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This paper proposes a fractional model for nonlinear waves in hyperelastic rods, which describes far-field, finite length, finite amplitude radial deformation waves in cylindrical compressible hyperelastic rods. In this model, fractional derivatives are described in the Caputo sense. The error analysis shows that our approximate solution converges very rapidly to the exact solution and the numerical solution is compared with the known analytical solution which is nearly identical with the exact solution. The method introduces a promising tool for solving time the fractional hyperelastic rod equation. doi:10.14456/WJST.2014.71