Asymptotic Methods for Solitary Solutions and Compactons

By Ji-Huan He. 2012

262 citations recorded by the source corpus.

This paper is an elementary introduction to some new asymptotic methods for the search for the solitary solutions of nonlinear differential equations, nonlinear differential-difference equations, and nonlinear fractional differential equations. Particular attention is paid throughout the paper to giving an intuitive grasp for the variational approach, the Hamiltonian approach, the variational iteration method, the homotopy perturbation method, the parameter-expansion method, the Yang-Laplace transform, the Yang-Fourier transform, and ancient Chinese mathematics. Hamilton principle and variational principles are also emphasized. The reviewed asymptotic methods are easy to be followed for various applications. Some ideas on this paper are first appeared.

Subfields: Computational Theory and Mathematics, Mechanics of Materials, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/6b78cf6cba5b97188a3259853fd5312b8fd67baf

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

In Mechanics of Materials

In Modeling and Simulation

Citations in the corpus

Local Fractional Sumudu Transform with Application to IVPs on Cantor Sets (2014, methodology): “The local fractional Laplace transforms were developed in [27–30].”
A Tutorial Review on Fractal Spacetime and Fractional Calculus (2014, unknown): “we have three iteration algorithms [50, 51, 57]: Variational iteration algorithm-I:”
A Local Fractional Integral Inequality on Fractal Space Analogous to Anderson’s Inequality (2014, methodology): “Some of the concepts of the local fractional derivative were established in [3–26].”
Generalizations of Hölder’s and Some Related Integral Inequalities on Fractal Space (2011, unknown): “Local fractional derivative had been introduced in [12–36]; that is the local fractional derivative was structured in [12–18, 26, 30–36], Jumarie gave the modified Riemann-Liouville derivative in [19, 20], the fractal derivative was considered in [21–25, 27–29], and the generalized fractal derivativ”
Homotopy Perturbation Method for Fractional Black-Scholes European Option Pricing Equations Using Sumudu Transform (2013, unknown): “, u n [43–45], and it can be calculated by the following formula:”
The Yang-Laplace Transform for Solving the IVPs with Local Fractional Derivative (2014, methodology): “The Yang-Laplace transform structured in 2011 [22] was suggested to deal with local fractional differential equations [31, 32].”
Challenge in the variational iteration method – A new approach to identification of the Lagrange multipliers (2013, methodology): “For a relatively comprehensive survey on the concepts, theory and applications of the method, readers are referred to review articles (He and Wu, 2007; He, 2012).”