A Unified Yang Transform Framework for Linear and Nonlinear Partial Differential Equations of Integer and Fractional Order

Abstract

Fractional calculus supports mathematical descriptions of phenomena involving memory and hereditary effects in physics, engineering, and biology. However, nonlinear fractional partial differential equations remain difficult to solve because fractional differentiation and nonlinear terms introduce simultaneous analytical challenges. This study presents two hybrid approaches, the Yang Transform–Adomian Polynomial (YTAP) method and the Yang Transform–New Iterative Method (Yang-NIM), for constructing series solutions to linear and nonlinear partial differential equations of integer and fractional order. Both approaches combine the Yang transform with techniques for handling nonlinear contributions. YTAP employs Adomian polynomials, while Yang-NIM uses an iterative formulation that avoids their explicit construction. The methods were applied to benchmark problems involving fractional diffusion equations, nonlinear Fornberg–Whitham equations, space-fractional telegraph equations, and other nonlinear fractional partial differential equations. Comparisons with exact solutions indicated close agreement, with absolute errors on the order of 10⁻⁸ to 10⁻¹¹ for the benchmark problems examined. The applications also indicated rapid convergence. Yang-NIM offered a procedural advantage by eliminating explicit Adomian polynomial generation while retaining high approximation accuracy. The proposed framework accommodates both integer and non-integer differential orders, providing a consistent basis for investigating these equation classes through related solution procedures. Its contribution lies in integrating transform-based reformulation with alternative approaches to nonlinear approximation and demonstrating their applicability across the selected benchmarks. Further research should compare computational costs and convergence behavior across broader problem sets and fractional-order settings to establish the extent of these advantages. The study concludes that YTAP and Yang-NIM provide promising approaches for obtaining accurate series approximations, with potential relevance to mathematical models in which fractional dynamics represent memory-dependent behavior. Applications to specific physical or biological systems would require evaluation against the characteristics and requirements of those models.