6.3 Euler method.ipynb

Kernel: SageMath 7.6

6.3 - The Euler Method ┬──┬ ノ( ゜-゜ノ)

Suppose $x(t)$ is differentiable, then $\frac{dx}{dt} = \lim_{\Delta t \to 0} \frac{x(t + \Delta T) - x(t)}{\Delta t}$

So when $\Delta t$ is very small $\frac{dx}{dt} \approx \frac{x(t + \Delta t) - x(t)}{\Delta t}$ $x(t+\Delta t) \approx x(t) + \Delta t \cdot \frac{dx}{dt}$

We can use a discrete-time dynamical system to model a continous-time dinamical system $\frac{dx}{dt} = f(x,t)$ $\Delta x = \Delta t \cdot f(x,t)$

OMG WHALES AGAIN (╯°□°）╯︵ ┻━┻

Variables $B =$ Number of blue whales $F =$ Number of fin whales

Assumptions $\Delta B = \Delta t (0.05B(1-\frac{B}{150000})-\alpha B F)$ $\Delta F = \Delta t (0.08F(1-\frac{F}{400000})-\alpha B F)$

B0 = 5000
F0 = 70000
a = 10^(-7)
dt = 0.25

f1(B,F) = 0.05*B*(1-B/150000) - a*B*F
f2(B,F) = 0.08*F*(1-F/400000) - a*B*F

L_B = [B0]
L_F = [F0]

for i in range(100):
    B_next = L_B[-1] + dt*f1(L_B[-1],L_F[-1])
    F_next = L_F[-1] + dt*f2(L_B[-1],L_F[-1])
    L_B.append(B_next)
    L_F.append(F_next)
    
p1 = list_plot(L_B)
p2 = list_plot(L_F, color="fuchsia")
show(p1+p2)

RLC Problem (ｏ・・)ノ”(ᴗ ᴗ。)

Variables: $v_C =$ voltage across capacitor $i_C =$ current through capacitor $v_R =$ voltage across resistor $i_R =$ current through resistor $v_L =$ voltage across inductor $i_L =$ current through inductor

Assumptions: $C\frac{dv_C}{dt} = i_C$ $v_R = f(i_R)$ $L\frac{di_L}{dt} = v_L$ $i_C = i_R = i_L$ $v_C + v_R + v_L = 0$ $L = 1$ $C = 1/5$ $f(x) = x^3 + 4x$

Dynamical system: $\frac{dv_C}{dt} = \frac{i_L}{C}$ So $\Delta v_C = \Delta t (\frac{i_L}{C})$ $\frac{di_L}{dt} = -\frac{v_C+f(iL)}{L}$ So $\Delta i_L = \Delta t (-\frac{v_C+f(iL)}{L})$

L = 1
C = 1
dt = 0.1

f(x) = x^3+4*x
f1(vC,iL) = iL/C
f2(vC,iL) = iL-iL^3-vC

vC0 = 0.3
iL0 = 0.1

L_vC = [vC0]
L_iL = [iL0]

for i in range(400):
    vC_next = L_vC[-1] + dt*f1(L_vC[-1],L_iL[-1])
    iL_next = L_iL[-1] + dt*f2(L_vC[-1],L_iL[-1])
    L_vC.append(vC_next)
    L_iL.append(iL_next)
    
p1 = list_plot(L_vC)
p2 = list_plot(L_iL, color="fuchsia")
show(p1+p2)

list_plot(zip(L_vC,L_iL))