First-Order Linear Systems of ODEs
Last edited: August 8, 2025Consider the case where there are two functions interacting with each other:
\begin{equation} y_1(t) \dots y_{2}(t) \end{equation}
So we have more than one dependent function, with functions \(y_1, y_1’, y_2, y_2’\) and so forth. To deal with this, we simply make it into a matrix system:
\begin{equation} y(t) = \mqty(y_1(t) \\ \dots \\ y_{n}(t)) \end{equation}
For instance, should we have:
\begin{equation} \begin{cases} y_1’ = 3y_1 - 2y_2 \\ y_2’ = -y_1 + 5y_2 \end{cases} \end{equation}
fixed-point iteration
Last edited: August 8, 2025Fixed point iteration is a method for finding a fixed point of a function, which is a value that remains unchanged when the function is applied to it (i.e., f(x) = x). The method works by repeatedly applying the function to an initial guess:
- Start with an initial approximation x₀
- Compute successive iterations: xₙ₊₁ = f(xₙ)
- Continue until convergence (|xₙ₊₁ - xₙ| < ε) or maximum iterations
The method converges if the function is a contraction mapping in the neighborhood of the fixed point (|f’(x)| < 1).
flexua
Last edited: August 8, 2025Floyd's Invariant Method
Last edited: August 8, 2025To prove properties on Finite State Machines, we can construct a proof:
- stating an invariant
- proving that the invarient is true for all states
- for all transitions: assume invarient is true before transition and prove that its true after
So, essentially induction.
flux
Last edited: August 8, 2025flux is the volume of flow per unit time: multiplying the speed of flow \(\frac{m}{s}\) against the area \(m^{2}\) gives you the volume flowed per second \(\frac{m^{3}}{s}\).
tilted flux
Flow, however, is not necessarily perpendicular to the plain. Therefore, we only analyze the perpendicular component of the flow: that is — \(\Phi = Av \cos \theta\). Why? If we tipped the plane (of certain area) up, the flow that used to cross the bottom of the plane now will not go through the plane, so we want to account for that.