View Single Post
Feb17-06, 02:52 PM
P: 23
We have the first order ODE

[tex] y'=4t \sqrt y,~y(0)=1, [/tex]

for which i have found the exact solution, namely a fourth order polynomial.

I want a numerical method to solve the problem exactly. This method has to be a fourth order method, since this implies that the local error vanishes.

Now we change the problem so it becomes

[tex] y'=4t \sqrt y - \lambda(y-(1+t^2)^2),~y(0)=a, [/tex]

and the question is: for which values of [itex]\lambda[/itex] and [itex]a[/itex] does a method that has the above mentioned property solve the new problem exactly.

Of course, the obvious case is [itex]\lambda=0[/itex] and [itex]a=1[/itex], because in this case the new problem reduces to the first problem.

My idea is that the solution must be a fourth order polynomial, since a fourth order numerical method has to solve the new problem exactly.

Although I want your view on this and a strategy to find the values of [itex]\lambda[/itex] and [itex]a[/itex] for which the new problem is solved exactly by a fourth order numerical method.
Phys.Org News Partner Science news on
Bees able to spot which flowers offer best rewards before landing
Classic Lewis Carroll character inspires new ecological model
When cooperation counts: Researchers find sperm benefit from grouping together in mice