Computing solution landscape of nonlinear space-fractional problems via fast approximation algorithm

TL;DR

This paper introduces a fast approximation algorithm to compute the complex solution landscapes of nonlinear space-fractional problems, revealing how solution structures change with diffusion parameters.

Contribution

It develops a novel fast approximation method for variable-order spectral fractional Laplacian and integrates saddle dynamics to analyze solution landscapes of space-fractional phase field models.

Findings

The algorithm accurately captures stationary solutions efficiently.

Solution landscapes can be reconfigured by adjusting diffusion coefficients.

Numerical results validate the method's effectiveness and reveal key solution features.

Abstract

The nonlinear space-fractional problems often allow multiple stationary solutions, which can be much more complicated than the corresponding integer-order problems. In this paper, we systematically compute the solution landscapes of nonlinear constant/variable-order space-fractional problems. A fast approximation algorithm is developed to deal with the variable-order spectral fractional Laplacian by approximating the variable-indexing Fourier modes, and then combined with saddle dynamics to construct the solution landscape of variable-order space-fractional phase field model. Numerical experiments are performed to substantiate the accuracy and efficiency of fast approximation algorithm and elucidate essential features of the stationary solutions of space-fractional phase field model. Furthermore, we demonstrate that the solution landscapes of spectral fractional Laplacian problems can be reconfigured by varying the diffusion coefficients in the corresponding integer-order problems.