Rational approximation to the fractional Laplacian operator in reaction-diffusion problems

TL;DR

A new numerical strategy to solve fractional in space reaction-diffusion equations on bounded domains under homogeneous Dirichlet boundary conditions using the matrix transform method, based on the approximation of this matrix by the product of two suitable banded matrices.

Abstract

This paper provides a new numerical strategy to solve fractional in space reaction-diffusion equations on bounded domains under homogeneous Dirichlet boundary conditions. Using the matrix transform method the fractional Laplacian operator is replaced by a matrix which, in general, is dense. The approach here presented is based on the approximation of this matrix by the product of two suitable banded matrices. This leads to a semi-linear initial value problem in which the matrices involved are sparse. Numerical results are presented to verify the effectiveness of the proposed solution strategy.

Figures (6)

• Figure 1: Example of function $p_{\tau }(\lambda )$ for $\lambda _{\min }=0.5$, $\lambda _{\max }=4.$ The choice of $\tau$ as in (\ref{['tauopt']}) ensures the symmetry of the set $Q_{\tau }.$ The minimum distance of the curve $p_{\tau }(\lambda )$ from the set $[-1,1]$ is given by $\gamma -1$ and is attained in either $\lambda =\lambda _{\min }$ or $\lambda =\lambda _{\max }.$
• Figure 2: Relative error of the rational approximation versus $k,$ the number of points of the Gauss-Jacobi rule, for some values of $\alpha.$ The one- and the two-dimensional cases are on the left and on the right, respectively. In the first case the dimension of the problem is $200$ and in the second one it is $400.$
• Figure 3: Comparison of the analytic solution of the problem in Example \ref{['es:Es1']} with the numerical solutions provided by the rational approach and the matrix transfer technique at $t=0.4$ (left) and corresponding errors (right).
• Figure 4: Comparison of the numerical solutions of the problem in Example \ref{['es:Es2']} provided by the MT and rational approaches at $t=0.3$ (left) and corresponding step-by-step maximum norm of their difference (right) for $\alpha =1.1$ (top) and $\alpha =1.9$ (bottom), respectively.
• Figure 5: Comparison of the errors provided by solving the problem of the Example \ref{['es:Es3']} using both rational with $k=1$ (blue dashed-dot-line), $k=3$ (red dashed-line) and $k=5$ (black dot-line) and MT (green solid-line).

Theorems & Definitions (11)

• Proposition 1
• Theorem 1
• Theorem 2
• proof
• Corollary 1
• proof
• Remark 1
• Example 1
• Example 2
• Example 3