A Hybrid Yang Transform and El-Kalla Polynomials Approach for Efficient Solutions of Nonlinear Differential Equations

Abstract

Nonlinear differential equations are central to mathematical modeling in science and engineering, but their nonlinear structure can make exact analytical solutions difficult to obtain. This study develops the Yang Transform–El-Kalla Polynomial (YT-EKP) method, a hybrid analytical–numerical approach for constructing approximate series solutions. The method combines the Yang transform with recursive El-Kalla polynomial decomposition to treat nonlinear terms while avoiding repeated differentiation of nonlinear operators during polynomial construction. The mathematical framework is formulated, recursive relations are derived, and the resulting algorithm is implemented using computational software. Error bounds are incorporated to support the assessment of approximate solutions. Benchmark applications include a first-order nonlinear differential equation, a Riccati equation, and the Duffing oscillator. Approximation accuracy, convergence behavior, and computational performance are examined through comparisons with the Kamal–Adomian Polynomial method and the Transform Adomian Method. The numerical results yield absolute errors ranging from 10⁻¹⁶ to 10⁻⁶ across the tested problems. Comparative analysis indicates faster convergence and reduced computational complexity relative to the methods considered. These findings suggest that integrating the Yang transform with El-Kalla polynomial recursion can produce accurate approximations while simplifying the treatment of nonlinear contributions. The benchmark results support the potential of YT-EKP as an effective approach for the equations examined and highlight the usefulness of derivative-free polynomial construction within a transform-based solution framework. Further evaluation should establish whether these advantages persist across broader classes of nonlinear equations and different parameter settings. Extensions to fractional and delay differential equations represent directions for future investigation. Overall, the study contributes a hybrid framework for obtaining series approximations, with demonstrated accuracy in the reported applications and opportunities for further assessment of its computational efficiency and generalizability.