Overview

• Growth/decay models

• Heat transfer

• Mechanics

• How might the quantity $Q\left(t\right)$ change in time?

  • Rate of change determined by the independent variable $t$: $Q^{\text{'}}=f\left(t\right)$

    ${\displaystyle Q\left(t\right)=Q_0+\underset{0}{\overset{t}{\int }}f\left(\tau \right)d\tau =f\left(0\right)t+\frac{1}{2}{f}^{\text{'}}\left(0\right)t^2+\cdots +\frac{1}{n!}{f}^{(n-1)}\left(0\right)t^n+\dots }$

  • Rate of change determined by the dependent variable $Q:Q^{\text{'}}=g\left(Q\right)$

    ${\displaystyle g\left(Q\right)\cong g\left(0\right)+g^{\text{'}}\left(0\right)Q\Rightarrow Q\left(t\right)=e^{g^{\text{'}}\left(0\right)t}{Q}_0+(e^{g^{\text{'}}\left(0\right)t}-1)\frac{g\left(0\right)}{g\left(1\right)}}$

  • Rate of change determined by both variables: $Q^{\text{'}}=h(t,Q)$

    ${\displaystyle h(t,Q)\cong h(0,0)+h_t(0,0)t+h_Q(0,0)Q\equiv h_{00}+h_{10}t+h_{01}Q\Rightarrow }$ ${\displaystyle Q\left(t\right)=e^{h_{01}t}{Q}_0+(e^{h_{01}t}-1)\frac{h_{00}}{h_{01}}+(e^{h_{01}t}-1-h_{01}t)\frac{h_{10}}{h_{01}^2}}$

Growth rate depends only on time $\Rightarrow$ future system states accumulate past growth

${\displaystyle Q\left(t\right)=Q_0+\underset{0}{\overset{t}{\int }}f\left(t\right)dt,f\left(t\right)=f_0+f_{1}t+f_2{t}^2+\cdots }$

Example 1

Constant growth rate $\Rightarrow$ linear growth, $Q\left(t\right)=Q_0+f_{0}t$

Example 2

Linearly-varying growth rate $\Rightarrow$ quadratic growth

${\displaystyle f\left(t\right)=f_0+f_{1}t\Rightarrow Q\left(t\right)=Q_0+f_{0}t+\frac{1}{2}{f}_{1}t}$

Example 3

Oscillatory growth rate $\Rightarrow$ delayed, oscillatory system state

${\displaystyle f\left(t\right)=a\sin t\Rightarrow Q\left(t\right)=Q_0+a(1-\cos t)}$

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• If $Q^{\text{'}}=g\left(Q\right)$, i.e., growth rate determined by dependent variable, the system exhibits exponential growth or decay $Q^{\text{'}}=rQ,Q\left(t_0\right)=Q_0\Rightarrow Q\left(t\right)=e^{r(t-t_0)}{Q}_0$

  • Radioactive decay $(r<0)$

  • Compound interest accumulation

  • Carbon dating $r=-(\ln 2)/5570$ (half life of ${}^{14}$C is 5570 years)

• Equilibrium temperature $T^{\text{'}}=-k(T-T_m)$ ($T_m$ ambient temperature)

  ${\displaystyle T\left(t\right)=T_m+(T_0-T_m)e^{-kt}.}$

• Mixing problems $Q^{\text{'}}=r-cQ$

  ${\displaystyle Q\left(t\right)=\frac{r}{c}+e^{-ct}(Q_0-\frac{r}{c}).}$

• Conservation of momentum

  ${\displaystyle \frac{d}{dt}(mv)=F\Rightarrow v^{\text{'}}=\frac{F}{m}.}$

• Examples:

  • Object falling with air resistance $v^{\text{'}}=g-cv$ (very low speed behavior)

    ${\displaystyle v\left(t\right)=\frac{g}{c}+e^{-ct}(v_0-\frac{g}{c}).}$

  • Object falling with air resistance $v^{\text{'}}=g-d{v}^2$