A Finite Difference Scheme for the Fractional Laplacian on Non-uniform Grids

In this study, we analyze the convergence of the finite difference method on non-uniform grids and provide examples to demonstrate its effectiveness in approximating fractional differential equations involving the fractional Laplacian. By utilizing non-uniform grids, it becomes possible to achieve higher accuracy and improved resolution in specific regions of interest. Overall, our findings indicate that finite difference approximation on non-uniform grids can serve as a dependable and efficient tool for approximating fractional Laplacians across a diverse array of applications.