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Without using conformal transformation, a simple type of five-dimensional$f(R)-$brane model is linearized directly in its higher-order frame. In thispaper, the linearization is conducted in the equation of motion approach. Wefirst derive all the linear perturbation equations without specifying a gaugecondition. Then by taking the curvature gauge we derive the master equations ofthe linear perturbations. We show that these equations are equivalent to thoseobtained in the quadratical action approach [Phys. Rev. D 95 (2017) 104060],except the vector sector, in which a constraint equation can be obtained in theequation of motion approach but absent in the quadratical action approach. Ourwork sets an example on how to linearize higher-order theories without usingconformal transformation, and might be useful for studying more complicatedtheories.

Linearization of a warped $f(R)$ theory in the higher-order frame II: the equation of motion approach