I have a differntial equation du/dx=f(x,u), where f(x,u)=d psi/dx+p(u−psi(x))^4, where psi is a given function of x independent of u.

pokvarilaap 2022-09-06 Answered
I have a differntial equation d u / d x = f ( x , u ), where
f ( x , u ) = d ψ d x + p ( u ψ ( x ) ) 4 ,
where ψ is a given function of x independent of u.
The backward Euler Method written for step i is
u i + 1 = u i + h f ( x i + 1 , u i + 1 ) .
So, at every step we have a 4th degree polynomial to solve for u i + 1 , which can have more than one root. I am told to employ a root-finding algorithm. But I don't understand how to find the one correct root and reject all others. Bisection, Newton's and other methods seem to only work when there is one root in an interval. Would appreciate any help!
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Answers (1)

xgirlrogueim
Answered 2022-09-07 Author has 13 answers
If h is small and f is Lipschitz you will have that u i + 1 is close to u i . Hence, your first guess for the Newton algorithm ( u i itself) will be always "close enough" to the fixed point for Newton's method to work in practice.
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The explicit Euler method is:
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The problem is, I don't know if I have to use somewhere Gronwall's lemma or if it is straight calculation. The solution to the ODE is y ( s ) = 1 1 s and here is what I've tried so far
y n + 1 = y n + h n f ( s n , y n )
becomes
y n = y n + 1 h n f ( s n , y n )
y n = y n + 1 ( s n + 1 s n ) y n 2
But then I get stucked, since I don't see how I can remove the y n + 1 and s n + 1 terms without getting new recursive terms. Or do I have to use the geometric series?
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L T E i = h 2 2 y ( ξ i )
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asked 2022-08-07
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1. Write this second order ode as a first order ode.
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Suppose we have a system of ODE's: a a = a 2 b and b = 2 a b with initial conditions a ( 0 ) = 1 and b ( 0 ) = 1.

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