Exact Solutions to the Bernoulli and Riccati Equations with Conformable Derivatives: An Application to Liquid Flow in Reservoirs, Tanks, and Funnels

Abstract

This study presents an explicit representation of the solution for a linear conformable differential system with variable coefficients, utilizing the method of variation of constants combined with the state-transition approach. To tackle the exact solutions of nonlinear fractional Bernoulli-type and Riccati-type differential equations involving conformable derivatives-as well as separable fractional differential equations-these are skillfully transformed into an equivalent linear conformable system through appropriate variable substitutions. Theoretical results are further substantiated by detailed numerical and simulated examples. Moreover, the practical applicability of the proposed method is demonstrated through modeling liquid flow in engineering structures such as reservoirs, tanks, and funnels. © 2025 The Author(s).