I got a question regarding finite difference method to solve boundary value problem on second order derivative equation. This is taken from the "Numerical Methods using MATLAB" by Mathews & Fink. This is the problem from the book:

Solve the boundary value problem:

$x^{''}(t) = \frac{2t}{1+t^2}x^{'}(t)-\frac{2}{1+t^2}x(t)+1$

with $x(0)=1.25$ and $x(4)=−0.95$ over the interval $[0, 4]$.

We're supposed to use the following equations to solve it:

$x^{''}(t) = p(t)x^{'}(t) + q(t)x(t) + r(t)$

$(\frac{-h}{2}p_j -1)x_{j-1}+(2+h^2q_j)x_j+(\frac{h}{2}p_j -1)x_{j+1}=-h^2 r_j$

The book only gives the $x_j$ calculation result in a table without any step-by-step calculation. I understand that $x_1=1.25$ but I cannot get the result for $x_2$, $x_3$, etc.

How do we calculate $x_j$ using the formula, if we don't know $x_{j+1}$ ? The information given is only for the initial and end value of $x$, but we need $x_{3}$ to calculate $x_{2}$.

Please find the screenshot from the book below: Problem & result table

  • $\begingroup$ A more detailed screenshot of the book & formula can be seen here imgur.com/a/GR7yiCJ $\endgroup$ – JIM BOY Oct 31 '18 at 17:13

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