What is Fractional Calculus?
Fractional calculus is a generalised form of the integer-order calculus. While an integer-order derivative is a local operator, a fractional derivative is a non-local operator. The notion of Brownian motion is extended to admit Levy stable processes in the case of fractional diffusion. Many different operators have been described as fractional derivatives and integrals, with different properties and behaviours, and there are also further generalisations within non-local calculus.
Real-world applications of non-local models can be found in turbulence, viscoelasticity, fracture mechanics, economics, electrical circuits, and plasma physics. Until recently, the theory and applications of fractional operators and equations did not receive much attention, so that many fundamental questions remain unanswered.
Introduction to Fractional Calculus
Development of Vector Calculus for two-sided operators of variable-order
General Fractional Calculus on finite interval
Introduced, pointwise Fractional Physics-informed Neural Networks for inverse problems
Introduced Jacobi convolution series, a new class of basis functions for Spectral (and element) methods
Introduced, General Fractional Physics-informed Neural Networks (GFPINNs)
Derived 3D Stream-function vorticity equations
Derived Time-averaged equation of motion for small particles - valid for turbulent flows