How to implement tightly coupled nonlinear odes using ode45 in matlab?

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Vikash Pandey 2018-11-12

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评论： Huy Nguyen 2022-12-7

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.

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Vikash Pandey 2018-11-12

编辑：Vikash Pandey 2018-11-12

在 MATLAB Online 中打开

Please see below and confirm if I am correct or wrong.

PLEASE check the expression for y-dots carefully. Is my understanding correct?

>
function ydot = f(t, y);
ydot(1) = y(2);
ydot(2) = t*y(4) - y(1)*y(5) - y(6);
ydot(3) = y(3);
ydot(4) = y(5);
ydot(5) = (t^2)*y(1) + y(4)*y(2) - y(3);
ydot(6) = y(6);
ydot = ydot';

------

clc;
clear all;
close all;
time_range = [0 2];
initial_conditions = [2 1 0.5 4 3 1]; % x1(0) y1(0) y1-dot(0) x2(0) y2(0) y2-dot(0)
[t, y] = ode45('bubble', time_range, initial_conditions);
x1=y(:,1);
x1dot=y(:,2);
x1ddot = y(:,3);
x2=y(:,4);
x2dot=y(:,5);
x2ddot = y(:,6);
figure(201);
h1 = plot(t, x1, 'b-', t, x2, 'k:');
set(h1,'linewidth',4);
legend(['x1'], ['x2'])
legend('Location','Northwest')
set(gca,'xscale','linear','FontSize',26)
set(gca,'yscale','linear','FontSize',26)
set(gca,'XMinorTick','on')
set(gca,'YMinorTick','on')
grid off
xlabel('Time, $t\mbox{ } (s)$','interpreter','latex','FontSize',34);
ylabel('Distance, $x_{1,2 \mbox{ }} (m)$','interpreter','latex','FontSize',34)
hold off
figure(202);
h2 = plot(t, x1dot, 'b-', t, x2dot, 'k:');
set(h2,'linewidth',4);
legend(['x1-dot'], ['x2-dot'])
legend('Location','Northwest')
set(gca,'xscale','linear','FontSize',26)
set(gca,'yscale','linear','FontSize',26)
set(gca,'XMinorTick','on')
set(gca,'YMinorTick','on')
grid off
xlabel('Time, $t\mbox{ } (s)$','interpreter','latex','FontSize',34);
ylabel('Speed, $\dot{x}_{1,2} \mbox{ } (ms^{-1})$','interpreter','latex','FontSize',34)
hold off
figure(203);
h3 = plot(t, x1ddot, 'b-', t, x2ddot, 'k:');
set(h3,'linewidth',4);
legend(['x1-ddot'], ['x2-ddot'])
legend('Location','Northwest')
set(gca,'xscale','linear','FontSize',26)
set(gca,'yscale','linear','FontSize',26)
set(gca,'XMinorTick','on')
set(gca,'YMinorTick','on')
grid off
xlabel('Time, $t\mbox{ } (s)$','interpreter','latex','FontSize',34);
ylabel('Acceleration, $\ddot{x}_{1,2} \mbox{ } (ms^{-2})$','interpreter','latex','FontSize',34)
hold off

-------

And my plots are:

Please check my approach and my code properly. I have always used FORTRAN to solve such equations, trying Matlab first time for such a problem.

Torsten 2018-11-12

在 MATLAB Online 中打开

ydot(3)=y(3) and ydot(6)=y(6) ? Looks wrong to me.

Vikash Pandey 2018-11-12

编辑：Vikash Pandey 2018-11-12

@ Torsten, you may be correct. How do I fix that? What worries me is that if I change the initial conditions to initial_conditions = [2 1 0 4 3 0]; which means acceleration is zero for both x1 and x2 at t= 0. Then I get the following plots. x1,2 and x1,2-dot plots change, but x1,2-ddot plots do not show any sign of change at all. How is that possible? Velocity is changing but not the acceleration? There must be some bug in the code.

I cannot post more images, so I am putting up the links below.

https://postimg.cc/xJJknsmz

https://postimg.cc/DJ9SZqQY

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采纳的回答

Torsten 2018-11-12

1
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/429363-how-to-implement-tightly-coupled-nonlinear-odes-using-ode45-in-matlab#answer_346494

在 MATLAB Online 中打开

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.

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