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Additional info for Differential Equations and Numerical Analysis: Tiruchirappalli, India, January 2015
From this bound, we see that f (x)/b(x) is indeed the reduced solution for the reaction-diffusion problem. , let k = 2 in (1)). However, we wish to generate approximations without imposing a constraint of the form ε ≤ C N −2 p on the set of problems being considered. Our interest lies in designing parameter-uniform numerical methods  for a large set of singularly perturbed problems. Parameter-uniform numerical methods guarantee convergence of the numerical approximations, without imposing a meshdependent restriction on the permissible size of the singular perturbation parameter.
This is necessary here | is unbounded as ε → 0, while |ε dy | because the first derivative of the solution | dy dt dt is bounded. This is seen at once from the exact solution y(t) = where λ± are the roots of ε(λ+ γ + − (eλ t − eλ t ), − −λ ) ελ2 + kλ + c. We assume that k 2 > 4εc to ensure that the roots are real and distinct. The plot of the exact solution for the moderate value ε = 21 is given in Fig. 1. Let us investigate now what happens as ε → 0. The plots of the exact solution for 1 1 , 32 are given in Fig.
3] v (i) Ω¯ ≤ C(1 + ε2−i ); |w (i) (x)| ≤ Cε−i e−α(1−x)/ε , i ≤ 3. Using simple stable finite difference scheme with a standard piecewise-uniform Shishkin mesh to produce a numerical approximation U N , one has for the convectiondiffusion problem (2) (see, for example, [1, Chap. 3]): If a, b, f ∈ C 3 (Ω) then u − U¯ N Ω ≤ C N −1 (ln N ) and for the reaction-diffusion problem (1) (see, for example, [3, Chap. 6]): If b, f ∈ C 4 (Ω) then u − U¯ N Ω ≤ C N −2 (ln N )2 , where U¯ N is the piecewise linear interpolant of the mesh function U N .