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May 8th, 2018, 05:17 PM   #1
ZMD
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Absolute stability

f(t,u) = -iu where i$^{2}$ = -1

find theta for which the function has a range of absolute stability
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May 8th, 2018, 07:44 PM   #2
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There is no theta in the expression.

What do you mean by absolute stability?
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May 9th, 2018, 05:41 AM   #3
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Absolute stability: the numerical method/function is absolutely stable if all its roots lie in a unit circle.

u(t$_{n+1}$) = u(t$_{n}$) + h[(1-theta) f(t$_{n}$,u(t$_{n}$) ) + (theta)f(t$_{n+1}$,u(t$_{n+1}$) )]
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May 9th, 2018, 02:42 PM   #4
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It would be helpful if you could explain your notation. What is $t_n$? What is $u(t_n)$?
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May 9th, 2018, 05:16 PM   #5
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Quote:
Originally Posted by ZMD View Post
Absolute stability: the numerical method/function is absolutely stable if all its roots lie in a unit circle.

u(t$_{n+1}$) = u(t$_{n}$) + h[(1-theta) f(t$_{n}$,u(t$_{n}$) ) + (theta)f(t$_{n+1}$,u(t$_{n+1}$) )]
...And is h[ ] a function or a constant?

If f(t, u) = -i u then why include the t in the notation?

You have
$\displaystyle u( t_{n+1} ) = u( t_{n} ) + h [ ( 1 - \theta) f( t_{n} , u( t_{n} ) ) + \theta f( t_{n+1},u( t_{n+1}) ) ] $

$\displaystyle u( t_{n+1} ) = u( t_{n} ) + h [ ( 1 - \theta) ( -i u( t_{n} ) + \theta ( -i u( t_{n+1}) ) ] $

$\displaystyle u( t_{n+1} ) = u( t_{n} ) + h [ -i u( t_{n} ) + i \theta \left \{ u( t_{n}) - ~ u( t_{n+1}) \right \} ] $

That's as far as I can help until you define the h[ ] thing.

-Dan

Last edited by topsquark; May 9th, 2018 at 05:32 PM.
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May 9th, 2018, 07:45 PM   #6
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h is a constant.

u(t$_{n}$) is technically u$_{n}$

General definition of this notation is under theta method (in numerical analysis) while calculating error
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May 10th, 2018, 01:41 PM   #7
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Quote:
Originally Posted by ZMD View Post
h is a constant.

u(t$_{n}$) is technically u$_{n}$

General definition of this notation is under theta method (in numerical analysis) while calculating error
Asking you to elaborate seems to be like pulling teeth. What is the theta method?
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