1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I Exponential decay: I need an expression for N at time t

  1. Apr 25, 2017 #1
    I have a large quantity N, which starts off equal to a determinable value N0.

    Over a short time ∆t, the value of N changes by -∆t*(B*N - C)

    where B and C are determinable constants. Am I correct in thinking I can turn this into:

    dN/dt = -(B*N - C)

    How do I get this into a formula for N at time t? The 'extra' constant C seems to be making it more difficult than the examples of exponential decay I've found on the net.
     
  2. jcsd
  3. Apr 25, 2017 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    There are various ways to solve the differential equation, but the easiest is to "guess" the answer: an exponential function plus a constant. You can then find the parameters of this ansatz.
     
  4. Apr 27, 2017 #3
    You can solve the equation in two steps:
    1) Solve the homogeneous equation: ##dN_h/dt=-B*N_h(t)##
    2) Then, you need to find a particular solution satisfying the full equation. In this case we can guess ##N_p(t)=A##, where ##A## is to be determined.
    3) The most general solution is ##N(t)=N_h(t)+N_p(t)##
     
  5. Apr 27, 2017 #4
    Unfortunately, my maths isn't yet advanced enough to do that.

    I've been using a computer program to iterate the value of N over shorter and shorter dt intervals:

    Num0 = 8.98249E+23 'Num0 = start value of N
    B = 1.3831E-3
    C = 1.21989E+21​

    Numleft = Num0 'Numleft is the value of N as it decreases through the iterations

    For time = 0 To (120 / dt) Step dt
    Numleft = Numleft - dt * (B * Numleft - C)
    Next time

    As I try smaller and smaller values of dt, the final value of Numleft converges to 8.82E23

    B and C will change in different situations. I'd like a non-iterative equation to find Numleft for any B and C (these will always be positive, if that helps).

    PS. I notice that the constant C is not multiplied by Numleft, so the calculation in the loop could be written as: Numleft = Numleft - dt * B * Numleft + dt * C

    Does this mean that dt * C can be ignored as dt -> 0?​
     
  6. Apr 27, 2017 #5

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    That is more complicated than finding the exact solution.
     
  7. Apr 27, 2017 #6
    I don't know what your background in Mathematics is. How to solve that differential equation is taught in a first course in differential equations, already in high school. Anyway, I can give some advice regarding how to solve it numerically. I don't know if you can find a non-iterative equation unless you solve it analytically.
    However, I think you have misunderstood the way it should be solved. To do this numerically the simplest way is to discretize the equation. That is,
    ##dN \rightarrow N_{k+1}-N_{k)}, dt\rightarrow\Delta_t=t_{k+1}-t_{k}##, where ##N_{k+1}=N(t_{k+1}), N_{k}=N(t_{k})##. Here, ##t_k=k*\Delta##, assuming a equidistant mesh in time.
    Then, you have
    ##N_{k+1}=N_{k}-(BN_k-C)##, where ##N_0=Num0##
    In this way you can compute ##N(t)## by starting from ##t=0## and use the aforementioned relation. This is the so-called Euler method, see e.g. https://en.wikipedia.org/wiki/Euler_method
     
  8. Apr 27, 2017 #7
    Yes, I completely agree with you.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Exponential decay: I need an expression for N at time t
  1. T=k1.H, T=k2.1/N? (Replies: 8)

Loading...