where $b_j = (j+1)^1-\alpha - (j)^1-\alpha$.
For spatial fractional derivatives (e.g., fractional Laplacian $(-\Delta)^\alpha/2$), spectral methods using Jacobi polynomials or Fourier expansions offer exponential convergence for smooth solutions. The fractional Laplacian in Fourier space is simply multiplication by $|\xi|^\alpha$, making spectral methods extremely efficient in periodic domains. For non-periodic problems, hierarchical matrices ($\mathcalH$-matrices) can approximate the dense stiffness matrices with $\mathcalO(N \log N)$ storage and operations. where $b_j = (j+1)^1-\alpha - (j)^1-\alpha$
Beyond RL and Caputo, several other fractional operators have been developed: requires the entire history of the function from
This elegant expression reduces an $n$-fold integral to a single convolution integral. The key insight for fractional calculus is to replace the factorial with the Gamma function $\Gamma(\alpha)$, which generalizes the factorial to real and complex numbers ($\Gamma(\alpha) = (\alpha-1)!$ for $\alpha \in \mathbbN$). this becomes prohibitive.
requires the entire history of the function from the starting point
, meaning they depend on the function's values over an entire interval rather than just at a single point. Two primary definitions dominate the field: The Riemann-Liouville (RL) Operator Riemann-Liouville fractional integral is defined as:
This first-order method is simple but has two major drawbacks: (i) its convergence rate is only $\mathcalO(h)$, and (ii) the computational cost at time step $n$ is $\mathcalO(n)$, leading to an overall $\mathcalO(N^2)$ complexity for $N$ steps. For large $N$ (e.g., long-time simulations), this becomes prohibitive.