Contact Us!
CoCalc Logo Icon
StoreFeaturesDocsShareSupportNewsAboutSign UpSign In

Real-time collaboration for Jupyter Notebooks, Linux Terminals, LaTeX, VS Code, R IDE, and more,
all in one place. Commercial Alternative to JupyterHub.

| Download
Views: 150
Tweet
Share
Share
Kernel: SageMath 7.6

6.3 - The Euler Method ┬──┬ ノ( ゜-゜ノ)

Suppose x(t)x(t) is differentiable, then dxdt=lim⁡Δt→0x(t+ΔT)−x(t)Δt\frac{dx}{dt} = \lim_{\Delta t \to 0} \frac{x(t + \Delta T) - x(t)}{\Delta t}

So when Δt\Delta t is very small dxdt≈x(t+Δt)−x(t)Δt\frac{dx}{dt} \approx \frac{x(t + \Delta t) - x(t)}{\Delta t} x(t+Δt)≈x(t)+Δt⋅dxdtx(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 dxdt=f(x,t)\frac{dx}{dt} = f(x,t) Δx=Δt⋅f(x,t)\Delta x = \Delta t \cdot f(x,t)

OMG WHALES AGAIN (╯°□°）╯︵ ┻━┻

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

Assumptions ΔB=Δt(0.05B(1−B150000)−αBF)\Delta B = \Delta t (0.05B(1-\frac{B}{150000})-\alpha B F) ΔF=Δt(0.08F(1−F400000)−αBF)\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)
Image in a Jupyter notebook

RLC Problem (ｏ・・)ノ”(ᴗ ᴗ。)

Variables: vC=v_C = voltage across capacitor iC=i_C = current through capacitor vR=v_R = voltage across resistor iR=i_R = current through resistor vL=v_L = voltage across inductor iL=i_L = current through inductor

Assumptions: CdvCdt=iCC\frac{dv_C}{dt} = i_C vR=f(iR)v_R = f(i_R) LdiLdt=vLL\frac{di_L}{dt} = v_L iC=iR=iLi_C = i_R = i_L vC+vR+vL=0v_C + v_R + v_L = 0 L=1L = 1 C=1/5C = 1/5 f(x)=x3+4xf(x) = x^3 + 4x

Dynamical system: dvCdt=iLC\frac{dv_C}{dt} = \frac{i_L}{C} So ΔvC=Δt(iLC)\Delta v_C = \Delta t (\frac{i_L}{C}) diLdt=−vC+f(iL)L\frac{di_L}{dt} = -\frac{v_C+f(iL)}{L} So ΔiL=Δt(−vC+f(iL)L)\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)
Image in a Jupyter notebook
list_plot(zip(L_vC,L_iL))
Image in a Jupyter notebook