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A singularly perturbed linear system of second order ordinary differentialequations of reaction-diffusion type with given boundary conditions isconsidered. The leading term of each equation is multiplied by a small positiveparameter. These parameters are assumed to be distinct. The components of thesolution exhibit overlapping layers. Shishkin piecewise-uniform meshes areintroduced, which are used in conjunction with a classical finite differencediscretisation, to construct two numerical methods for solving this problem. Itis proved that the numerical approximations obtained with these methods areessentially first, respectively second, order convergent uniformly with respectto all of the parameters.

A parameter--uniform finite difference method for a singularly perturbed  linear system of second order ordinary differential equations of  reaction-diffusion type