Pseudo-parameter Iteration Method (PIM): A semi-analytic solution procedure for nonlinear problems

•A new iterative method is proposed for integrating both nonlinear ordinary differential equations and partial differential equations.•It is a semi-analytical approach (or an analytic approximation) to nonlinear differential equations, called the pseudo-parameter iteration method (PIM), providing a powerful tool to analyze strongly nonlinear problems. We present an iterative semi-analytical method for solving not only nonlinear ordinary differential equations but partial differential equations. In this method, a system of nonlinear integral equations, equivalent to an original differential equation together with (initial and) boundary conditions, is constructed based on an artificially introduced (non-zero) parameter. The method finds a solution for the differential equation with the conditions by solving the integral equation system via the Banach contraction principle. The parameter, referred to as the pseudo-parameter, is a non-zero auxiliary (non-physical) parameter that provides a key to arrive at the integral equations from the differential equation. Further, the parameter can be viewed as a control parameter, which can control the performance of the method, e.g., its accuracy and convergence speed, etc. Especially, the present method, different from other nonlinear semi-analytic techniques such as the perturbation approach, does not depend on a small (perturbation) parameter, so that it can find a wide application in (strongly) nonlinear physical problems without a proper linearization strategy under small perturbations.