Solve Differential Equation with Euler's Method

[youcef]

Hi evry body
i would like to have an help to resolve this exercice below
the followin differential equation with its initial condition
dy/dt=-lambda t y(t) t>=0
avec y(0)=y0
where lambda is damping coeficient strictly positive.
-find the solution of this equation with Euler's explicite and implicite methode
-find analytically the values of h in order to euler methode (explicite) being applicable and obviously stable ( lim IynI=0 where n --->infini .and find the superior borne of time lag h according lambda>0
thanks
warmest Regards

 

[BvU]

Hello youcef, bienvenu a PF !

$${dy\over dt } = - \lambda \, t \, y(t) \\
y(0) = y_0$$ is what you want to solve ? Or have to solve (in that case it should be in the homework section!)

Or is it $-\lambda(t) \, y(t)$ or is it just $- \lambda \, y(t)$ ?

What would make $\lambda$ a damping coefficient ? (I am used to damping coefficients in forms like ${d^2y\over dt^2 } = - \lambda \, { dy\over dt}\$ so I thought I'd better ask first.)

 

[youcef]

Hello youcef, bienvenu a PF !

$${dy\over dt } = - \lambda \, t \, y(t) \\
y(0) = y_0$$ is what you want to solve ? Or have to solve (in that case it should be in the homework section!)

Or is it $-\lambda(t) \, y(t)$ or is it just $- \lambda \, y(t)$ ?

What would make $\lambda$ a damping coefficient ? (I am used to damping coefficients in forms like ${d^2y\over dt^2 } = - \lambda \, { dy\over dt}\$ so I thought I'd better ask first.)
Thanks

$${dy\over dt } = - \lambda \, t \, y(t) \\
y(0) = y_0$$

 

[BvU]

OK, so let's get started on the first part: for Euler explicit you get $$ { y_{k+1}-y_k\over \Delta t} = -\lambda \, t \, y_k $$ and for Euler implicit you have to solve $$
{ y_{k+1}-y_k\over \Delta t} = -\lambda \, t \, y_{k+1}
$$to get $y_{k+1}$ as a function of $y_k$, $t$, and $\Delta t$.

Agree ?

--

 

[youcef]

Thanks BvU .I Agree.let's continue

 

[BvU]

Well, where do you have a problem when you do continue ?

 

[BvU]

Wow, I don't follow. Is this for explicit Euler ?
So how do you come from $$
{ y_{k+1}-y_k\over \Delta t} = -\lambda \, t \, y_k
$$ to your ...(1) ? I don't see a square appearing at all !

 

[youcef]

Wow, I don't follow. Is this for explicit Euler ?
So how do you come from $$
{ y_{k+1}-y_k\over \Delta t} = -\lambda \, t \, y_k
$$ to your ...(1) ? I don't see a square appearing at all !

sorry
for implicite method y_k+1=y_k/(1+Δtλt)
for explicit
y_k+1=y_k(1-Δtλt)

 

[BvU]

What happened to your post ? If you edit it away completely, no one else can follow the thread later on !

for implicit method y_k+1 = y_k / (1 + Δt λ t )
for explicit y_k+1 = y_k (1 - Δt λ t )

Good. Any further problems ? If not then part one is ready ?

 

[youcef]

What happened to your post ? If you edit it away completely, no one else can follow the thread later on !Good. Any further problems ? If not then part one is ready ?

you are very kind .yes no problem.let's go to second part

 

[BvU]

IF part 1 is ready, then what does your solution look like ? Any differences between implicit and explicit methods ?
Do you know the error both methods give when compared to the exact solution ?
What choices of delta t and lambda did you make ? I tried lambda = 0.5 and delta t up to 0.5 (0.501 went bang for the explicit Euler...)

But in fact the stability limit is exceeded a lot earlier. 0.32 also crashes

 

[youcef]

IF part 1 is ready, then what does your solution look like ? Any differences between implicit and explicit methods ?
Do you know the error both methods give when compared to the exact solution ?
What choices of delta t and lambda did you make ? I tried lambda = 0.5 and delta t up to 0.5 (0.501 went bang for the explicit Euler...)

But in fact the stability limit is exceeded a lot earlier. 0.32 also crashes

i don't understand what do you mean.is that is wrong solution

 

[BvU]

So far, I haven't seen your solution of the differential equation, so I don't know...

 

[youcef]

good morning
so anyone can't resolve it?

 

[BvU]

good morning
so anyone can't resolve it?

I don't understand. How far are you really with part 1? What results do you have to show ? See questions in post #11

 

[youcef]

nderstand. How far are you really with part 1? What results do you have to show ? See question

I have no idea if yes i do it by my self.