Nonlinear phenomena appear everywhere in our daily life and our scientific works, and today nonlinear science represents one of the most challenging, promising, and romantic fields of research in science and technology Citation1 Citation4 Citation5. It was very difficult to solve nonlinear problems effectively either numerically or analytically, and even more difficult to establish models for real-world nonlinear problems. Many assumptions had to be made artificially or unnecessarily to make practical engineering problems solvable, leading to loss of important information. Now things are changing, for we have some new mathematical tools and mathematical software, which increase tantalizingly the possibility of analytically seeking the approximate solutions to more complex problems and revealing various features of series of obtained solutions. The analytical approaches make it possible to achieve a realistic, eventually utilitarian, representation of an actual nonlinear system that connects the experimentally significant behaviour with its nonlinear properties.
One of our chief aims with this collection of papers is to establish more reasonable nonlinear models for practical engineering problems using useful mathematical concepts, such as fractional derivation (see papers by Momani and Yıldırım; and Yıldırım and Momani), optimal control (see Yousefi et al.’s paper), optimal design (Geng and Cui's paper), soliton (see papers by Taşcan and Bekir; Chiu and Chow; and Triki and Wazwaz), symbolic algorithm (see Baldwin and Hereman's paper), integral and integro-differential equation (see papers by Wazwaz; Aminikhah and Biazar; and Aminikhah and Salahi); particular interest is placed on contributions concerned with the application of new analytical methods (e.g. the variational iteration method and the homotopy perturbation method) to real-life problems to reveal hidden pearls in nonlinear phenomena.
There are many results on the exact solutions of nonlinear equations where the initial or boundary conditions are not considered; these solutions are called mathematical solutions (trial solutions) because the physical constraints on the real-world problem that is being modelled are not accounted for; however, this section focuses on approximate solutions of nonlinear equations.
In Citation2, the variational iteration method is used to solve the solution and determine the initial condition simultaneously. This section has access to the basic concept and last development of the variational iteration method (see papers by Yousefi et al.; Sweilam and Khader; and Wazwaz). Optimal initial/boundary identification can be used in the optimal variational iteration algorithm proposed by Herişanu and Marinca Citation3. In this special issue, Geng and Cui suggest a multi-point boundary value problem which is an ill-posed problem in view of mathematics and admits no solution. Actually, the solution does exist, which is obtained using the variational iteration method.
This special section conveys a strong, reliable, efficient, and promising development of nonlinear science. Included herein is a collection of original refereed research papers by well-established researchers in the field of nonlinear science. We hope that these papers will prove to be a timely and valuable reference for researchers in this area.
Finally, we would like to express our appreciation to all reviewers who took the time to review articles in a very short time.
References
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- He, J.-H. , Wu, G. C. , and Austin, F. , 2010. The variational iteration method which should be followed , Nonlinear Sci. Lett. A 1 (2010), pp. 1–30.
- Herişanu, N. , and Marinca, V. , 2010. A modified variational iteration method for strongly nonlinear problems , Nonlinear Sci. Lett. A 1 (2010), pp. 183–192.
- Krumhansl, J. A. , 1993. Nonlinear science: Toward the next frontiers , Phys. D 68 (1993), pp. 97–103.
- Xu, L. , He, J.-H. , and Wazwaz, A. M. , 2007. Variational iteration method – reality, potential, and challenges , J. Comput. Appl. Math. 207 (2007), pp. 1–2.