How to solve this nonhomogeneous heat equation

Question

I don't know how to solve it $$ \left\lbrace \begin {array}{lcc} u_{t} \left( x,t \right) =u_{{\it xx}} \left( x,t \right) +2\,{{\rm e}^{-t}} \left( x-1+\sin \left( \pi\,x \right) \right) &0<x<1&t>0\\ u \left( 0,t \right) =2\,{{\rm e}^{-t}}&t \geq 0 & \\u \left( 1 ,t \right) =3&t \geq 0 & \\u \left( x,0 \right) =x+2 &0<x<1& \end {array}\right.$$

Answer 1

I suppose that you could solve the PDe if it was homogeneous. So, first make it homogeneous. This requires to find a particular solution.

Search for a particular solution :

Taking account of the right term of the PDE, we search a solution on the form $2e^{-t}(ax+b+c\sin(\pi x))$ . Putting this function into the PDE, it is very easy to identify the coefficients : $a=-1 \:;\: b=1 \:;\: c=\frac{1}{\pi^2-1}$

A particular solution is $y=2e^{-t}\left(-x+1+\frac{1}{\pi^2-1}\sin(\pi x)\right)$

The change of function $u(x,t)=v(x,t)+2e^{-t}\left(-x+1+\frac{1}{\pi^2-1}\sin(\pi x)\right)$ leads to the homogeneous PDE :

$$ \left\lbrace \begin {array}{lcc} v_{t} \left( x,t \right) =v_{{\it xx}} \left( x,t \right) \\ v \left( 0,t\right) =0 \\v \left( 1,t \right) =3 \\v \left( x,0 \right) =3x-2\frac{1}{\pi^2-1}\sin(\pi x) \end {array}\right.$$ I suppose that you can continue from here.