A universal solution scheme for fractional and classical PDEs

by   Yixuan Wu, et al.

We propose a unified meshless method to solve classical and fractional PDE problems with (-Δ)^α/2 for α∈ (0, 2]. The classical (α = 2) and fractional (α < 2) Laplacians, one local and the other nonlocal, have distinct properties. Therefore, their numerical methods and computer implementations are usually incompatible. We notice that for any α≥ 0, the Laplacian (-Δ)^α/2 of generalized inverse multiquadric (GIMQ) functions can be analytically written by the Gauss hypergeometric function, and thus propose a GIMQ-based method. Our method unifies the discretization of classical and fractional Laplacians and also bypasses numerical approximation to the hypersingular integral of fractional Laplacian. These two merits distinguish our method from other existing methods for the fractional Laplacian. Extensive numerical experiments are carried out to test the performance of our method. Compared to other methods, our method can achieve high accuracy with fewer number of unknowns, which effectively reduces the storage and computational requirements in simulations of fractional PDEs. Moreover, the meshfree nature makes it free of geometric constraints and enables simple implementation for any dimension d ≥ 1. Additionally, two approaches of selecting shape parameters, including condition number-indicated method and random-perturbed method, are studied to avoid the ill-conditioning issues when large number of points.



page 18

page 19

page 20

page 21


A unified meshfree pseudospectral method for solving both classical and fractional PDEs

In this paper, we propose a meshfree method based on the Gaussian radial...

Fast Fourier-like Mapped Chebyshev Spectral-Galerkin Methods for PDEs with Integral Fractional Laplacian in Unbounded Domains

In this paper, we propose a fast spectral-Galerkin method for solving PD...

Highly accurate operator factorization methods for the integral fractional Laplacian and its generalization

In this paper, we propose a new class of operator factorization methods ...

nPINNs: nonlocal Physics-Informed Neural Networks for a parametrized nonlocal universal Laplacian operator. Algorithms and Applications

Physics-informed neural networks (PINNs) are effective in solving invers...

A Direct Sampling Method for the Inversion of the Radon Transform

We propose a novel direct sampling method (DSM) for the effective and st...

Efficient Monte Carlo Method for Integral Fractional Laplacian in Multiple Dimensions

In this paper, we develop a Monte Carlo method for solving PDEs involvin...

Numerical Approximation of the Fractional Laplacian on R Using Orthogonal Families

In this paper, using well-known complex variable techniques, we compute ...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.