dTdt=−k(T−Ta) k=0.4,Ta=25∘, initial condition T(0)=T0=75∘C .how to solve using ode45 and report the temperature obtained at times t=1,2,3 and 4 . report T(1.0)

ORIGINAL:

dTdt=−k(T−Ta)

k=0.4,Ta=25∘C, k=0.4,Ta=25° and the initial condition T(0)=T0=75∘

The true solution of the ODE is given by:

T(t)=Ta+(T0−Ta)e−kt

Mathematically translated by @Sam Chak (in the original way) using LaTeX:

, °C, and the initial condition °C

The true solution of the ODE is given by:

[Ramesh]

Please help me to solve this problem

[Dyuman Joshi]

How is your question related to MATLAB?

[Chuguang Pan]

you can refer to the ode45 documentation, there are many good examples.

[Sam Chak]

Hi @Ramesh,

The Law of Cooling is shivering COOL 🥶! However, two of your previous questions are still open. Would you like to follow up and provide feedback? This will allow others to update their answers or submit new ones, so that you can evaluate and close them satisfactorily.

Question 1 (opened on 2 Feb 2024):

https://www.mathworks.com/matlabcentral/answers/2077436-minimum-value-from-a-vector-for-x-in-the-range-x-4-0-5-4-calculate-f-x-x2-1-3x-2?s_tid=srchtitle

Question 2 (opened on 5 Feb 2024):

https://www.mathworks.com/matlabcentral/answers/2078176-how-to-solve-this-function?s_tid=srchtitle

Hi @Chuguang Pan

Since an analytical solution exists for a first-order linear system and its mathematical structure is simple enough for display, it would be nice to straightforwardly use another popular command from the Symbolic Math Toolbox.

[Sam Chak]

Edit: The code is edited to reflect on the true solution is corrected by @Chuguang Pan. However, @Ramesh should verify it mathematically.

k   = 0.4;
Ta  = 25;
T0  = 75;
e   = exp(1);                       % Numeric Representation of e
% T = @(t) Ta + (T0 -Ta)*e - k*t;   % true solution supplied by the OP
T   = @(t) Ta + (T0 -Ta)*e.^(- k*t);% true solution corrected by Chuguang Pan

t   = 0:0.1:20;                     % simulate from time 0 to 20
plot(t, T(t)), grid on, xlabel t, ylabel T(t)
title('Temperature decay based on the Analytical Solution')

For the numeric representation of e, please see https://www.mathworks.com/help/matlab/ref/exp.html#bt_a2dn-1.

[Ramesh]

I am not able to get the answer kindly help me

[Sam Chak]

Hi @Ramesh,

I understand that you are unable to obtain the answer, which might be due to a possible error in entering the equation in the MATLAB code when using the ode45() command. To help troubleshoot the issue, please click on the indentation icon and copy/paste the MATLAB code into the grey field. This will allow us to assist you further.

[Chuguang Pan]

Hi @Sam Chak,

I think the true solution of T should be , so the formula of T in your code should be:

T=@(t) Ta + (T0-Ta)*exp(-k*t);

[Sam Chak]

Thanks @Chuguang Pan. @Ramesh could have checked all equations before posting the problem.

Answer 1

[Udit06]

Hi Ramesh,

You can follow the following steps to solve a differential equation using "ode45" function

1) Define the ODE function:Create a function that represents the right-hand side of the ODE.

2) Solve the ODE using ode45: "ode45" function is used for solving initial value problems for ordinary differential equations. Give the function handle, time span and initial conditions as input to the function.

3) Extract the Temperatures at desired times: After solving the ODE, you can extract the temperature values at the desired times using interpolation

Here is the code implementation for the same:

% Define the parameters
k = 0.4;
Ta = 25;
T0 = 75;

% Define the ODE function
odefun = @(t, T) -k * (T - Ta);

% Time span (from t=0 to t=4)
tspan = [0 4];

% Initial condition
T_initial = T0;

% Solve the ODE
[t, T] = ode45(odefun, tspan, T_initial);

% Interpolate or find the closest values to t=1, 2, 3, and 4
T1 = interp1(t, T, 1.0); % Interpolating for T at t=1.0

% Display the result for T(1.0)
fprintf('The temperature at t=1.0 is approximately %.2f°C\n', T1);

The temperature at t=1.0 is approximately 58.52°C

You can refer to the following MathWorks documentation to understand more about "ode45" function.

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

I hope this helps.