I have already posted question where I was asking about sketching Euler method.The explicit Euler...

Lilliana Livingston

Lilliana Livingston

Answered

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?

Answer & Explanation

Deacon Nelson

Deacon Nelson

Expert

2022-07-21Added 13 answers

You find x by solving the differential equation. It has separeted variables, and the solutiion is easily found to be x ( t ) = 1 t 2 . Then x ( 1 ) = 0.

To apply Euler's method we have t 0 = 0, T = 1, x 0 = 1, h = 0.5 and hence N = 2. Then
x k + 1 = x k h k h x k .
Find x 1 and then x 2 .

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