Suppose we have a system of ODE's: a′=−a−2b and b′=2a−b with initial conditions a(0)=1 and b(0)=−1. How can we find the maximum value of the step size such that the norm a solution of the system goes to zero (if we apply the forward Euler formula)?

musicbachv7 2022-08-10 Answered
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.

How can we find the maximum value of the step size such that the norm a solution of the system goes to zero (if we apply the forward Euler formula)?

Edit: the main part is to calculate the eigenvalues of the following matrix, based on the Euler method, this becomes
( 1 h 2 2 h 2 + 2 h 1 h )
The eigenvalues are ( 1 + 2 i ) ( 1 + h ) and ( 1 2 i ) ( 1 + h )
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Answers (2)

Madilyn Dunn
Answered 2022-08-11 Author has 16 answers
The Euler discretizations of a differential system
{ a ( t ) = f ( a ( t ) , b ( t ) ) b ( t ) = g ( a ( t ) , b ( t ) )
are based on the difference systems
{ a n + 1 = a n + h f ( a n , b n ) b n + 1 = b n + h g ( a n , b n )
for some positive step size h. In the present case, this blackuces to
( a n + 1 b n + 1 ) = M h ( a n b n ) ,
where
M h = ( 1 h 2 h 2 h 1 h ) .
The eigenvalues of M h are
1 h ± 2 i h ,
hence the square of their common modulus is
( 1 h ) 2 + ( 2 h ) 2 = 1 h ( 2 5 h ) .
When both eigenvalues of M h have modulus less than 1, then ( a n , b n ) ( 0 , 0 ) for every starting point ( a 0 , b 0 ). When this modulus is at least 1, then ( a n , b n ) ( 0 , 0 ) never happens except when ( a 0 , b 0 ) = ( 0 , 0 ) (this is because in the present situation both eigenvalues have the same modulus).

Thus, ( a n , b n ) ( 0 , 0 ) for every starting point ( a 0 , b 0 ) when
0 < h < 2 5 .
Note that the eigenvalues of the linear differential system are λ = 1 ± 2 i such that λ = 1 and | λ | 2 = 5. More generally, for a linear differential system with eigenvalues λ such that λ < 0 for every λ, Euler discretizations yield sequences with limit 0 for every starting point and every positive step size h such that
h < min λ ( 2 λ | λ | 2 ) .

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Nica2t
Answered 2022-08-12 Author has 4 answers
You need to compute the eigenvalues of the system. As a shortcut consider that
( a + i b ) = ( 2 i 1 ) ( a + i b )

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asked 2022-08-12
The problem I have is the initial value problem
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that should be solved with Eulers method using the step length, h = 1 2 .

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And it's here I'm stuck, I don't understand how I should start iterate with the step-length from here, I have only encounteblack first-order problem with Eulers method so I would love if someone could point me in the right direction.
asked 2022-08-14
Use the two second-order multi-step methods
ω i + 1 = ω i + h 2 ( 3 f i f i 1 )
and
ω i + 1 = ω 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 ) = 3 t y , y ( 0 ) = 1.
Use Euler’s method to start.

I do not understand how to use these methods to approximate y ( 0.3 ). Moreover I am not sure how Euler's method fits into this question. Could someone clarify this question please?
asked 2022-04-07
For: F ( 0 ) = 0 and F ( x ) = f ( x )
Euler's method: F ( 0 + h ) = F ( 0 ) + h F ( 0 ) = 0 + h f ( 0 )
Continuing the process, F ( 10 h ) = h f ( 0 ) + h f ( h ) + h f ( 2 h ) + . . . . . h f ( 9 h )
This resembles the Riemann sum: Σ i = 1 n f ( x i ) ( x i x i 1 )
Therefore my professor used Euler's method to solve integral problems.

Example: 3 3.09 f ( x ) d x = 0.81. Approximate f ( 3 ) . F ( x ) = h f ( x )
0.81 = 0.09 f ( x )
f ( x ) = 3
My question: How did F ( x + h ) = F ( x ) + h f ( x ) become F ( x ) = h f ( x ) ?
asked 2022-07-20
I have already posted question where I was asking about sketching Euler method.
The explicit Euler method for numerically solving the begining values of differential equation x = f ( t , x ) , x ( t 0 ) = x 0 on the interval I = [ t 0 , T ] is given by
x k + 1 = x k + h f ( t k , x k ) , k = 0 , , N 1 with h = ( T t 0 ) / N , N N .
X k is an approximation of the exact solution x ( t ) of the begining values at time t k := t 0 + k h , k = 0 , . . . , N . By linear interpolation between the points ( t k , x k ) and ( t k + 1 , x k + 1 ) , k = 0 , . . . , N 1 , we obtain a approximation solution x h ( t ).
I need to calculate an approximation of the solution of the begining values at the point t = 1
x = t / x , x ( 0 ) = 1
I need to use h = 0.5. Specify x h (1) and calculate the error, that is difference x h (1)− x(1), where x(1) is the value of the exact solution.
I really don't know how to start. How can I calulate x or x h at all?
asked 2022-08-22
Solve the first-order system that satisfies the given initial conditions using the Euler Method for y ( 0.5 ) and z ( 0.5 ), using a mesh size of h = 0.1:
1. y 6 z 2 z y x 3 y = 0 ; y ( 0 ) = 1 ,   y ( 0 ) = 1.5
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asked 2022-07-21
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with either the explicit Euler method or the implicit Euler method.

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λ 1 = 401 600 h ,
λ 2 = 399 600 h .
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h 4 6 = 2 3 .
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If I instead were to use the implicit Euler method I would get the updating scheme
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asked 2022-07-16
Y = A Y
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Y ( 0 ) = ( 1 0 0 )
I have to solve this system using Eulers method, but what's Eulers method, like I know how to approach this using the eigenvalue method, but the problem explicitly states that it has to be solved using that method, yet my textbook doesn't provide an algorithm for such a method.

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