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I'm taking a (fairly basic, but graduate level) numerical analysis class this semester and came across the below question:

Construct a Taylor Table for:

$$a_0(u_{xx})_{j-1} + (u_{xx})_j + a_2(u_{xx})_{j+1} = \frac{1}{\Delta x^2} (b_0 u_{j-1} + b_1 u_j + b_2 u_{j+1}) + ...$$

and find the optimum value for the parameters and the resulting Taylor series error, $er_t$.

I'm struggling to wrap my head around the meaning of the $a_0(u_{xx})_{j-1}$ and $ a_2(u_{xx})_{j+1}$ terms on the left-hand side. Normally a problem like this would be framed as estimating the second derivative of $u$, and the resulting Taylor table would look something like:

Typical Taylor Table

where the coefficients can be pull out and unknowns solved for as a system of equations like:

Typical Taylor Table Coefficients

but these extra terms would make the last equations go to:

$$ \frac{1}{2} b_0 + \frac{1}{2} b_2 = 1 + \frac{a_0}{2} + \frac{a_2}{2} $$

With this being the only equation containing either $a_0$ or $a_2$ (ie the system can't be solved)?

Questions:

  1. I can't find any reference to something like this in the textbooks or through Google - what approach should I be taking to solve this?
  2. What is the actual meaning of these extra terms? Are they somehow related to an error correction for $u_{xx}$ or are we no longer estimating $u_{xx}$?
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