# Using an Auxillary Function to Solve a Partial Differential Equation

I've been practicing on PDEs, since I haven't needed to work with them in a long time until now, when I got stumped on how to solve this problem. The problem asks to use the change of variable $$u(x,t) = w(x,t) + q(x)$$ to solve the PDE

$$\frac{\partial u}{\partial t} = 3\frac{\partial^2 u}{\partial x^2} + 3,\space 0<x<\pi, and\space t> 0$$

which is subject to the boundary condition $$u(0,t) = u(\pi,t) = 1$$ and initial condition $$u(x,0) = 1$$

Any help is greatly appreciated.

• What are your thoughts? – YoTengoUnLCD Dec 13 '15 at 19:53

## 1 Answer

Hints

$1.$ Try to find a function $q(x)$ in such a way that $w(0,t) = w(\pi,t) = 0$.

$2.$ According to hint $1$, we should have $q(0)=q(\pi)=1$. What is the simplest function $q(x)$ with this property?

$3.$ Rewrite the boundary value problem (BVP) in terms of $w(x,t)$.

$4.$ Use the usual separation of variables method to solve the BVP for $w(x,t)$.

$5.$ Once you are finished, the final answer will be $u(x,t) = w(x,t) + q(x)$.

• Thanks a lot for the hints, they really helped determine where I needed to start, and where to go from there. – TachyonArray Dec 14 '15 at 2:24
• You are welcome! and Welcome to MSE. :) – H. R. Dec 14 '15 at 9:27