MATLAB: How to implement tightly coupled nonlinear odes using ode45 in matlab

differential equationsMATLABnumerical integrationodeode45

I am solving a problem from fluid dynamics; in particular tightly coupled nonlinear ordinary differential equations. The following is a scaled-down version of my actual problem. I have solved system of coupled odes many times in the past but this case is different since double derivatives of one variable depends on the double derivative of another variable. How do I implement it in ode45? I need 3 x 2 = 6 plots of x, x-dot and x-ddot versus time for t, 0 to 2. All required initial conditions have zero values at t = 0 How do I store the updated value of the double derivatives as the ode45 code runs? The way ode45 works, I get x and x-dot as output but not the double derivatives. Any help will be highly appreciated.

Best Answer

Solve the equations
t*x2 - x1*x2' = t^2*x1 + x2*x1'
x1'' + x2'' = t*x2 - x1*x2'
Setting
y1 = x1
y2 = x1'
y3 = x2
y4 = x2'
you arrive at the system
y1' = y2
y2' + y4' = t*y3 - y1*y4
y3' = y4
y1'*y3 + y1*y3' = t*y3 - t^2*y1
Now either solve for y1', y2, y3' and y4' in each time step or use the mass matrix facility of the ODE solvers by writing your system as
M(t,y)*y' = F(t,y)
with
M = [1 0 0 0; 0 1 0 1; 0 0 1 0; y3 0 y1 0]
F = [y2;t*y3-y1*y4;y4;t*y3-t^2*y1]
Best wishes
Torsten.
Related Question