Abstract
We develop a collocation method for solving the singularly perturbed convection-diffusion equation and reaction-diffusion equation of elliptic type partial differential equations. When the singularly perturbed solution is expressed in terms of a set of fractional powers 2D exponential trial functions, and collocate points to satisfy the boundary conditions and the governing equation, we can obtain a small scale normal linear system, which is solved by the Gaussian elimination method or the conjugate gradient method to determine the expansion coefficients. The numerical algorithm with complexity O(n) is very time saving, effective and accurate to find the solutions of highly singularly perturbed reaction-convection-diffusion problems, which are defined in a rectangular domain as well as in an arbitrary domain, and the numerical examples together with an experimental result confirm the assessments.
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