By: A Fellow Math Enthusiast 

Target Audience: Control Systems Engineers, Material Scientists 

Introduction: In Between Dimensions 

You know the first derivative (velocity) and the second derivative (acceleration). But what is the half-derivative? 

Fractional Calculus allows us to take the $0.5^{th}$ derivative. This is not just a math trick; it is the best way to model “memory” in materials, like viscoelastic polymers or batteries, where the current state depends on the entire history of the system. 

The Syllabus Map 

Your Bachelor’s Subject  The Implementation 
Calculus II  The Gamma Function (Factorials for fractions) 
Complex Analysis  Cauchy Integral Formula (The definition) 
Transform Calculus  Laplace Transforms (Solving the equations) 
Diff Eqs  Convolution Integrals (Memory effects) 
Series & Sequences  Power Series (Mittag-Leffler Function) 

The Breakdown 

  • Calculus II (The Gamma Function) 
    1. The Component: Generalizing the factorial. 
    2. The Math: The power rule \frac{d^n}{dx^n} x^k involves factorials (k!). To do this for non-integers, we replace the factorial with the Gamma Function \Gamma(z). 
    3. This allows us to calculate things like \frac{d^{0.5}}{dx^{0.5}} x. 
  • Diff Eqs (Convolution) 
    1. The Component: Memory. 
    2. The Math: Standard derivatives are “local” (slope right now). Fractional derivatives are “non-local.” They are defined by an integral (Riemann-Liouville definition) that sums up the function’s behavior from time t=0 to t=now. This is technically a Convolution, a concept you likely used in Signals and Systems. 
  • Transform Calculus (Laplace) 
    1. The Component: Solving the system. 
    2. The Math: Solving fractional differential equations in the time domain is hard. But in the Laplace Domain (s-domain), the fractional derivative $D^\alpha simply becomes multiplication by s^\alpha. You solve it algebraically, just like you did in Control Theory 101.