Fractional calculus provides us with a set of axioms and methods to extend the concept of a derivative operator from integer order n to arbitrary order α, where α is a real or complex value.

Fractional calculus is a challenging area in many aspects.
A significant aspect is complexity: in most cases fractional
calculus immediately leads to generalised views, requiring a broader understanding of advanced methods and tools. For example already the step from factorial to the gamma function demands substantial insight into non-trivial mathematical techniques.

As we delve deeper, we quickly encounter functions such as the Mittag-Leffler, Jonqui\`ere’s,
Hurwitz Zeta or Meijer-G function to name a few — many of which a bachelor’s, master’s, or even PhD physics student may have never heard of.

Moreover, we quickly face technical challenges. Integrals with weakly singular kernels often resist direct solutions.

For these reasons, it is crucial to gain a working knowledge of numerical methods which applied to our findings step by step increase our confidence in the correctness of our approach.
Numerics serve as a powerful, indispensable and often the only guide to validate or falsify assumptions promptly helping us to circumnavigate
the dangerous cliffs and shallows of fractional calculus.

Doing computer physics in the fields of fractional calculus presents a wide range of challenges:
We need methods to evaluate complex integrals, solve
fractional extensions of ordinary or partial differential equations, manage intricate algebraic expressions, and importantly, generate graphical representations of derived results.

Our primary goal is to raise awareness among scientists about the critical importance of sustainability and the responsible use of resources and energy. In scientific research and technology, we have become too accustomed to wasting hardware and resources. There is still much progress to be made in promoting practices like reusing components rather than prematurely discarding them.

The examples we provide share a common goal: to prove that, instead of relying on simple library calls, minimal intelligence and creativity can lead to smart solutions for problems – not only within the framework of fractional calculus, but also for broader application areas – while carefully conserving resources.

With that in mind, we hope you enjoy exploring the numerical challenges in fractional calculus.

GitGub repository:

https://github.com/gigahedron-book/fractionalCalculus.git