How do I solve an ODE with time-dependent parameters in MATLAB?

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Consider the following ODE with time-dependent parameters:
y'(t) + f(t)y(t) = g(t)
and the given initial condition:
y(0) = 1
This is an example of an ODE with time-dependent terms. Suppose the time-dependent terms are defined only through the set of data points given in two vectors. Which of the MATLAB ODE solvers should I use, and how do I set up this problem?

Accepted Answer

MathWorks Support Team
MathWorks Support Team on 5 Jun 2023
Edited: MathWorks Support Team on 2 Aug 2023
To solve this ODE, you must pass the data sets to the derivative function as additional parameters. When the ODE solver calls the derivative function, it will pass a specified time as the first input argument. You must then interpolate the datasets to obtain the value of the time-dependent terms at the specified time. This is performed within the following function, called myODE.
function dydt = myODE(t, y, ft, f, gt, g)
f = interp1(ft, f, t); % Interpolate the data set (ft, f) at times t
g = interp1(gt, g, t); % Interpolate the data set (gt, g) at times t
dydt = -f.*y + g; % Evalute ODE at times t
The function f is defined through the n-by-1 vectors tf and f, and the function g is defined through the m-by-1 vectors tg and g.
Now, you can refer to myODE within a call to a MATLAB ODE solver. Assume that the time-dependent parameters ft and gt are defined within the data sets generated by the following code.
ft = linspace(0, 5, 25); % Generate t for f
f = ft.^2 - ft - 3; % Generate f(t)
gt = linspace(1, 6, 25); % Generate t for g
g = 3*sin(gt - 0.25); % Generate g(t)
The following code uses the ODE45 function to solve this time-dependent ODE.
TSPAN = [1 5]; % Solve from t=1 to t=5
IC = 1; % y(t=0) = 1
[T Y] = ode45(@(t,y) myODE(t, y, ft, f, gt, g), TSPAN, IC); % Solve ODE
Note that if you are using a version of MATLAB prior to MATLAB 7.0 (R14), you will need to pass the four additional parameters ft, f, gt, and g, into the ODE solver as follows.
TSPAN = [1 5]; % Solve from t=1 to t=5
IC = 1; % y(t=0) = 1
[T Y] = ode45(@myODE, TSPAN, IC, [], ft, f, gt, g); % Solve ODE
Now you can plot the solution y(t) as a function of time.
plot(T, Y);
title('Plot of y as a function of time');
xlabel('Time'); ylabel('Y(t)');
  4 Comments
Steven Lord
Steven Lord on 27 Jan 2021
Please show your updated scripts or functions.
As a guess, if you have a script file (your call to ode45) that contains a function (myODE) put the script content before the function rather than after.
Steven Lord
Steven Lord on 27 Jan 2021
As I suspected, you put the function before the script content. Move the function to after the script content. But you need to do the interpolation inside myODE.
ft = linspace(0, 5, 25); % Generate t for f
f = ft.^2 - ft - 3; % Generate f(t)
gt = linspace(1, 6, 25); % Generate t for g
g = 3*sin(gt - 0.25); % Generate g(t)
TSPAN = [1 5]; % Solve from t=1 to t=5
IC = 1; % y(t=0) = 1
[T Y] = ode45(@(t,y) myODE(t, y, ft, f, gt, g), TSPAN, IC); % Solve ODE
plot(T, Y);
title('Plot of y as a function of time');
xlabel('Time'); ylabel('Y(t)')
function dydt = myODE(t, y, ft, f, gt, g)
% You probably want to change one of the uses of the variables f and g
% to a different name
f = interp1(ft, f, t); % Interpolate the data set (ft, f) at times t
g = interp1(gt, g, t); % Interpolate the data set (gt, g) at times t
dydt = -f.*y + g; % Evalute ODE at times t
t = [1 5]; % this isn't necessary
end

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