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A variational approach is formulated and implemented for numerically solving a system of nonlinear two-point boundary value problem (BVP) with coupled boundary conditions modeling the power evolution in cascaded fiber Raman laser with the fiber Bragg gratings at the ends of the cavity. The nonlinearity is treated by successive linearization and the coupled boundary conditions are naturally incorporated into the system through integration in the variational setting. A global approximation of the dependent variables in terms of Legendre polynomials is used to provide a stable Lagrangian interpolation representation as well as the Legendre-Gauss quadrature for accurate numerical evaluation of integrals in the variational formulation. An initial approximate solution is constructed for the delicate convergence to the solution. The approach is validated against an approximate analytic solution and some exact integrals of the variables. The numerical experiments show exponential (spectral) accuracy achieved with much lower resolution in comparison to a widely available BVP solver. Further numerical experiments are performed to reveal the physical characteristics of the underlying model.