How do I use RungeKutta 4/shooting method for this problem?

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In summary, the problem is looking for help in setting up a RungeKutta method for a system of equations involving B, L, K, U, and V as constants and X and Y as functions of T. The equations can be rearranged into two independent systems of two equations each, and then converted into first order systems to be solved using RK4. The boundary conditions for B, L, X', and Y' can be used to set the limits for the summation series in RK4. More assistance can be found on the website "www.mathworld.com".
  • #1
shaqychan
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X''[T]=K (1-L) sinB
Y''[T]=K (1-L) cosB -1
X[T]=sinB0+L sinB
Y[T]=-cosB0+L cosB

Boundary conditions (B0, U, V are constants)
L[0]=L[End]=1
B[0]=-B0, B[End]=B0
X'[0]=X'[End]=U
Y'[0]=-V, Y'[End]=V

I don't know how to set up RungeKutta for this? Please help if you can.
Thx,
 
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  • #2
Your notation/equations confuse me. Are B and L functions of T? What is K? And is there a physical meaning to your problem/equations?
 
  • #3
[0] y' = f(t,y) (1st order) is what RK4 solves as a system.(multiple equations)
[1] y" = f(t,y,y') (2nd order)is what you have.
[2] your eq'ns can be arranged to be independent set(x,y independent of each other)

step1->make the 2 independent systems
step2->you need to convert these 2 systems of 2nd Order into 1st order systems.
step3->then you use RK4 on all the equations you have. Should be 2 systems of 2eq'n = 4.


RK4 is a summation series so your B.Cs will give the limits to which you some over.

need more help "www.mathworld.com" greatest site ever =]
 

What is the RungeKutta 4 method?

The RungeKutta 4 method is a numerical integration technique used to solve ordinary differential equations (ODEs). It is a fourth-order method, meaning it has a higher accuracy compared to lower-order methods. It uses a series of intermediate steps to approximate the solution of the ODE.

How do I use the RungeKutta 4 method for my problem?

To use the RungeKutta 4 method, you will need to have a set of initial conditions and a differential equation that describes the behavior of your system. You will also need to choose a step size, which determines the spacing of the intermediate steps. Then, you can use the formula for the RungeKutta 4 method to compute the solution at each step until you reach your desired endpoint.

What is the shooting method?

The shooting method is a numerical technique used to solve boundary value problems, which involve finding a solution that satisfies certain conditions at both the initial and final points. It involves guessing an initial condition and iteratively adjusting it until the solution satisfies the boundary conditions.

How do I combine the RungeKutta 4 method with the shooting method?

To use the RungeKutta 4 method with the shooting method, you will need to first apply the RungeKutta 4 method to generate a numerical solution for your ODE. Then, you can use this solution as your initial guess for the shooting method. You can continue to iterate and adjust the initial guess until the solution satisfies the boundary conditions.

What are the advantages of using the RungeKutta 4 method with the shooting method?

The RungeKutta 4 method combined with the shooting method offers several advantages. It allows for a more accurate and efficient solution to boundary value problems compared to other numerical techniques. It also allows for more flexibility in choosing the initial conditions, as it does not require them to be on the boundary. Additionally, it can handle a wider range of boundary value problems compared to other methods.

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