Solving a nonlinear ODE

Hi all,

I am trying to use Matlab to solve the following equation obtain from the growth of a bubble with a liquid (fluid mechanics) where R is R(t):

R*Rdoubledot + ((3/2)(Rdot)^2) = Pgas/rhoo

With my two initial conditions at t=0 being:

R(t=0) = R0 and Rdot(t=0) = 0

I have basic experience with Matlab, and I only know that I need to use the syntax [t,y] = ode45(odefun , tspan, y0) to solve the given problem.

I would really appreciate any help you guys can offer with this script.

Thank you,

AJ

Answer 1

[James Tursa]

Step 1: Solve your ODE for the highest order derivative, in this case Rdoubledot (on paper)

Rdoubledot = stuff (you figure this out with simple algebra)

Step 2: Rewrite your ODE as a system of 1st order ODE's (on paper)

In your case you have a 2nd order ODE, so your state vector will be two elements. Call it R:

R(1) = R

R(2) = Rdot

Step 3: FIgure out the equations for the derivatives of Step 3 (on paper)

RDOT(1) = R(2); % this one is easy

RDOT(2) = some expression involving R(1) and R(2) obtained from Step 1 with appropriate substitutions

Step 4: Generate the function handle that you will feed to ode45 (actual code)

odefun = @(t,R) some expression involving t, R(1), R(2)

In this case, the expression needs to be a 2-element column vector using the RDOT stuff from Step 3

Also, the variables Pgas and rhoo will be part of this expression (you need to define them first before generating odefun)

Step 5: Generate the initial condition variable (actual code)

Since you have a two element state, this will be a two element column vector

R0 = [initial R value; initial Rdot value]; % you fill in these values

Step 6: Feed the odefun, R0, and time span to ode45 (actual code)

Give all this a try and come back with any questions you still have. The doc for ode45 gives an example of a 2nd order ODE, so you might look at that.