MathStudio Link: The M&M Experiment.

• This is a simulator for the M&M population experiment. Instead of M&Ms that can be up or down we have a row of boxes that can contain a 0 or 1. M&M down is a 0, M&M up is a 1.
• The experiment starts with an initial number boxes all in 1 except the first box, which counts how many 1 we have in that row.
• The second row is constructed from the first by randomly choosing between 0 or 1. Again, the first box counts how many 1 we got.
• The third row is constructed from the second, if we had a box with a 0, it remains a 0. If we had a 1, we randomly choose a between 0 or 1. As before, the first box counts the number of 1 in that row.
• We keep going in this way till we have a number of rows given by the constant in Trials.
• We can also add a fix number of boxes on each round, we call these added boxes the immigrants.

Finally, we can generalize the M&M experiment by changing the mortality coefficient from 50% to any percent from 0 to 100.

• The simulation has one button and four slider:
• The button Run runs the simulation with the chosen parameters.
• The slider Initial fixes the initial amount of M&Ms. Default Initial = 30
• The slider Imm fixes the number of immigrants M&M added on every trial. Default Imm = 0
• The slider Trials sets the number of rounds we have in the simulation. Default Trials = 10
• The slider Mortality sets the mortality percent. Default Mortality = 50%

We also can plot the first column with the counts of boxes with 1.

MathStudio Link: Going Viral Simulation.

This is a simulator of a viral infection spreading in a population with the following rules:

• The total population where the virus can spread is fixed at 100.
• Each person in the population has a fixed Identification Number in the interval \( [1,100]\).
• At day zero there is only one person infected.
• Each day, each infected person pass the virus to only one, random, person in the population.
• Infected persons remain always infected.
• Nobody dies.

MathStudio Link: Direction Field.

We plot the direction field for the differential equation \(y’ = \sin(y)\). For the segments you can change:

• The width.
• The length.
• The \(x\) separation and density.
• The \(y\) separation and density.

MathStudio Link: Picard Iteration vs Taylor Expansion: Linear Equations.

• We graph in purple the functions \(y(t)\) solutions of the initial value problem
  \[
  y^{\prime}(t) = 2 \,y(t) +3,
  \qquad
  y(0) = 1.
  \]
• The slider Function turns on-off the graph of the solution \(y(t)\).
• We graph in blue approximate solutions \(y_n\) of the differential equation constructed with the Picard iteration up to order \(n=10\). The slider Picard_App_Blue turns on-off the Picard approximate solution.
• We graph in green the \(n\)-order Taylor expansion centered \(t=0\) of the solution of the differential equation up to order \(n=10\). The slider Taylor_App_Green turns on-off the Taylor approximation of the solution.

We conclude that the Picard iteration is identical as the Taylor expansion for solutions of the linear differential equation above.

MathStudio Link: Picard Iteration vs Taylor Expansion: Non-Linear Equations – Explicit Solution.

• We graph in purple the functions \(y(t)\) solutions of the initial value problem
  \[
  y^{\prime}(t) = y^2(t),
  \qquad
  y(0) = -1.
  \]
• The slider Function turns on-off the graph of the solution \(y(t)\).
• We graph in blue approximate solutions \(y_n\) of the differential equation constructed with the Picard iteration up to order \(n=5\). The slider Picard_App_Blue turns on-off the Picard approximate solution.
• We graph in green the \(n\)-order Taylor expansion centered \(t=0\) of the solution of the differential equation up to order \(n=5\). The slider Taylor_App_Green turns on-off the Taylor approximation of the solution.

We conclude that the Picard iteration is a different and better approximation than the Taylor expansion for solutions of the non-linear differential equation above.

MathStudio Link: Picard Iteration vs Taylor Expansion: Non-Linear Equations – No Explicit Solution.

• In this second example we study approximate solutions of the initial value problem
  \[
  y^{\prime}(t) = y^2(t) + t,
  \qquad
  y(0) = -1.
  \]
• In this case we do not have an explicit expression for the solution \(y(t)\).  We only have the approximate solutions.
• We graph in blue approximate solutions \(y_n\) of the differential equation constructed with the Picard iteration up to order \(n=5\). The slider Picard_App_Blue turns on-off the Picard approximate solution.
• We graph in green the \(n\)-order Taylor expansion centered \(t=0\) of the solution of the differential equation up to order \(n=5\). The slider Taylor_App_Green turns on-off the Taylor approximation of the solution.

We conclude, one more time, that the Picard iteration is a different (hopefully better) approximation than the Taylor expansion for solutions of the non-linear differential equation above.

MathStudio Link: Beating and Resonance on an LC-series Circuit.

• An LC-series circuit with a voltage source is described by Krichhoff equation
  \[
  L \, I'(t) + \frac{1}{C} \int I(t)\, dt = V(t).
  \]
  • Consider the case that \(V(t) = L \sin(\omega t)\). In this case, computing one time derivative and dividing by \(L\) we get
    \[
    I” + \omega_0^2 I = \omega \cos(\omega t).
    \] where \(\displaystyle\omega^2 = \frac{1}{\sqrt{LC}}\). The solution with initial conditions \(I(0)=0\) and \(I'(0)=0\) is
    \[
    I(t) = \frac{t}{2}\,\sin(\omega t).
    \]
  • Consider the case that \(V(t) = L \sin(\nu t)\). In this case, computing one time derivative and dividing by \(L\) we get
    \[
    I” + \omega_0^2 I = \nu \cos(\nu t).
    \] where \(\displaystyle\omega^2 = \frac{1}{\sqrt{LC}}\). The solution with initial conditions \(I(0)=0\) and \(I'(0)=0\) is
    \[
    \tilde I(t) = \frac{\nu}{(\omega^2-\nu^2)}\bigl( \cos(\nu t)-\cos(\omega t) \bigr)
    \]
• Click on the interactive graph link here to see how the function changes when \(\nu \to \omega_0\), exhibiting the beating phenomenon.

MathStudio Link: Dirac Delta Sequences.

We show a few different sequence of functions that have the same Dirac’s delta as their limit. The sequences are:

• In red: \(\displaystyle\Bigl. \delta_n(t) = n \,\bigl(u(t) – u(t- \frac{1}{n})\bigr)\).
• In blue: \(\displaystyle\Bigl.\delta_n(t) = \frac{n}{2} \,\bigl(u(t+\frac{1}{n}) – u(t-\frac{1}{n})\bigr)\).
• In green: \(\displaystyle\Bigl.\delta_n(t) = \sqrt{\frac{n}{\pi}} \, e^{-n^2 t^2}\).
• In purple: \(\displaystyle\Bigl.\delta_n(t) = \frac{1}{\pi} \frac{n}{(1 + n^2 t^2)}\).
• In gray: \(\displaystyle\Bigl.\delta_n(t) = \frac{\sin(n t)}{\pi \,t}\).

Convolution Graphs.
We compute the convolution of the functions \(f\) and \(g\),
\[
(f*g)(t) = \int_0^t f(\tau) \,g(t-\tau)\, d\tau,
\] and we plot \(f\) in blue, \(g\) in green, and \(f*g\) in red.

We see that the convolution is a measure of (although not equal to) the area of the overlap of the two functions, \(f\) and \(g\), which is shown in gray.

• MathStudio Link: Convolution: Example 1 (Slow).
  In this graph we choose
  \[
  f(x) = u(x) -u(x-1),
  \qquad
  g(x) = u(x) -u(x-1).
  \]
• MathStudio Link: Convolution: Example 2.
  In this graph we choose
  \[
  f(x) = u(x) \, e^{-x},
  \qquad
  g(x) = u(x) \sin(x).
  \]
• MathStudio Link: Convolution, Example 3. (Slow).
  In this first graph we choose
  \[
  f(x) = u(x) -u(x-1),
  \qquad
  g(x) = 2\,u(x) \, e^{-x}.
  \]

MathStudio Link: Linear Pendulum.

MathStudio Link: Predator-Prey System.

MathStudio Link: Predator-Prey System: Finite Food.

MathStudio Link: Infectious Disease Models (SIR: Susceptible, Infected, Recovered).

MathStudio Link: Eigenvectors of a Matrix.

MathStudio Link: 2×2 Linear Differential System: Real Eigenvectors.

MathStudio Link: 2×2 Linear Differential System: Complex Eigenvectors.

MathStudio Link: The Nonlinear Pendulum.

MathStudio Link: Competing Species: Extinction.

MathStudio Link: Competing Species: Coexistence.

MathStudio Link: BVP and Eigenfunctions.

In the first picture we show the solution to the BVP
\[
y”+ \pi^2 y =0,
\qquad
y(0)=1,
\quad
y(1)=-1
\] This BVP has infinitely many solutions, given by
\[
y(x) = \cos(\pi x) + k \sin(\pi x),
\qquad
k\in \mathbb{R}.
\]

• We plot the fundamental solution \(y_1(x)= \cos(\pi x)\) in red.
• We plot the fundamental solution \(y_2(x)= \sin(\pi x)\) in red.
• We plot the solution \(y_k(x) = k \sin(\pi x)\) in purple.
• We plot the solution \(y(x) = \cos(\pi x) + k\sin(\pi x)\) in blue.

In the second picture we plot the function
\[
y_n(x) = \sin(n\pi x),
\qquad
n\in \mathbb{R}.
\] In the case that \(n\) is an integer, these functions are eigenfunctions solutions
\[
y” + \lambda y =0,
\qquad
y(0)=0,
\quad
y(1)=0.
\] In this problem the eigenvalues are \(\lambda_n = (n\pi)^2\) for \(n = 1, 2, 3, \cdots\), and the eigenfunctions are the functions \(y_n\) above for \(n=1,2,3,\cdots\).

MathStudio Link: Fourier Series.

MathStudio Link: Cosine and Sine Series of Even and Odd Extensions.

MathStudio Link: Heat Equation: Dirichlet BC.

MathStudio Link: Heat Equation: Neumann BC.