Non-dimensionalization of diff. system

[LouiseLøcke]

I don't have any experince in normalizing when the equations depend on each other, so was hoping someone could tell me if what I have done is correct, and if not what the problem is :)

Homework Statement

I need to non-dimensional the following differential equations.

L1:= dn/dt = (a-b-k*y)*n
L2:= dy/dt = l*n-d*y

to
dN/dT= (D-Y)*N
dY/dT = N-Y
knowing that D = (a-b)/d.

The Attempt at a Solution

I start with inserting

n = k[n]*N,
y = k[y]*Y,
t = k[t]*T

L1:= d(k[n]*N))/d(k[t]*T) = (a -b-k*(k[y]*Y)*k[n]*N

L2:= d(k[y]*Y)/d(k[t]*T)= l*(k[n]*N) - d*k[y]*Y

I look at L2 first, and divide with k[y]/k[t] on both sides.

Then I set l= k[y]/(k[n]*k[t]) and d=1/k[t]

getting dY/dT = N-Y

next I look at L1, and divide with k[n]/k[t] on both sides and inserting k[t] = 1/delta ( which I found above)

getting
dN/dT = ((a-b)/(d) - (k*(k[y]*Y))/(d))*N

I now put D = (a-b)/d and k = d/k[y]
getting

dN/dT= (D-Y)*N (which is what I wanted)

Is this the correct way of approaching it?
Thanks Louise

 

[mighty2000]

Dear Louise,

Thank you for reaching out and seeking clarification on your approach to non-dimensionalizing the given differential equations. It appears that your approach is correct and you have successfully non-dimensionalized the equations. However, it is always a good practice to double check your work and make sure that all the units and dimensions match up in the final equations.

In general, the process of non-dimensionalizing involves replacing the original variables with dimensionless variables, such as the ones you have defined (k[n]*N, k[y]*Y, k[t]*T). These dimensionless variables are chosen in a way that the resulting equations are simpler and easier to work with. You have correctly identified the constants D and k in your final equations, and it is important to note that these constants have specific units and dimensions that must match up with the variables in the equations.

Overall, it seems like you have a good understanding of the non-dimensionalization process and have applied it correctly in this case. Keep up the good work and always remember to double check your work to ensure accuracy.