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Fractional Laplacian in bounded domains
1Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA. andrea.zoia@polimi.it
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 13, 2007
Summary
We present a numerical method for the fractional Laplacian operator, essential for modeling physical systems like Lévy flights. This approach effectively handles boundary conditions on finite intervals, enabling eigenvalue calculations.
Area of Science:
- Mathematical Physics
- Numerical Analysis
- Fractional Calculus
Background:
- The fractional Laplacian operator, denoted -(-delta)(alpha/2), is crucial in diverse physical phenomena such as Lévy flights and stochastic interfaces.
- Existing methods face challenges in implementing boundary conditions for this operator on finite domains.
Purpose of the Study:
- To develop a discretized fractional Laplacian operator suitable for finite intervals with boundary conditions.
- To numerically compute eigenvalues and eigenfunctions for this operator under various boundary conditions.
- To provide analytical insights into the eigenvalue spectrum.
Main Methods:
- Discretization of the fractional Laplacian operator.
- Implementation of boundary conditions inspired by hopping particle and elastic spring models.
- Numerical computation of eigenvalues and eigenfunctions.
- Analytical investigation of eigenvalue spectrum structure.
Main Results:
- A well-suited discretized fractional Laplacian operator for finite intervals with boundary conditions.
- Numerical determination of eigenvalues and eigenfunctions for different boundary conditions.
- Derivation of analytical results on the eigenvalue spectrum.
Conclusions:
- The proposed discretization effectively handles boundary conditions for the fractional Laplacian operator.
- The numerical and analytical results offer valuable insights into the behavior of this operator in bounded domains.
- This work provides a foundation for further research in physical systems described by fractional operators.
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