Analytical solution 1 Analytical solutions Conformable fractional derivative 1 Fractional derivatives 1 Fractional order 1 Long water waves 1 Modified equal width equation 1 Modified equal-width equations 1 Nonlinear equations 1 Nonlinear medium 1 One-dimensional 1 Soliton solutions 1 Solitons 1 Water wave propagation 1 Water waves 1 Wave propagation 1