A Robust Analytical Framework for Solving Nonlinear Differential Equations Arising in Physical Sciences

Abstract

In this study, we propose an effective analytical framework for solving nonlinear differential equations, termed the quasi-linearized Picard iteration method (QPIM). This method combines Newton's quasilinearization technique, which is known for its quadratic convergence, with the Picard iteration method, which offers a straightforward iterative framework. The proposed approach converts nonlinear differential equations into a sequence of linear problems via quasilinearization, which are then solved analytically using Picard iterations. A detailed convergence analysis confirms the quadratic rate of convergence under suitable conditions. Several examples are provided to illustrate the applicability and accuracy of the QPIM. Numerical comparisons revealed that QPIM achieved higher accuracy with fewer iterations. Furthermore, the QPIM provides monotonic sequences of approximations, enabling pointwise upper and lower bounds for solutions, which is especially useful for problems with unique solutions. These advantages establish the QPIM as a reliable tool for solving a broad class of nonlinear differential equations.