Numerical Approximation of Riesz-Feller Operators on $\mathbb R$

Abstract

In this paper, we develop an accurate pseudospectral method to approximate numerically the Riesz-Feller operator $D_γ^α$ on $\mathbb R$, where $α\in(0,2)$, and $|γ|\le\min\{α, 2 - α\}$. This operator can be written as a linear combination of the Weyl-Marchaud derivatives $\mathcal{D}^α$ and $\overline{\mathcal{D}^α}$, when $α\in(0,1)$, and of $\partial_x\mathcal{D}^{α-1}$ and $\partial_x\overline{\mathcal{D}^{α-1}}$, when $α\in(1,2)$. Given the so-called Higgins functions $λ_k(x) = ((ix-1)/(ix+1))^k$, where $k\in\mathbb Z$, we compute explicitly, using complex variable techniques, $\mathcal{D}^α[λ_k](x)$, $\overline{\mathcal{D}^α}[λ_k](x)$, $\partial_x\mathcal{D}^{α-1}[λ_k](x)$, $\partial_x\overline{\mathcal{D}^{α-1}}[λ_k](x)$ and $D_γ^α[λ_k](x)$, in terms of the Gaussian hypergeometric function ${}_2F_1$, and relate these results to previous ones for the fractional Laplacian. This enables us to approximate $\mathcal{D}^α[u](x)$, $\overline{\mathcal{D}^α}[u](x)$, $\partial_x\mathcal{D}^{α-1}[u](x)$, $\partial_x\overline{\mathcal{D}^{α-1}}[u](x)$ and $D_γ^α[u](x)$, for bounded continuous functions $u(x)$. Finally, we simulate a nonlinear Riesz-Feller fractional diffusion equation, characterized by having front propagating solutions whose speed grows exponentially in time.

Figures (4)

• Figure 1: An integration contour example, with the two poles $z = i - x$ and $z = x + i$, corresponding to a given $x>0$.
• Figure 2: Left: Errors in the numerical approximation of $D_\gamma^\alpha[\mathop{\mathrm{erf}}\nolimits](x)$ versus $L\in\{0.01, 0.02, \ldots, 10\}$, for $\alpha = 1.37$, $\gamma = 0.58$, $l_{lim} = 100$ and $N\in\{2^3, 2^4, \ldots, 2^{14}\}$. Right: Errors in the numerical approximation of $\partial_x\mathcal{D}^{\alpha-1}[\ln(1+x^2)]$ versus $L\in\{0.1, 0.2, \ldots, 1000\}$, for $\alpha = 1.37$, $l_{lim} = 100$ and $N\in\{2^3, 2^4, \ldots, 2^{14}\}.$
• Figure 3: Left: $x_0(t)$ versus $t$. Right: $1 - \rho$ (blue dots) versus $t$, where $\rho$ is the Pearson correlation coefficient, and $|\sigma - 1/\alpha|$ (black stars) versus $t$, where $\sigma$ is the slope of the corresponding straight line on the left-hand side, such that the speed $c(t)\sim e^{\sigma t}$. In both sides, $\alpha = 1.37$, $\gamma \in \{-0.63, -0.625, \ldots, 0.625, 0.63\}$, $L = 2100$, $l_{lim} = 100$.
• Figure 4: Evolution of \ref{['e:fisher']} for $\alpha = 1.37$, $\gamma = -0.63$, $t\in\{0, 0.5, \ldots, 21.5, 22\}$, $L = 2100$, $l_{lim} = 100$. Left: $x\in[-100, 100]$. Right: $x\in[1, 10^6]$, and $u(x, t)$ appears in semilogarithmic scale.

Theorems & Definitions (18)

• Lemma 1.1: Equivalent representations of $\mathcal{D}^{\alpha}$ and $\partial_x \mathcal{D}^{\alpha}$
• proof
• Lemma 1.2: Equivalent representations of $\overline{\mathcal{D}^{\alpha}}$ and $\partial_x \overline{\mathcal{D}^{\alpha}}$
• proof
• Theorem 2.1
• proof
• Corollary 2.1
• proof
• Corollary 2.2
• proof