- #1

- 15

- 0

## Homework Statement

By using Taylor expansion, derive the following two-step backward differentiation which has second

order accuracy:

[tex]\frac{3y_{j+1}-4y_j+y_{j-1}}{2h}=f(t_{j+1},y_{j+1})[/tex]

## Homework Equations

Taylor expansion

ODE

[tex]y^{\prime}=f(t,y) , y(0)=\alpha[/tex]

## The Attempt at a Solution

I find the expansion for [tex] y_{j+1}=y_j+hy^{\prime}_j+\frac{h^2}{2!}y^{\prime \prime}+\cdots [/tex]

and

[tex]y_{j-1}=y_j-hy^{\prime}_j+\frac{h^2}{2!}y^{\prime \prime}+\cdots [/tex]

This is where I get stuck. If I multiply [itex]y_{j+1}[/itex] by 3 and add [itex]y_{j-1}[/itex] I get the needed left hand side but the right hand side is [itex]f(t_{j},y_{j})=y^{\prime}_j[/itex]. How can I have an expansion that includes[itex]f(t_{j+1},y_{j+1})[/itex] that will yield the LHS of the derivation? Am I going about this all wrong? This problem seems relatively simple yet I think I am missing an important step.