In this paper, the belief propagation (BP) based approximation methods for low-density parity-check codes are adapted to the Luby transform soft decoder structure in order to reduce its computational complexity. Moreover, the log-likelihood ratio based adaptive demodulation algorithm is combined with the BP and BP-based algorithms to further reduce the computational complexities and optimum key parameters are determined for normalized min-sum and offset min-sum algorithms. The bit error rate performances of the algorithms over the binary input additive white Gaussian noise channel are obtained by both theoretically and simulations. For theoretical analysis, the Monte-Carlo based density evolution method is used. In addition, computational complexity analyzes of methods are presented. Results show that the computational complexity can be reduced significantly by using combined methods which cost limited signal to noise ratio loss.