An analytic approximate solution for nonlinear drift-diffusion equation

Abstract The drift-diffusion equation is a fundamental model used for modeling phenomena in physics, finance, biology, and engineering. Its applications span particle dynamics, molecular diffusion, asset pricing, population dynamics, and noise control systems. The wide range of its applications motivated us to address the non-linear problem semi-analytically via two advanced schemes: the Accelerated Homotopy Perturbation Elzaki Transformation Method (AHPETM) and the Optimized Decomposition Method (ODM). AHPETM combines the accelerated homotopy perturbation method with the Elzaki transform, while ODM employs linear approximations and Adomian polynomials. Through diverse test cases in 1-, 2- and 3-dimensions, we demonstrate AHPETM’s superior accuracy, computational efficiency, and capability to provide closed-form solutions as compared to the ODM and the well-established semi-analytical schemes, i.e., variational iteration as well as Temimi and Ansari methods. Furthermore, rigorous convergence analysis and derivation of worst-case error bounds validate the reliability of the proposed methods, establishing AHPETM as a robust and versatile tool for tackling complex nonlinear systems