Utilizing differential quadrature-based RBF partition of unity collocation method to simulate distributed-order time fractional Cable equation

The field of distributed-order fractional calculus is rapidly expanding, and it has significant applications in the design of complex systems. In this work, we propose a novel approach to simulate the non-homogeneous two-dimensional distributed-order time-fractional Cable equation. Our proposed approach involves using the RBF partition of unity collocation method, which is based on the differential quadrature technique, in conjunction with the Composite M-point Gauss–Legendre quadrature rule. To obtain the numerical solution, we adopt a Crank–Nicolson type discretization for the temporal variable. The unconditionally stability and second-order rate of convergence of the time-discrete technique are demonstrated. We demonstrate the flexibility and effectiveness of our approach by applying it to several examples with varying geometric domains, including regular domains with Halton points and irregular domains, in both one- and two-dimensional scenarios. Through these examples, we showcase the capability of our proposed method to handle a wide range of problems with different geometric configurations.