From b7652c078a74ec0fd8419c4e0d8f9dc1d7b28020 Mon Sep 17 00:00:00 2001 From: Christian Kolset Date: Wed, 1 Oct 2025 14:04:14 -0600 Subject: Re-structure of Module 3 to make each file a seperate lecture --- tutorials/module_3/3_5_ode.md | 20 ++++++++++++++++++++ 1 file changed, 20 insertions(+) create mode 100644 tutorials/module_3/3_5_ode.md (limited to 'tutorials/module_3/3_5_ode.md') diff --git a/tutorials/module_3/3_5_ode.md b/tutorials/module_3/3_5_ode.md new file mode 100644 index 0000000..6249ad6 --- /dev/null +++ b/tutorials/module_3/3_5_ode.md @@ -0,0 +1,20 @@ +# Numerical Solutions of Ordinary Differential Equations + +Fundamental laws of the universe derived in fields such as, physics, mechanics, electricity and thermodynamics define mechanisms of change. When combining these law's with continuity laws of energy, mass and momentum we get differential equations. + +| Newton's Second Law of Motion | Equation | Description | +| ----------------------------- | -------------------------------- | ---------------------------------------------------- | +| Newton's Second Law of Motion | $$\frac{dv}{dt}=\frac{F}{m}$$ | Motion | +| Fourier's heat law | $$q=-kA\frac{dT}{dx}$$ | How heat is conducted through a material | +| Fick's law of diffusion | $$J=-D\frac{dc}{dx}$$ | Movement of particles from high to low concentration | +| Faraday's law | $$\Delta V_L = L \frac{di}{dt}$$ | Voltage drop across an inductor | + +In engineering ordinary differential equation's (ODE) are very common in the thermo-fluid science's, mechanics and control systems. By now you've solve many ODE's however probably not using numerical methods. Suppose we have an initial value problem of the form +$$ +\frac{dy}{dt}=f(t,y), \quad y(t_0)=y_0 +$$ +where $f(t,y)$ describes the rate of change of $y$ with respect to time $t$. + +[[3_6_Explicit_Methods]] +[[3_7_Implicit_Method]] +[[3_8_Systems_of_ODEs]] \ No newline at end of file -- cgit v1.2.3