consequences in the two PDEs.