A Hybrid Yang Transform–Adomian Polynomial Method for Nonlinear Partial Differential Equations

Abstract

Nonlinear partial differential equations (PDEs) require solution techniques that handle nonlinear terms while maintaining accuracy and practical implementability. This study presents the Yang Transform–Adomian Polynomial (YTAP) method, a hybrid numerical approach combining the Yang transform with Adomian polynomials to approximate solutions of nonlinear differential equations. The Yang transform reformulates the governing equations in the transform domain, while Adomian polynomials systematically represent nonlinear terms. Integrating these components provides a framework for constructing approximate solutions and examining their accuracy through benchmark applications. Three benchmark problems were solved to evaluate the proposed method, including linear PDEs and a nonlinear ordinary differential equation (ODE). Performance was assessed through reported approximation errors and comparison with the Kamal method. The results showed that YTAP achieved errors as low as 10⁻¹⁷ for linear PDEs. For the nonlinear ODE comparison at t = 1, the approximation error was 2.45 × 10⁻³ using YTAP, compared with 9.07 × 10⁻² using the Kamal method. These results indicate high accuracy for the tested linear problems and a lower approximation error for the nonlinear benchmark. The applications also indicated rapid convergence and straightforward implementation, supporting the potential of combining integral transformation with polynomial treatment of nonlinearities. The findings establish the method’s performance for the examples considered and provide a basis for further investigation of its applicability to nonlinear PDEs. Additional benchmarking should examine convergence behavior, computational cost, and robustness across a broader range of equations and parameter settings. Overall, YTAP offers a promising framework for obtaining accurate approximate solutions, with its wider computational advantages requiring evaluation beyond the reported benchmark cases.