In the present paper, we employ the generalized Mittag-Leffler function to investigate several fuzzy differential subordination results associated with suitable families of admissible functions in the open unit disk. By utilizing a refined analytic framework, we derive new inclusion relationships and establish sufficient conditions for fuzzy subordinating functions defined via Mittag-Leffler-type operators. Furthermore, the obtained results unify and extend a number of earlier findings in the theory of fuzzy analytic functions. These developments provide a deeper insight into the interaction between generalized special functions and the structure of fuzzy differential subordinations, offering potential applications to broader subclasses of analytic and bi-univalent functions, as well as to various operator-defined families in geometric function theory.

In this study, we proposed a hybrid methodology combining Physics-Informed Neural Networks (PINNs) and Immersed Finite Element (IFE) methods to address transmission problems in complex geometries, with a focus on Helmholtz-type equations. The technique addressed the challenge of solving wave equations in domains with circular interfaces, where material properties differ across the interface. The hybrid model leverages the strengths of PINNs to enforce the governing physical equations and IFE to provide a coarse initial solution, which is then corrected by the neural network using a signed-distance function to the interface. This correction was trained on a combination of supervised loss from data, physics-informed residual predictions, and interface conditions. Numerical experiments demonstrated high precision of the proposed technique when compared with manufactured exact solutions, achieving low error levels in both subdomains. High-frequency tests at $\omega =100$ further validated the method's accuracy and robustness across material configurations including normal and inverted contrast cases. Its mesh-free character provides flexibility and versatility for a wide range of transmission problems in computational physics, making it a promising method for solving interface-related issues in wave propagation.

The purpose of this paper is to define and prove that the Riemann–Liouville and Caputo fractional derivatives can be computed for tempered distributions, such that the fractional derivative of a tempered distribution remains a tempered distribution. The fact that the Fourier transform operator is an isomorphism in the dual of the Schwartz space is used, and we found that the fractional Riemann–Liouville and Caputo derivatives can be written as a Fourier transform composition and inverse. In this way, we are able to generalize both fractional Riemann–Liouville and Caputo derivatives for the tempered distributions. Moreover, certain examples of fractional derivatives for some tempered distributions are provided, such as the distribution of Dirac, the distribution of Heaviside, and the distribution of principal value.