## Unraveling Complex Dynamics: A Novel Technique for Nonlinear Fractional Partial Differential Equations### IntroductionFractional calculus has gained prominence in modeling and analyzing complex systems with memory and hereditary characteristics. This mathematical tool extends classical calculus to non-integer orders, enhancing the accuracy and comprehensiveness of modeling.### A Novel Approach: The Elzaki Residual Approach (ERA)To address nonlinear fractional differential equations (NFPDEs), this study introduces ERA, which combines the Elzaki transform (E-T), modified fractional power series (FPS), and the residual function. ERA employs a two-step iterative process that first transforms the given equation into the E-T space. Then, it utilizes the novel FPS to represent the solution in E-T space. The coefficients of this expansion are determined using residual functions and the limit principle.### Convergence AnalysisA novel convergence criterion is established for the revised fractional power series. This criterion ensures the convergence of the series solution under specific conditions.### Applications of ERAThe effectiveness of ERA is demonstrated by solving five diverse nonlinear problems:1. Nonlinear fractional gas dynamics equation2. Nonlinear fractional Fokker-Planck equation3. Nonlinear fractional Swift-Hohenberg equation4. Nonlinear time-fractional fractional gas dynamics equation5. Nonlinear time-fractional fractional Fokker-Planck equation### Numerical and Graphical AnalysisThe accuracy of ERA is evaluated using three error measurements: absolute error, relative error, and residual error. Numerical and graphical results confirm the high precision and effectiveness of ERA. The App-Ss obtained from ERA rapidly approach the Ex-Ss as the fractional order approaches 1.0.### ConclusionERA presents a straightforward and accurate approach for solving nonlinear fractional models. It offers several advantages:1. It can be applied to both weakly and strongly nonlinear problems.2. It does not require any assumptions or parameters in the equation.3. It can solve nonlinear problems without relying on polynomials like Adomian and He, enhancing its superiority over other methods.4. It employs the limit principle at zero to determine coefficients, avoiding the challenges faced by integration required in other techniques.### Future DirectionsThe authors plan to expand the applications of ERA to nonlinear fractional models in biological systems and engineering domains. They also intend to explore its application in solving equations involving the conformable fractional derivative.