The use of fractional integro-differential models has gained prominence in describing complex dynamical systems with memory and hereditary properties. A set of nonlinear descriptor systems with multi-fractional integro-differential operators is studied. The mathematical model uses the Caputo fractional derivative and Riemann-Liouville fractional integral to model the underlying dynamics. A Homotopy Perturbation Method (HPM)-based analytical framework is formulated to obtain approximate solutions to the proposed system. A number of illustrative examples are given to demonstrate the method’s efficiency and convergence behavior and to attest to its ability to serve as an efficient analytical tool for studying nonlinear multi-fractional dynamical models arising in applied mathematics and other fields. Secondly, the paper proposes a new statistical distribution that is based on the sine transformation framework. The proposed model extends the Alpha Leg-Exponential (ALE) distribution to construct a new composite distribution, called the Sine-Alpha Leg-Exponential Distribution (SALED). Significant distributional functions are calculated and discussed, such as probability density function, cumulative distribution function, survival function, and hazard rate function. Parameter estimation is performed via simulation experiments to assess the model’s flexibility and performance. The results show that the SALED distribution offers a competitive alternative to available sine-based distributions with a similar parameter structure.