MATLAB: How to use ode45 for an equation with space dependent coefficients

mathieuode45

Consider an equation of form:

where

so that after each time iteration

I have to update

to be used in the next time iteration. I have hard coded Runge-Kutta scheme but I believe that there are some tolerance problems. Therefore, I want to try ODE45 and check the difference.

Best Answer

p = @(x) sin(x).*sqrt(tan(x));
% x(1) == x
% x(2) == x'
f = @(t,x) [x(2); -p(x(1)).*x(1)];
tspan = [0 2];
x0 = [0.5 0.1];
[t, x] = ode45(f,tspan,x0);
plot(t,x(:,1),'r',t,x(:,2),'b')

Related Solutions

MATLAB: Trying to plot numerical solutions for systems of DEs

The reason is that ‘y’ is not defined in your function.

One option is to code ‘x’ as ‘x(1)’ and ‘y’ as ‘x(2)’:

f = @(t,x) [-2*x(1) + 4*x(2); x(1) + 2*x(2)];
tspan = [0 5];
x0 = [0;0]+eps;
[t,x] = ode45(f,tspan,x0);
figure
plot(t,x,'-o')

That works, and may do what you want.

MATLAB: 4th order Runge-Kutta code that can solve for several intial conditions

Here's a version:

% Initial conditions
t0=0;
x0=0:0.2:1;
y0=1:0.2:2;
dt=0.1;
tz=1;
t_range= t0:dt:tz;
X = zeros(numel(x0),numel(t_range));
Y = zeros(numel(y0),numel(t_range));
x_rk=zeros(1,numel(t_range));
y_rk= zeros(1,numel(t_range));
for n = 1:numel(x0)
    x_rk(1)=x0(n);
    y_rk(1)=y0(n);
    for i=1:length(t_range)-1
        [x_rk(i+1), y_rk(i+1)]=rk_method(x_rk(i),y_rk(i),dt);
    end
    X(n,:) = x_rk;
    Y(n,:) = y_rk;
    %% PLots
    subplot(numel(x0)/2,2,n)
    plot (t_range, x_rk,'b-o',t_range, y_rk,'y-o','MarkerSize',5);
    grid
    title([num2str(x0(n)),'   ',num2str(y0(n))])
    legend('x','y')
end
disp('X = '), disp(X)
disp('Y = '), disp(Y)
%% Functions
function [dxdt, dydt]= f(x,y)
dxdt =  5*x-3*y;
dydt = -6*x+2*y;
end 
function [x1, y1]=rk_method(x0,y0,dt)
[k11, k12]=f(x0,y0);     
[k21, k22]=f(x0+0.5*dt*k11,y0+0.5*dt*k12); 
[k31, k32]=f(x0+0.5*dt*k21,y0+0.5*dt*k22); 
[k41, k42]=f(x0+dt*k31, y0+dt*k32); 
x1=x0+dt*((k11/6)+((k21+k31)/3)+(k41/6));
y1=y0+dt*((k12/6)+((k22+k32)/3)+(k42/6));
end

Best Answer

Related Solutions

MATLAB: Trying to plot numerical solutions for systems of DEs

MATLAB: 4th order Runge-Kutta code that can solve for several intial conditions

Related Question