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## Homework Statement

Solve the Cauchy problem

u

_{t}=ku

_{xx}, 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.

## Homework Equations

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