Abstract
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.
© 2010 IEEE
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