How to solve this differential equation using matlab?

I know the value of n_(j-1), T_(j-1) and n_(j).

I want to solve this differential equation and get the numerical solution.

How can I do it in MATLAB?

[Torsten]

What do you mean by

I know the value of ndot_(j-1), T_(j-1) and ndot_(j)

?

Do you have time-dependent analytic functions for them ? Or only time-dependent vectors ? Or something else ?

[Anup]

I think I am having some problems here, I am not able to explain to you how i know those values.

I know the temperature T until a point where I have to use the differential equation. Let's say that is T(j-1).

Let's consider 'n' terms some constants.

How would you solve this?

If you haven't understand what I am saying, ignore all, just suggest me how would you approach the solution? Just your way if you are given equation only

Answer 1

[Simran]

Hi @Anup,

As I understand, you want to solve the following differential equation:

[ \frac{\partial T_j}{\partial t} = \dot{n}{j-1} T{j-1} - \dot{n}_j T_j ] using MATLAB. What you provided is a first-order differential equation for T_j.

As you mentioned, the values for - ( \dot{n}{j-1} ), ( T{j-1} ), and ( \dot{n}_j ) are known. All we need is an initial condition for ( T_j) at ( t = 0 ) . Lets say (T_j(0) = T_{j0} ) .

Next, create a function in MATLAB to represent the ODE (first-order differential equation) and name it odefun.m. Below is odefun.m:

function dTdt = odefun(t, Tj, nj_minus_1, Tj_minus_1, nj) 
dTdt = nj_minus_1 * Tj_minus_1 - nj * Tj; 
End 

We can solve this ODE using the inbuilt MATLAB function “ode45”. You can refer to this link to understand about it more:

https://www.mathworks.com/help/matlab/ref/ode45.html

Below is the script to solve it:

% first define the known parameters 
nj_minus_1 = 1.0; % example value, you can use your own value 
Tj_minus_1 = 100; % example value 
nj = 0.5; % example value 
% Set the initial condition by defining T_{j0}. 
Tj0 = 50; % example initial value 
% Define time span for the solution 
tspan = [0 10]; 
% Use MATLAB’S ODE solver (ode45) to solve ODE (first order differential equation) 
[t, Tj] = ode45(@(t, Tj) odefun(t, Tj, nj_minus_1, Tj_minus_1, nj), tspan, Tj0); 
% Lastly plot the solution 
plot(t, Tj); 
xlabel('Time'); 
ylabel('T_j'); 
title('Solution of the Differential Equation'); 

The graph of the ODE should look something like this:

The solution can be tested with some different parameter values to see if it behaves as expected.

Below are the links attached that maybe relevant to the discussion:

https://www.mathworks.com/help/matlab/ref/ode45.html

https://www.mathworks.com/help/matlab/math/differential-equations.html