English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We propose a unified meshless method to solve classical and fractional PDEproblems with $(-\Delta)^{\frac{\alpha}{2}}$ for $\alpha \in (0, 2]$. Theclassical ($\alpha = 2$) and fractional ($\alpha < 2$) Laplacians, one localand the other nonlocal, have distinct properties. Therefore, their numericalmethods and computer implementations are usually incompatible. We notice thatfor any $\alpha \ge 0$, the Laplacian $(-\Delta)^{\frac{\alpha}{2}}$ ofgeneralized inverse multiquadric (GIMQ) functions can be analytically writtenby the Gauss hypergeometric function, and thus propose a GIMQ-based method. Ourmethod unifies the discretization of classical and fractional Laplacians andalso bypasses numerical approximation to the hypersingular integral offractional Laplacian. These two merits distinguish our method from otherexisting methods for the fractional Laplacian. Extensive numerical experimentsare carried out to test the performance of our method. Compared to othermethods, our method can achieve high accuracy with fewer number of unknowns,which effectively reduces the storage and computational requirements insimulations of fractional PDEs. Moreover, the meshfree nature makes it free ofgeometric constraints and enables simple implementation for any dimension $d\ge 1$. Additionally, two approaches of selecting shape parameters, includingcondition number-indicated method and random-perturbed method, are studied toavoid the ill-conditioning issues when large number of points.

A universal solution scheme for fractional and classical PDEs