Use the two second-order multi-step methods omega_(i+1)=omega_i+h/2(3f_i−f_(i−1)) and omega_(i+1)=omega_i+h/2(f_i+1+f_i) as a pblackictor-corrector method to compute an approximation to y(0.3), with stepsize h=0.1, for the IVP; y′(t)=3ty,y(0)=−1. Use Euler’s method to start.
Use the two second-order multi-step methods
as a pblackictor-corrector method to compute an approximation to , with stepsize , for the IVP;
Use Euler’s method to start.
I do not understand how to use these methods to approximate . Moreover I am not sure how Euler's method fits into this question. Could someone clarify this question please?
Answer & Explanation
is an explicit two-step (using three points , and ) method. In contrast, the second one
is an implicit one-step (using two points and ) method. It is implicit since you can not easily solve it for , you have an equation for it
Instead of solving that equation you're advised to use a pblackictor-corrector scheme, i.e. use , obtained from the first scheme and plug it into term of the second one. I'll rewrite it using tilde notation ( is a pblackicted value, while is a corrected one):
Now the evaluation is really straitforward, you just take , compute and finally . The only problem is that you cannot start with that. The only initial value you have is which is not sufficient to compute using pblackictor-corrector pair (it would need also ). To overcome this problem you can use some other method to compute , explicit Euler for example: