# Solve a PDE, modified Heat $u_t+u_{xx}/2 +xu = 0$; Where $u(T,x)=1$.

$$u$$ is a function of space $$x$$ and time $$t$$. Assume that it has smooth derivatives and all of the requirements to have a solution.

Also initial/final time condition $$u(T,x)=1$$. Where $$T$$ is fixed.

Solve the following equation:

$$u_t+u_{xx}/2 +xu = 0$$

What is the analytical solution to this? Even an Ansatz would help

• I know that I could solve a normal heat equation. This thing could be solved just add some particular solutions to the homogenous heat equation correct? What would be your idea for a particular solution? – Book Book Book May 15 at 21:43
• Sorry I meant $u(T,x) = 1$; edits made – Book Book Book May 16 at 1:16
• By a change of variable $s = t - T$ this is the initial condition; Also you could view this as a final time condition. $T$ is known and fixed. – Book Book Book May 16 at 1:29

Hint:

Let $$\begin{cases}t_1=t-T\\x_1=x\end{cases}$$ ,

Then $$u_t=u_{t_1}(t_1)_t+u_{x_1}(x_1)_t=u_{t_1}$$

$$u_x=u_{t_1}(t_1)_x+u_{x_1}(x_1)_x=u_{x_1}$$

$$u_x=(u_{x_1})_x=(u_{x_1})_{t_1}(t_1)_x+(u_{x_1})_{x_1}(x_1)_x=u_{x_1x_1}$$

$$\therefore u_{t_1}+\dfrac{u_{x_1x_1}}{2}+x_1u=0$$ with $$u(0,x_1)=1$$

$$u_{t_1}+\dfrac{u_{xx}}{2}+xu=0$$ with $$u(0,x)=1$$

Let $$u=e^{-xt_1}v$$ ,

Then $$u_{t_1}=e^{-xt_1}v_{t_1}-xe^{-xt_1}v$$

$$u_x=e^{-xt_1}v_x-t_1e^{-xt_1}v$$

$$u_{xx}=e^{-xt_1}v_{xx}-t_1e^{-xt_1}v_x-t_1e^{-xt_1}v_x+t_1^2e^{-xt_1}v=e^{-xt_1}v_{xx}-2t_1e^{-xt_1}v_x+t_1^2e^{-xt_1}v$$

$$\therefore e^{-xt_1}v_{t_1}-xe^{-xt_1}v+\dfrac{e^{-xt_1}v_{xx}}{2}-t_1e^{-xt_1}v_x+\dfrac{t_1^2e^{-xt_1}v}{2}+xe^{-xt_1}v=0$$ with $$v(0,x)=1$$

$$e^{-xt_1}v_{t_1}=-\dfrac{e^{-xt_1}v_{xx}}{2}+t_1e^{-xt_1}v_x-\dfrac{t_1^2e^{-xt_1}v}{2}$$ with $$v(0,x)=1$$

$$v_{t_1}=-\dfrac{v_{xx}}{2}+t_1v_x-\dfrac{t_1^2v}{2}$$ with $$v(0,x)=1$$