Some basic theorems and formulas for building fractal nonlinear wave models

Abstract

As a tool to deal with the phenomenon in layered or porous media, many theoretical problems need to be improved urgently. In this paper, the definitions of two kinds of fractal derivatives based on scale transformation are developed from the perspective of mathematics, and the four operation rules, chain rules, Rolle theorem, mean value theorem, Taylor formula, variable upper limit integral derivation rule, definite integral basic principle, partial integral formula and double partial integral formula of the fractal version are given. Better still, we derive the fractal Euler-Lagrangian equation by using the variational method, and demonstrate the steps and effectiveness of deriving the fractal nonlinear wave equation based on the fractal Euler-Lagrangian equation through the example of the functionally graded beam.

Keywords

• Fractal derivative
• Fractal Euler-Lagrange equation
• Fractal integral
• Functional gradient beam