A robust B-spline method for two parameter singularly perturbed parabolic differential equations with discontinuous initial condition

A singularly perturbed parabolic problem with two parameters and a discontinuous initial condition is investigated. A singular function is identified to describe the nature of the singularity that arises from this discontinuity and a remainder function is generated by deducting this singular function from the solution of the given problem. The temporal variable is discretized on a uniform mesh using the implicit Euler method while the spatial variable is discretized on a piecewise uniform Shishkin mesh using the cubic B-spline collocation method. Through error analysis, it is shown that the associated numerical scheme is uniformly convergent with almost second order in space and first order in time. Two test examples are provided to corroborate the findings.