I have n recurrence relations of the following form:
For $i=1$:
$v_1(t) = \frac{1}{2}v_1(t-1) + \frac{1}{2}v_2(t-1)$
for $1<i<n$:
$v_i(t) = \frac{1}{2}v_{i-1}(t-1) + \frac{1}{2}v_{i+1}(t-1)$
For $i=n$:
$v_n(t) = \frac{1}{2}v_{n-1}(t-1) + \frac{1}{2}v_n(t-1)$
Additionally, for all $i$:
$v_i(0) = x_i$, a variable representing an initial value.
Does anyone know how to solve this?
$$ v_{i,t} = \frac{1}{2}\left(v_{i-1,t-1}+v_{i+1,t-1}\right) $$
Making $v_{i,t} = \alpha^i\beta^t$ and substituting we get
$$ \alpha + \alpha^{-1} = 2\beta\Rightarrow \alpha = \beta \pm \sqrt{\beta^2-1} $$
or
$$ v_{i,t} = C_1\alpha_1(\beta)^i\beta^t + C_2\alpha_2(\beta)^i\beta^t $$
etc.