In this work, three variant Time Domain Random Walk (TDRW) algorithms were developed for the problem of solute transport in a single fracture-matrix system, where an arbitrary inlet boundary condition can be applied. One approach performs an additional evaluation of integral in terms of injection boundary and the solution to a Dirac delta case. One method makes use of two functions, dependent on specific boundary conditions, to estimate the particle arrival time. The other additionally introduces the concept of solute injection time, resulting from the injection boundary, into the calculation of particle arrival time. To validate the developed variant algorithms, two benchmark cases are considered with respect to a general Dirichlet injection mode and a Robin injection boundary, respectively. The results from three approaches all make a good agreement with those of inverse Laplace transform method. However, the Monte Carlo nature of the TDRW algorithm implies that the accuracy of the computational result is highly dependent on the number of particles applied in the simulation.