Given
$(\textrm{Pe}-1)u_{i+1} + 2u_i - (\textrm{Pe}-1)u_{i-1} = \frac{h^2}{\mu}$
where $\textrm{Pe}$ stands for Péclet number, $h$ is the step size of the mesh and $\mu$ is the diffusion (constant)
How to solve such equation? I'm trying to solve it to understand why there are no oscillations in a finite element method approximation with the upwind schema.
UPDATE 1
The roots I got are: $r_1 = 1$ and $r_2 = \frac{1+\textrm{Pe}}{1-\textrm{Pe}}$
So I have the following guess for the general solution
$u_i = C_1r_1^i + C_2r_2^i$
To find the constants $C_1$ and $C_2$ I need to initial conditions. What are those? Are they related to the boundary conditions? I have $u(0) = 1$ and $u'(1) = 1$. How to proceed?
UPDATE 2
According to this text, I replace the roots in my guess for the solution to the homogeneous part.
$u_i = C_1(1)^i + C_2\Big(\frac{1+\textrm{Pe}}{1-\textrm{Pe}}\Big)^i$
Applying the initial conditions I have
$u_0 = 1 = C_1(1)^0 + C_2\Big(\frac{1+\textrm{Pe}}{1-\textrm{Pe}}\Big)^0$ $1-C_1 = C2$
for the other initial condition, what should I do, I mean I don't know how to proceed with that Neumann condition there.
UPDATE 3
Before applying the initial conditions to find the values of $C_1$ and $C_2$ I tried to solve for the particular case following the example on the information of this site
So I came up with using a guess like $f_i = ki$. Replacing on the original equation I have
$(\textrm{Pe}-1)k(i+1) + 2ki - (\textrm{Pe}-1)k(i-1) = \frac{h^2}{\mu}$
which after rearranging gives me
$\big[(\textrm{Pe}-1)+2-(\textrm{Pe}-1)\big]ki + \big[(\textrm{Pe}-1)+(\textrm{Pe}+1)\big]k = \frac{h^2}{\mu}$
finally $k = \frac{h^2}{\mu \big[(\textrm{Pe}-1)+(\textrm{Pe}+1)\big]}$
Hence, the complete (I think) equation is:
$u_i = C_1(1)^i + C_2\Big(\frac{1+\textrm{Pe}}{1-\textrm{Pe}}\Big)^i+\frac{h^2 i}{\mu \big[(\textrm{Pe}-1)+(\textrm{Pe}+1)\big]}$
Am I right? I just need to apply initial conditions. The $u(0) = 1$ is easy but the other one $u'(1) = 1$ how to apply that?