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We present a finite difference method for a system of two singularly perturbed initial-boundary value semilinear reaction-diffusion equations. The highest order derivatives are multiplied by small perturbation parameters of different magnitudes. The problem is discretized using a central difference scheme in space and backward difference scheme in time on a Shishkin mesh. The convergence analysis has been given, and it has been established that the method enjoys almost second-order parameter-uniform convergence in space and first-order in time. Numerical experiments are conducted to demonstrate the efficiency of the method.

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