Exploring new routes in fractional modeling: analytical solutions of Burgers-type systems via Caputo-Hadamard and ϕ-Caputo derivatives

The study of fractional-order partial differential equations has gained significant attention due to its ability to model complex physical systems with memory effects and hereditary properties. Among these systems is the Burgers equation, which serves as a fundamental model for describing nonlinear waves, turbulence, and the behavior of viscous fluids. Recent advancements in fractional calculus have led to the development of generalized fractional operators, such as the Caputo-Hadamard and ϕ-Caputo fractional derivatives. These operators offer greater flexibility and precision in capturing temporal and spatial nonlocal effects. This paper provides a comprehensive review of analytical and semi-analytical methods for solving these systems, with a particular emphasis on the Residual Power Series Method (RPSM) and the New Iterative Method (NIM). Both methods demonstrate superior convergence rates and accuracy. By combining generalized fractional operators with iterative algorithms, new approaches can be developed to model real-world phenomena in fields like physics, engineering, and applied mathematics. This work not only summarizes recent progress but also establishes a strong foundation for future research into complex fractional-order systems.