Recent development of the homotopy perturbation method

The homotopy perturbation method is extremely accessible to non-mathematicians and engineers. The method decomposes a complex problem under study into a series of simple problems that are easy to be solved. This note gives an elementary introduction to the basic solution procedure of the homotopy perturbation method. Particular attention is paid to constructing a suitable homotopy equation. This special issue on “the homotopy perturbation method and its application” of Topological Methods in Nonlinear Analysis consists mainly of a collection of recently obtained results and various new interpretations of earlier conclusions pertinent to the application of the homotopy perturbation method for real-life nonlinear problems, ranging from advanced calculus to fractional calculus(see Momani and Odibat’s contributions), from peridic problems to solitary problems (see J. C. Lan, Ozis, Z. L. Tao’s papers), from biology to engineering applications (see L. Xu, Sadighi, Chowdhury’s papers), from stochastic system to scalar images (see El-Tawil and Q. Ma’s papers). The aim of this special issue is to bring to the fore the many new and exciting applications of the homotopy perturbation iteration method, thereby capturing both the interest and imagination of the wider communities in various fields, such as in mathematics, physics, information science, computational science, biologics, medicine, and others. 2000 Mathematics Subject Classification. 35B20, 35B25, 35A15.