Stability analysis of multi-spatial Riesz and multi-fractional non-linear damped wave equations involving Caputo-Fabrizio derivative

This research investigates a multi-fractional non-linear generalized diffusion wave equation involving a multi-spatial Riesz operator and a damping term, analyzed using the Caputo–Fabrizio fractional derivative in the Caputo sense. We employ the fractional Laplace operator in symbolic form, denoted by $$(-\Delta )^{\frac{\alpha }{2}}$$ , where the fractional order $$\alpha \in (0,2)$$ . The operator is widely used in dynamic systems. In bounded domains, it effectively handles boundary conditions. The Caputo-Fabrizio derivative is incorporated to capture the damping effects and explore the behavior of the non-linear generalized diffusion-wave equation. The outcomes concerning existence and uniqueness are obtained by employing fixed-point theorems. Furthermore, we examine the Hyers-Ulam stability of the proposed multi-fractional model, establishing that small perturbations in the system do not result in divergence. Moreover, this study improves the accuracy of model approximations for non-integer order systems. Multiple case studies are explored in the one- and two-dimensional spatial domains to support the theoretical findings. Visual representations illustrate how the error evolves over space and time, consistently showing that it stays within bounded limits. The 2D and 3D plots of the simulation results further demonstrate that the system remains stable in the presence of perturbations. Furthermore, the stability properties are thoroughly established over various time and space domains, underscoring the robustness and reliability of the proposed approach.