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High-order uniformly convergent method for nonlinear singularly perturbed delay differential equations with small shifts

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In this paper, we propose and analyze a high-order uniform method for solving boundary value problems (BVPs) for singularly perturbed nonlinear delay differential equations with small shifts (delay and advance). | Turk J Math (2016) 40: 1144 – 1167 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ doi:10.3906/mat-1508-75 Research Article High-order uniformly convergent method for nonlinear singularly perturbed delay differential equations with small shifts Abdelhay SALAMA, Dirhem AL-AMERY∗ Department of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt Received: 27.08.2015 • Accepted/Published Online: 19.01.2016 • Final Version: 21.10.2016 Abstract: In this paper, we propose and analyze a high-order uniform method for solving boundary value problems (BVPs) for singularly perturbed nonlinear delay differential equations with small shifts (delay and advance). Such types of BVPs play an important role in the modeling of various real life phenomena, such as the variational problem in control theory and in the determination of the expected time for the generation of action potentials in nerve cells. To obtain parameter-uniform convergence, the present method is constructed on a piecewise-uniform Shishkin mesh. The error estimate is discussed and it is shown that the method is uniformly convergent with respect to the singular perturbation parameter. Moreover, a bound of the global error is also derived. The effect of small shifts on the solution behavior is shown by numerical computations. Several numerical examples are presented to support the theoretical results, and to demonstrate the efficiency and the high-order accuracy of the proposed method. Key words: Singularly perturbed, nonlinear differential equations, high-order method, delay differential equations, small shifts 1. Introduction In this paper, we consider the following singularly perturbed nonlinear delay differential equation (DDE) with small shifts: ( ) Lu(x) ≡ εu′′ (x) + a(x)u′ (x) = f x, u(x), u(x − δ), u(x + η) , (1.1) on Ω = (0, 1) with the interval conditions u(x) = ϕ(x), −δ ≤ x ≤ 0, u(x) = ψ(x), 1 ≤ x ≤ 1 + η, (1.2) where 0 0 on Ω, fu (x, u, v, w) > .