Solving a System of Differential Equations
System of 2 second-order coupled differential equations:
Equation 1:
Equation 2:
Convert 2nd order into first order form:
Actual method for converting a system of DEs
We define the variables θ1, θ2 and their derivatives using a single state vector as follows:
you could equally well have had:--->>
It doesn't matter as long as the DE references the correct elements
And then rewrite Equations 1 and 2 in this form
First re-arrange the equations to separate out the second differential
Equation 1:
Equation 2:
Then substitute for θ1 and θ2 with θ
Equation 1:
Equation 2:
Defining the DE
If we now differentiate θ we get
So let's put this into practice
Define the constants used within the DE
Now define the DE in terms of the new variable θ
Pick some arbitrary initial conditions
These are the initial conditions for θ1, θ2 and their first derivatives
in vector form for DE solver
Define the solution parameters
Apply the DE Solver
Extract t, θ1 and θ2
Passing extra parameters via the state vector
It's somewhat of a nuisance having to define the parameters A, B and K before defining the DE.
A possible solution is to have the worksheet read in a file of the parameters, but this means running the worksheet for each set of values.
A neater solution is to include the parameters as part of the state vector describing the system of equations; to indicate that they are constants, simply set their derivatives to zero.
Convert 2nd order into first order form:
As before, we convert the DEs into first order form using a single state vector
Now add the constants used within the DE
Let θ4
giving a new state vector of
Giving a revised DE in terms of the extended variable θ:
(I've written the DE as row vector simply to save vertical space!)
Pick some arbitrary initial conditions
These are the initial conditions for θ1, θ2 and their first derivatives
in vector form for DE solver
Define the solution parameters
Apply the DE Solver
Extract t, θ1 and θ2
But now we can simply change the values of A, B and K by redefining them in the initial conditions