Question

Given the partial differential equation 𝑈𝑡 + 𝑈𝑥𝑥 = 0, 0 < 𝑥 < 𝜋,𝑡 > 0. Use the method of separation of variables to solve the given equation with the following conditions. 𝑈𝑥 (0,𝑡) = 0,𝑡 > 0

𝑈𝑥 (𝜋,𝑡) = 0,𝑡 > 0 𝑈(𝑥, 0) = 𝑥(𝜋 − 𝑥), 0 < 𝑥 < 𝜋

Answer 1

Given, $U_t+U_{xx}=0,0<x<\pi,t>0$

So, $\dfrac{du}{dt}=-\dfrac{d^2u}{dx^2}$

Taking fourier sine transform of Above equation-

$\dfrac{d}{dt}(usinsxdx)=-\dfrac{d^2}{dx^2}(sinsxdx)$

$\dfrac{du_s}{dt}=-(-s^2u(0,t)-su_s) \text{, Where } u_s= \int_0^{\infty}usinstdt$

$\dfrac{du_s}{dt}=s^2(0)+su_s$

$\dfrac{du_s}{u_s}=-sdt$

Integrating and we get,

$logu_s=-st+logc$

$log\dfrac{u_s}{c}=-st$

$u_s=ce^{-st}~~~~~~-(1)$

As $u_s= \int_0^{\pi}u(x,0)sinsxdx$

$=\int_0^{\pi}x(\pi-x)sinsxdx=1$

From 1 we have, c=1

So $u_s=e^{-st}$

Now taking inverse fourier transform and we get-

$u(x,t)=\int_0^{\pi}e^{-st}sinstds$