Variational principle and periodic wave solutionals for elastic rod equation with fractal derivative

The present research paper will demonstrate the variational principle and periodic wave solutions of the elastic rod equation. First, we will illustrate the generalized variational principle in two examples. Secondly, we consider a fractal non-linear elastic rod equation with an unsmooth boundary. Based on two-scale fractal theory and the semi-inverse method, we successfully establish the fractal variational principle for the non-linear elastic rod equation. This is helpful for studying symmetry, finding conserved quantities, and revealing possible traveling solution structures of the equation. Finally, we investigate periodic wave solutions of the non-linear elastic rod equation