Solve the Cauchy problem
ut =kuxx, x ∈ R, t>0, u(x, 0) = φ(x),
for the following initial conditions.
(a) φ(x)=1if |x|<1 and φ(x)=0 if |x|>1.
Write the solutions in terms of the erf function.
u(x,t)=∫G(x-y,t)*φ(y)dy from -∞, to ∞
where G(x,t) is the heat kernel or fundamental solution to heat equation.
The Attempt at a Solution
I am not sure if this correct:
Separate the integral into different parts according above condition and then plugin φ(x) value for φ(y) in the integral. And then proceed from there on