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Thread: Runge-Kutta of Order 4

  1. #1
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    Unhappy Runge-Kutta of Order 4

    Good day everyone,

    Until now I cannot move on with these two problems. I don't even know where to start . Anywhere, here goes:

    1.) At time t, a pendulum makes an angle x(t) with the vertical axis. Assuming there is no friction, the equation of motion is

    $\displaystyle mlx''(t)=-mg sin(x(t))$


    where m is the mass and l is the length of the string. Use the Runge-Kutta method to solve the differential equation over the interval [0, 2] using M = 40 steps and h = 0.05 if g = 32 ft/secē and
    (a)$\displaystyle l= 3.2$ ft and $\displaystyle x(0) = 0.3$ and $\displaystyle x'(0) = 0$
    (a)$\displaystyle l= 0.8$ ft and $\displaystyle x(0) = 0.3$ and $\displaystyle x'(0) = 0$

    My concerns for this problem are that
    (1) Am i supposed to integrate
    $\displaystyle mlx''(t)=-mg sin(x(x))$ so that x'' becomes x' and then proceed with the iteration process?
    (2) I am having a hard time trying to figure out the iteration for this problem since x(t) is not represented as an equation.



    2.) Predator-Prey model. An example of a system of nonlinear differential equations is the predator-prey problem. Let x(t) and
    y(t) denote the population of rabbits and foxes, respectively at time t. The predator-prey model asserts that x(t) and y(t) satisfy


    $\displaystyle x'(t)=Ax(t) - Bx(t)y(t)$

    $\displaystyle y'(t)=Cx(t)y(t) - Dy(t)$

    A typical computer simulation might use the coefficients
    A = 2, B = 0.02, C = 0.0002, D = 0.8.
    Use the Runge-Kutta method to solve the differential equation over the interval [0, 5] using M = 50 steps and h = 0.1 if
    (a) $\displaystyle x(0) = 3,000$ rabbits and $\displaystyle y(0) = 120$ foxes
    (b) $\displaystyle x(0) = 5,000$ rabbits and $\displaystyle y(0) = 100$ foxes

    I basically have the same concern as the first problem. Both x(t) and y(t) are not represented as equations, but as functions alone.

    I just need someone to help me start with the two problems .
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  2. #2
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    Quote Originally Posted by zeugma View Post
    Good day everyone,

    Until now I cannot move on with these two problems. I don't even know where to start . Anywhere, here goes:

    1.) At time t, a pendulum makes an angle x(t) with the vertical axis. Assuming there is no friction, the equation of motion is

    $\displaystyle mlx''(t)=-mg sin(x(t))$



    where m is the mass and l is the length of the string. Use the Runge-Kutta method to solve the differential equation over the interval [0, 2] using M = 40 steps and h = 0.05 if g = 32 ft/secē and


    (a)$\displaystyle l= 3.2$ ft and $\displaystyle x(0) = 0.3$ and $\displaystyle x'(0) = 0$


    (a)$\displaystyle l= 0.8$ ft and $\displaystyle x(0) = 0.3$ and $\displaystyle x'(0) = 0$



    My concerns for this problem are that


    (1) Am i supposed to integrate $\displaystyle mlx''(t)=-mg sin(x(x))$ so that x'' becomes x' and then proceed with the iteration process?


    (2) I am having a hard time trying to figure out the iteration for this problem since x(t) is not represented as an equation.

    Numerical ODE solvers usually work on ODE's of the form:

    $\displaystyle x'(t)=f(t,x)$

    The way you get them to operate on higher order ODE's is to turn the ODE into a system of first order ODE's.

    Here you have:

    $\displaystyle x''(t)=f(x)$ ,

    so we put $\displaystyle \bold{x}={\bold{x}_1 \brack \bold{x}_2}={x \brack x'}$

    Then:

    $\displaystyle
    \bold{x}'={x' \brack x''}={x' \brack f(x)}={\bold{x}_2 \brack f(\bold{x}_1)}
    $

    Now you can use RK4 on this equation.

    CB
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  3. #3
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    Quote Originally Posted by zeugma View Post
    2.) Predator-Prey model. An example of a system of nonlinear differential equations is the predator-prey problem. Let x(t) and y(t) denote the population of rabbits and foxes, respectively at time t. The predator-prey model asserts that x(t) and y(t) satisfy





    $\displaystyle x'(t)=Ax(t) - Bx(t)y(t)$





    $\displaystyle y'(t)=Cx(t)y(t) - Dy(t)$





    A typical computer simulation might use the coefficients





    A = 2, B = 0.02, C = 0.0002, D = 0.8.


    Use the Runge-Kutta method to solve the differential equation over the interval [0, 5] using M = 50 steps and h = 0.1 if


    (a) $\displaystyle x(0) = 3,000$ rabbits and $\displaystyle y(0) = 120$ foxes


    (b) $\displaystyle x(0) = 5,000$ rabbits and $\displaystyle y(0) = 100$ foxes





    I basically have the same concern as the first problem. Both x(t) and y(t) are not represented as equations, but as functions alone.





    I just need someone to help me start with the two problems .









    Same idea here:

    $\displaystyle
    \bold{x}={\bold{x}_1 \brack \bold{x}_2}={x \brack y}
    $

    Then:

    $\displaystyle \bold{x}'={A \bold{x}_1 - B \bold{x}_1 \bold{x}_2 \brack C \bold{x}_1 \bold{x}_2 -D \bold{x}_2}$

    CB
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  4. #4
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    Quote Originally Posted by CaptainBlack View Post
    Numerical ODE solvers usually work on ODE's of the form:

    $\displaystyle x'(t)=f(t,x)$

    The way you get them to operate on higher order ODE's is to turn the ODE into a system of first order ODE's.

    Here you have:

    $\displaystyle x''(t)=f(x)$ ,

    so we put $\displaystyle \bold{x}={\bold{x}_1 \brack \bold{x}_2}={x \brack x'}$

    Then:

    $\displaystyle
    \bold{x}'={x' \brack x''}={x' \brack f(x)}={\bold{x}_2 \brack f(\bold{x}_1)}
    $

    Now you can use RK4 on this equation.

    CB
    Hi CaptainBlack,

    Thank you so much for replying. I just have one concern though. Call me stupid, but I just don't know how to do it.
    Anyways, how do I plug in the Runge-Kutta equation?

    $\displaystyle f_{n+1}=f_n+\frac{h(f_1+2f_2+2f_3+f_4)}{6}$

    where

    $\displaystyle f_1=f(t_n,y_n)$

    $\displaystyle f_2=f(t_n+\frac{h}{2},y_n+\frac{h}{2}f_1)$

    $\displaystyle f_3=f(t_n+\frac{h}{2},y_n+\frac{h}{2}f_2)$

    $\displaystyle f_4=f(t_n+h,y_n+hf_3)$

    I'm sure I can solve for #2 if I just know how to do it in #1
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  5. #5
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    Quote Originally Posted by zeugma View Post
    Hi CaptainBlack,

    Thank you so much for replying. I just have one concern though. Call me stupid, but I just don't know how to do it.
    Anyways, how do I plug in the Runge-Kutta equation?

    $\displaystyle f_{n+1}=f_n+\frac{h(f_1+2f_2+2f_3+f_4)}{6}$

    where

    $\displaystyle f_1=f(t_n,y_n)$

    $\displaystyle f_2=f(t_n+\frac{h}{2},y_n+\frac{h}{2}f_1)$

    $\displaystyle f_3=f(t_n+\frac{h}{2},y_n+\frac{h}{2}f_2)$

    $\displaystyle f_4=f(t_n+h,y_n+hf_3)$

    I'm sure I can solve for #2 if I just know how to do it in #1
    Write the ODE as the vector first order vector ODE:

    $\displaystyle \bold{y}'=\bold{f}(t,\bold{y})$

    Then the 4th order RK stepping algorithm is:

    $\displaystyle \bold{y}_{n+1}=\bold{y}_n+\frac{h(\bold{k}_1+2\bol d{k}_2+2\bold{k}_3+\bold{k}_4)}{6}$

    where

    $\displaystyle \bold{k}_1=\bold{f}(t_n,\bold{y}_n)$

    $\displaystyle \bold{k}_2=\bold{f}(t_n+\frac{h}{2},\bold{y}_n+\fr ac{h}{2}\bold{k}_1)$

    $\displaystyle \bold{k}_3=\bold{f}(t_n+\frac{h}{2},\bold{y}_n+\fr ac{h}{2}\bold{k}_2)$

    $\displaystyle \bold{k}_4=\bold{f}(t_n+h,\bold{y}_n+h\bold{k}_3)$

    Here a bold charater denotes a vector quantity, and the starting value of $\displaystyle \bold{y}_0$ is given by the initial conditions.

    From here it is just (vector) arithmetic.

    CB
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  6. #6
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    By the way, I just would like to confirm regarding the given equation in #1,

    $\displaystyle mlx''(t)=-mg sin(x(t))$


    Am I right in canceling the "m" in both sides of the equation
    ? I am also assuming that the x'' would be isolated at the left-hand side of the equation,

    $\displaystyle x''=-\frac{g sinx(t))}{l}$
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  7. #7
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    Quote Originally Posted by zeugma View Post
    By the way, I just would like to confirm regarding the given equation in #1,

    $\displaystyle mlx''(t)=-mg sin(x(t))$


    Am I right in canceling the "m" in both sides of the equation? I am also assuming that the x'' would be isolated at the left-hand side of the equation,

    $\displaystyle x''=-\frac{g sinx(t))}{l}$
    Other that the mismatch in opening and closing brackets, that looks OK

    $\displaystyle x''=-\frac{g\ \sin(x(t))}{l}$



    CB
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  8. #8
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    From the RK4 equation, I'm supposed to solve for $\displaystyle k_1,k_2,k_3,$ and $\displaystyle k_4$ to use the equation below:


    $\displaystyle y_{n+1}=y_n+\frac{h(k_1+2k_2+2k_3+k_4)}{6}$

    I understand that $\displaystyle k_1$ is solved by getting$\displaystyle f(t_n,y_n)$ and that
    $\displaystyle f(t_n,y_n)=-\frac{gsin(x(t))}{l}$. From here I don't know how to proceed since I don't know what values to fill in for $\displaystyle x(t)$. Just looking at the equations make me so confused.


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  9. #9
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    Quote Originally Posted by zeugma View Post
    From the RK4 equation, I'm supposed to solve for $\displaystyle k_1,k_2,k_3,$ and $\displaystyle k_4$ to use the equation below:


    $\displaystyle y_{n+1}=y_n+\frac{h(k_1+2k_2+2k_3+k_4)}{6}$

    I understand that $\displaystyle k_1$ is solved by getting$\displaystyle f(t_n,y_n)$ and that
    $\displaystyle f(t_n,y_n)=-\frac{gsin(x(t))}{l}$. From here I don't know how to proceed since I don't know what values to fill in for $\displaystyle x(t)$. Just looking at the equations make me so confused.

    Go back and read the second post in this thread.

    CB
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