Solving 2D distributed-order Riesz space-fractional PDEs in complex geometries using a cubic spline and meshless method with non-singular kernels

Abstract

This paper introduces an efficient and fully discrete numerical scheme for solving two-dimensional time-space fractional partial differential equations (PDEs) of distributed order, incorporating Riesz space-fractional operators. The temporal fractional derivative (FD) is modeled using the distributed-order (DO) Caputo definition, which effectively captures the intricate memory effects typical of multi-scale diffusion processes. To approximate the distributed time-fractional integral with higher accuracy, a cubic spline-based high-order discretization is employed, enhancing temporal resolution. For the spatial component, a meshless approach is developed using radial basis functions (RBFs) in combination with moving least squares (MLS) techniques. This method is well-suited for handling Riesz FDs across both standard and complex irregular domains, including shapes like butterfly, six-petal flower, heart, and rectangle. The proposed scheme integrates both temporal and spatial discretizations into a fully discrete framework that offers high accuracy and adaptability. A detailed stability analysis carried out using the discrete energy method confirms that the scheme is unconditionally stable. Convergence analysis further reveals a temporal accuracy order nearing 1.95 and a spatial convergence rate of approximately 1.6 in the fractional Sobolev norm. Numerical experiments using exact solutions support the theoretical findings, demonstrating strong agreement between numerical and analytical results. Computational performance is also assessed, with error norms and CPU times reported to highlight the method's scalability. This work provides a solid groundwork for future developments, including adaptive time-stepping strategies, three-dimensional extensions, and large-scale parallel implementations for DO fractional diffusion problems.