A short review on analytical methods for a fully fourth-order nonlinear integral boundary value problem with fractal derivatives

Abstract

Purpose

This paper aims to review some effective methods for fully fourth-order nonlinear integral boundary value problems with fractal derivatives.

Design/methodology/approach

Boundary value problems arise everywhere in engineering, hence two-scale thermodynamics and fractal calculus have been introduced. Some analytical methods are reviewed, mainly including the variational iteration method, the Ritz method, the homotopy perturbation method, the variational principle and the Taylor series method. An example is given to show the simple solution process and the high accuracy of the solution.

Findings

An elemental and heuristic explanation of fractal calculus is given, and the main solution process and merits of each reviewed method are elucidated. The fractal boundary value problem in a fractal space can be approximately converted into a classical one by the two-scale transform.

Originality/value

This paper can be served as a paradigm for various practical applications.

Citation

He, J.-H. (2020), "A short review on analytical methods for a fully fourth-order nonlinear integral boundary value problem with fractal derivatives", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 30 No. 11, pp. 4933-4943. https://doi.org/10.1108/HFF-01-2020-0060