MATLAB: Solving a differential equation

ddedifferential equationsmass matrixMATLABode

Hello,

I am working at a mathematical problem and I wanted to solve it numerically through MATLAB. The equation has the form:

y'(t) = g(t,y(t)) – a y(t)

and

g(t,y(t)) = b sinh[(g(t-dt, y(t-dt))] y(t) + c

with a,b,c in R.

I already tried using the ode and dde function in MATLAB without luck. Therefore I hope you can help me with the implementation in the code.

Best Answer

If your equations contain only linear combinations of the derivatives, you could formulate it as a mass matrix problem M(t,y)y' = f(t,y) . For the two equations you have provided in the comment,

dF = - R*dI/u_a;
dI = const2*cosh(insinh*F)*insinh*dF+vor*sinh(insinh*F)*dn*exp(-beta*dEC2);

you could set up the problem as follows:

y = [F; I];
M = [1 - R/u_a; const2*cosh(insinh*F)*insinh 1];

and f(t,y) would output

[0; vor*sinh(insinh*y(1))*dn*exp(-beta*dEC2)]

See the ode45 documentation for how to include a mass matrix.

(edited to include the relevant part of betlor5's comment)

Related Solutions

MATLAB: Writing a differential equation for iteration using for loop.

This is known as simple Euler integration. It arises from

% dx/dt = (a - x^2 - y^2)*x - omega*y   
% Let dx/dt be approximated by (x(t+dt) - x(t))/dt
% and the right hand side by (a - x(t)^2 - y(t)^2)*x(t) - omega*y(t)
% so (x(t+dt) - x(t))/dt = (a - x(t)^2 - y(t)^2)*x(t) - omega*y(t)        approximately
% Multiply both sides by dt to get
% x(t+dt) - x(t) = ((a - x(t)^2 - y(t)^2)*x(t) - omega*y(t))*dt
% Add x(t) to both sides to get
% x(t+dt) = x(t) + ((a - x(t)^2 - y(t)^2)*x(t) - omega*y(t))*dt

MATLAB: Using ode45 for coupled equations

You are mixing syntax here:

@(t,x)myfun

Either use the syntax @(t,x)expression, where expression is the derivative

or use the syntax @myfun, where myfun is a function that returns the derivative.

E.g., using y as the input state where we define y(1) = x and y(2) = z

@(t,y)[t+y(1)+y(2);t+y(1)-(3*y(1))]

But if that last x(i) in your z derivative was supposed to be z(i), then it would be this:

@(t,y)[t+y(1)+y(2);t+y(1)-(3*y(2))]

Or have a separate function for this in a file called myfun.m and use the @myfun syntax:

function dy = myfun(t,y);
dy = [t+y(1)+y(2);t+y(1)-(3*y(2));  % either y(1) or y(2) for that last term, you need to check this
end

Finally, that last input argument for ode45 needs to be the initial values of x and z, or in this case y(1) and y(2). So that last argument should be [1;1], not 5.

Best Answer

Related Solutions

MATLAB: Writing a differential equation for iteration using for loop.

MATLAB: Using ode45 for coupled equations

Related Question