Numerical Approximation of Riesz-Feller Operators on $\mathbb R$
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866929209986777088 |
|---|---|
| author | Cuesta, Carlota M. de la Hoz, Francisco Girona, Ivan |
| author_facet | Cuesta, Carlota M. de la Hoz, Francisco Girona, Ivan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_07140 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Numerical Approximation of Riesz-Feller Operators on $\mathbb R$ Cuesta, Carlota M. de la Hoz, Francisco Girona, Ivan Numerical Analysis 26A33, 33C05, 65T50 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. |
| title | Numerical Approximation of Riesz-Feller Operators on $\mathbb R$ |
| topic | Numerical Analysis 26A33, 33C05, 65T50 |
| url | https://arxiv.org/abs/2401.07140 |