A Novel Approach for Modeling Non-Local Systems with Scale-Dependent Fluctuations and Material Heterogeneities

This study makes a novel contribution to the field of fractional calculus through the introduction of the Sachit Hussein Jassim (SHJ) fractional derivative. This is a newly defined operator with a non-singular, smooth kernel that provides a unified two-sided formulation for temporal and spatial variables. Methodologically, the paper develops a rigorous mathematical formulation and theoretical foundation for the SHJ derivative, establishing proofs for existence, uniqueness, and transform compatibility. The analytical properties of the operator are examined through the use of Laplace, Sumudu, and Jafari transforms, thereby confirming its integration within established analytical techniques. In order to demonstrate its applicability, the Laplace–Adomian Decomposition Method (LADM) is employed to obtain approximate solutions for fractional partial differential equations incorporating the SHJ derivative. Examples of such equations include Burgers' equation and heat-like models. The numerical findings demonstrate that the SHJ derivative attains considerably higher accuracy and faster convergence in comparison to the Caputo–Fabrizio and Atangana–Baleanu formulations. Quantitative comparisons and graphical analyses reveal minimal numerical errors and enhanced stability across varying fractional orders, thereby emphasising the robustness of the new operator in modelling nonlocal and multiscale phenomena. The study concludes that the SHJ derivative constitutes a substantial advancement in the theory and application of fractional derivatives, providing a reliable analytical and computational tool for simulating nonlocal dynamics, material heterogeneities, and complex diffusion processes. The implications of this phenomenon extend to a variety of disciplines, including applied mathematics, physics, and engineering, where nonlocal modelling is imperative. It is suggested that future work should explore multidimensional extensions and numerical optimisation, with a view to broadening the potential impact across scientific and industrial domains.