Numerical methods for fractional partial differential equations

C Li, A Chen - International Journal of Computer Mathematics, 2018‏ - Taylor & Francis
In this review paper, we are mainly concerned with the finite difference methods, the
Galerkin finite element methods, and the spectral methods for fractional partial differential …

[کتاب][B] Theory and numerical approximations of fractional integrals and derivatives

C Li, M Cai - 2019‏ - SIAM
Fractional calculus, which has two main features—singularity and nonlocality from its origin—
means integration and differentiation of any positive real order or even complex order. It has …

Numerical approaches to fractional integrals and derivatives: a review

M Cai, C Li - Mathematics, 2020‏ - mdpi.com
Fractional calculus, albeit a synonym of fractional integrals and derivatives which have two
main characteristics—singularity and nonlocality—has attracted increasing interest due to its …

Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations

D Baleanu, GC Wu, SD Zeng - Chaos, Solitons & Fractals, 2017‏ - Elsevier
This paper investigates chaotic behavior and stability of fractional differential equations
within a new generalized Caputo derivative. A semi–analytical method is proposed based …

Analysis and approximation of a fractional Cahn--Hilliard equation

M Ainsworth, Z Mao - SIAM Journal on Numerical Analysis, 2017‏ - SIAM
We derive a fractional Cahn--Hilliard equation (FCHE) by considering a gradient flow in the
negative order Sobolev space H^-α, α∈0,1, where the choice α=1 corresponds to the …

Analysis and application of new fractional Adams–Bashforth scheme with Caputo–Fabrizio derivative

KM Owolabi, A Atangana - Chaos, Solitons & Fractals, 2017‏ - Elsevier
Recently a new fractional differentiation was introduced to get rid of the singularity in the
Riemann-Liouville and Caputo fractional derivative. The new fractional derivative has then …

A novel high order space-time spectral method for the time fractional Fokker--Planck equation

M Zheng, F Liu, I Turner, V Anh - SIAM Journal on Scientific Computing, 2015‏ - SIAM
The fractional Fokker--Planck equation is an important physical model for simulating
anomalous diffusions with external forces. Because of the nonlocal property of the fractional …

[HTML][HTML] Numerical solution of the time fractional Black–Scholes model governing European options

H Zhang, F Liu, I Turner, Q Yang - Computers & Mathematics with …, 2016‏ - Elsevier
When considering the price change of the underlying fractal transmission system, a
fractional Black–Scholes (BS) model with an α-order time fractional derivative is derived. In …

A two-grid mixed finite element method for a nonlinear fourth-order reaction–diffusion problem with time-fractional derivative

Y Liu, Y Du, H Li, J Li, S He - Computers & Mathematics with Applications, 2015‏ - Elsevier
In this article, we develop a two-grid algorithm based on the mixed finite element (MFE)
method for a nonlinear fourth-order reaction–diffusion equation with the time-fractional …

A fourth-order approximation of fractional derivatives with its applications

Z Hao, Z Sun, W Cao - Journal of Computational Physics, 2015‏ - Elsevier
A new fourth-order difference approximation is derived for the space fractional derivatives by
using the weighted average of the shifted Grünwald formulae combining the compact …