Differential Equations

Overview

Differential equations are equations expressed in the language of calculus (usually as equations consisting of the derivatives of functions). They form the core of many modeling paradigms.

Sometimes differential equations are exactly solvable, meaning there is an analytic solution. When no such solution exists, we have to use numerical methods to calculate an approximation.

Topics

• Model Types
  • Discrete vs Continuous
• Types of Differential Equations
  • Ordinary Differential Equations (ODE)
  • Partial Differential Equations (PDE)
  • Stochastic Differential Equations
• Solution Methods
  • Finite Difference
  • Fourier Analysis : is a technique used in the solution of 2nd order differentional equations with a specified boundary condition. Fourier series can represent arbitrary (suitably smooth) function as a series of cosine and sine functions summed.
  • Greens Functions
• Equations
  • Heat Equation:
  • Wave Equation:
  • Navier Stokes: