An Innovative Laplace-Based Hybrid Method for the Helmholtz PDE: Application of the Daftardar–Jafari Iteration

In this paper, a new hybrid framework is proposed by combining the Laplace Transform with the Daftardar–Jafari Method for solving the Helmholtz differential equation, which is a partial differential equation widely used in modeling wave-related phenomena and is typically defined under a set of initial conditions. The proposed technique has demonstrated strong effectiveness in producing accurate and stable solutions. It has been successfully applied to several practical models arising in physics and engineering mathematics. The results confirm the efficiency and reliability of this hybrid approach in handling various types of differential equations with complex structures.