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Improved Eulers Method Problem

Athanasios Paraskevopoulos

Athanasios Paraskevopoulos, Hellenic Open University

Posted 2 years ago

POSTED BY: Athanasios Paraskevopoulos

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Athanasios Paraskevopoulos

Athanasios Paraskevopoulos, Hellenic Open University

Posted 2 years ago

Another Way I have thought is the following

POSTED BY: Athanasios Paraskevopoulos

Athanasios Paraskevopoulos

Athanasios Paraskevopoulos, Hellenic Open University

Posted 2 years ago

That works but it is not accurate as your solution

POSTED BY: Athanasios Paraskevopoulos

Henrik Schachner

Henrik Schachner, Radiation Therapy Center, Weilheim, Germany

Posted 2 years ago

Athanasios, sorry - obviously I misunderstood your question! As far as I could understand on the quick the "improved Euler method" I simply would do it like so: \[Alpha] = 0.9; \[Beta] = 0.2; \[Gamma] = 1.2; f[{x_, y_, z_}] := {z + (y - \[Alpha])x, 1 - \[Beta]y - x^2, -x - \[Gamma]*z} h = 0.1; impEuler[v_List] := v + h/2 (f[v] + f[v + h f[v]]) data = NestList[impEuler, {1, 2, 3}, 1000]; Graphics3D[{Arrowheads[{.03, #} & /@ Range[0, 1, .05]], Arrow[data]}, BoxRatios -> Automatic, Axes -> True, AxesLabel -> {"X", "Y", "Z"}] I do hope that this now is more helpful! At least my "solution" above can be used as reference. Best regards -- Henrik

Athanasios,

sorry - obviously I misunderstood your question!

As far as I could understand on the quick the "improved Euler method" I simply would do it like so:

\[Alpha] = 0.9;
\[Beta] = 0.2;
\[Gamma] = 1.2;

f[{x_, y_, z_}] := {z + (y - \[Alpha])*x, 1 - \[Beta]*y - x^2, -x - \[Gamma]*z}

h = 0.1;
impEuler[v_List] := v + h/2 (f[v] + f[v + h f[v]])

data = NestList[impEuler, {1, 2, 3}, 1000];
Graphics3D[{Arrowheads[{.03, #} & /@ Range[0, 1, .05]], Arrow[data]}, 
 BoxRatios -> Automatic, Axes -> True, AxesLabel -> {"X", "Y", "Z"}]

enter image description here

I do hope that this now is more helpful! At least my "solution" above can be used as reference.

Best regards -- Henrik

POSTED BY: Henrik Schachner

Athanasios Paraskevopoulos

Athanasios Paraskevopoulos, Hellenic Open University

Posted 2 years ago

Thank you that was very useful for sure. At my master with have an assignment and we study the Dynamic analysis and control of a new hyperchaotic finance system. For this reason we use Improved Euler's Method

POSTED BY: Athanasios Paraskevopoulos

Cornel B.

Posted 2 years ago

Hello, impEuler[v_List] := v + h/2 (f[v] + f[v + h f[v]]) - Henrik Schachner, Radiation Therapy Center, Weilheim, Germany So, initially we have a function with 3 parameters: x, y, and z (f[{x , y , z _}]), and then we call the function only with one single parameter: v (f[v]). Could you explain what is the meaning of v List, and how f[v] is related to f[{x , y , z }]? Formula for improved Euler method:

POSTED BY: Cornel B.

Rohit Namjoshi

Posted 2 years ago

So, initially we have a function with 3 parameters: x, y, and z (f[{x , y , z _}]) No, it is a function that takes a single parameter, which is a `List` of 3 elements that are locally bound to the symbols `x`, `y`, and `z`. f[{x_, y_, z_}] If it took three parameters it would be f[x_, y_, z_] This question is unrelated to this thread. You should have asked a new question.