Uniqueness Theorem and Midpoint Numerical Technique for Fractal-Fractional Differential Equations With Power Law Kernel

In this paper, we obtained conditions for the uniqueness of solution and a Crank–Nicholson–type algorithm for fractal-fractional differential equations with a power law kernel in Riemann–Liouville sense. For the uniqueness analysis, Kooi's conditions are imposed on the problem. The Kooi conditions involve a larger class of functions than the Lipschitz condition; hence, the uniqueness result applies to a wider class of problems. The key objectives of the numerical pursuit are to avoid repeated use of function evaluations, computation of special functions, and unnecessary grid restrictions, while achieving truly high-order accuracy and ease of implementation. All existing methods are based on an integral representation; hence, they cannot avoid all the challenges listed above. In this work, we base our formulation on the fact that the integral in the fractal-fractional operator is differentiable; hence, we collocate the original problem at the midpoint between grid points. Then, we apply linear Lagrange interpolation to derive a very easy-to-implement algorithm. We propose a discrete Gronwall lemma and then use it to prove several theorems for consistency, stability, and convergence. Finally, four numerical experiments are used to demonstrate the second-order accuracy and practicability of the method.