[Paper Review] Numerical solutions to boundary value problem for anomalous diffusion equation with Riesz-Feller fractional operator

Mariusz Ciesielski, Jacek Leszczyński|arXiv (Cornell University)|Jul 5, 2006

Fractional Differential Equations Solutions8 references28 citations

TL;DR

This paper presents a fractional finite difference method (FFDM) for solving a boundary value problem involving the Riesz-Feller fractional derivative in one-dimensional space, enabling numerical simulation of anomalous diffusion. The method generalizes the classical finite difference scheme, with key results showing non-linear temperature profiles for α < 2 and accurate fitting of experimental nanotube thermal data using α = 0.35 and θ = -0.055.

ABSTRACT

In this paper, we present a numerical solution to an ordinary differential equation of a fractional order in one-dimensional space. The solution to this equation can describe a steady state of the process of anomalous diffusion. The process arises from interactions within complex and non-homogeneous background. We present a numerical method which is based on the finite differences method. We consider a boundary value problem (Dirichlet conditions) for an equation with the Riesz-Feller fractional derivative. In the final part of this paper, same simulation results are shown. We present an example of non-linear temperature profiles in nanotubes which can be approximated by a solution to the fractional differential equation.

Motivation & Objective

• To develop a numerical method for solving boundary value problems involving the Riesz-Feller fractional derivative in anomalous diffusion.
• To model steady-state temperature profiles in complex, non-homogeneous media such as nanotubes where standard diffusion fails.
• To extend classical finite difference methods to fractional-order derivatives that capture long-range spatial dependence and heavy-tailed particle jumps.
• To validate the method by comparing numerical solutions with experimental data from nanotubes.
• To investigate the influence of the skewness parameter θ and order α on solution symmetry and shape.

Proposed method

• Uses the finite difference method (FDM) to discretize the Riesz-Feller fractional derivative operator in space.
• Derives a numerical approximation for the Riesz-Feller derivative using weighted sums of function values across the domain, with coefficients dependent on α and θ.
• Constructs a linear system A·T = B, where matrix A encodes the fractional derivative stencil and vector B incorporates Dirichlet boundary conditions.
• For α = 2 and θ = 0, the scheme reduces to the classical central difference method for the second derivative.
• The method accounts for non-local behavior: each point’s derivative depends on values across the entire domain, not just local neighbors.
• The solution is computed by solving the linear system, with boundary conditions influencing all interior nodes due to the non-local nature of fractional derivatives.

Experimental results

Research questions

• RQ1 How can the Riesz-Feller fractional derivative be accurately approximated in a boundary value problem using a finite difference scheme?
• RQ2 What is the effect of varying the fractional order α and skewness parameter θ on the shape of the steady-state temperature profile?
• RQ3 Can the proposed fractional finite difference method accurately reproduce experimental temperature profiles in nanotubes where standard diffusion fails?
• RQ4 How does the solution behavior change as α approaches 1 and θ approaches ±1, and what physical interpretation does this correspond to?
• RQ5 What is the relationship between the fractional derivative parameters and the emergence of long-tailed probability distributions in anomalous diffusion?

Key findings

• For α = 2 and θ = 0, the solution is linear, confirming consistency with the classical heat equation.
• For α < 2, the solution exhibits non-linear profiles, indicating anomalous diffusion behavior.
• As α → 1+ and θ → ±1+, the solution approaches the steady state of the first-order wave equation.
• The skewness parameter θ introduces asymmetry in the solution, with θ ∈ (0,1) producing non-symmetric profiles.
• The model with α = 0.35 and θ = -0.055 provides the best fit to experimental nanotube temperature data from Zhang and Li (2005).
• The fractional derivative captures long-tailed particle jumps, enabling modeling of rare but extreme events not possible with Gaussian statistics.