Discretization problem help

[TSN79]

If I have a logged temperature change over time which makes up a plottet graph. I denote this change

$${{dT} \over {dt}}$$

People have told me this can be solved using discretization, but I have no idea what that is. Apparently it is something like this:

$${{T_1 - T_2 } \over {\Delta t}}$$

but I don't really know what it means. Also, isn't a differential equation supposed to end up as a function of some variable? If so, how can an approximation like this wind up as one? Could someone explain it to me? Thx

 

[vatant]

If I have a logged temperature change over time which makes up a plottet graph. I denote this change

$${{dT} \over {dt}}$$

People have told me this can be solved using discretization, but I have no idea what that is. Apparently it is something like this:

$${{T_1 - T_2 } \over {\Delta t}}$$

but I don't really know what it means. Also, isn't a differential equation supposed to end up as a function of some variable? If so, how can an approximation like this wind up as one? Could someone explain it to me? Thx

Say you have logged time T = [T1,T2,T3...TN] over time values t = [t1,t2,t3,...tN]
now, if you do a plot of T vs t, you would get a curve. If all goes well, your plot may represent a function, say like Newton's law of cooling ..some exponential form may be available. Now, a continuous function like exp is continuously differentiable. Hence, assume you got a solution

T =f(t),
at any time t, you can get dT/dt leading to

dT/dt = f'(t) evaluated at some time t.

Now, this is straightforward.

Also, thiink abt a curve made up of discrete points.

So to get a change or derivative (the slope), we can also do

dT/dt at t1 = (T2-T1)(t2-t1) like forward differenced

or dT/dt at t1 = (T1-T2)/(t1-t2) backward differenced.

However, since these are first order approximations of the original functions, f(t) , your solution of evaluating dT/dt is first order accurate.

Higher order can be achieved by taking more grid points. Look at finite difference approximations.

the smaller the difference between t1 and t2, the better approximation of your time derivative.

 

[tacman]

Discretization is a method used to approximate continuous functions or data by dividing them into smaller, discrete intervals. In your case, the temperature change over time is a continuous function, but by using discretization, we can approximate it by dividing the time interval into smaller time steps. This allows us to represent the temperature change as a series of discrete values instead of a continuous function.

The notation you mentioned, {{T_1 - T_2 } \over {\Delta t}}, is an example of discretization. Here, T_1 and T_2 represent two temperature values at different time points, and Δt represents the time interval between those two points. This can be interpreted as the average rate of change in temperature over that time interval.

As for your question about differential equations, discretization is used to solve differential equations numerically. It allows us to break down a continuous function into smaller, simpler equations that can be solved using numerical methods. This approximation may not give an exact solution, but it can provide a good estimate of the behavior of the function over time.

I hope this helps to clarify the concept of discretization for you. If you need further assistance, you can consult with a math tutor or do some additional research on the topic. Best of luck!