Abstract
This paper is concerned with two-point boundary value problems for singularly perturbed nonlinear ordinary differential equations. The case when the solution only has one boundary layer is examined. An efficient method so called Successive Complementary Expansion Method (SCEM) is used to obtain uniformly valid approximations to this kind of solutions. Four test problems are considered to check the efficiency and accuracy of the proposed method. The numerical results are found in good agreement with exact and existing solutions in literature. The results confirm that SCEM has a superiority over other existing methods in terms of easy-applicability and effectiveness.
Highlights
Nonlinear problems have always been more attractive than linear ones for scientists
We study on an efficient asymptotic method called Successive Complementary Expansion Method (SCEM) that generates uniformly valid approximations (UVA) to the solution of singularly perturbed nonlinear boundary value problems
If a regular expansion can be constructed in 1, we can write n ya(x, ε) = E1φ = φi(1)(ε)y(i1)(x). This inner expansion operator E1 is defined in 1 at the same order φ(ε) as the outer expansion operator E0;y(x, ε) − E1y(x, ε) = o(φ(ε)) and so ya(x, ε) = E0y(x, ε) + E1y(x, ε) − E1E0y(x, ε) is clearly uniformly valid approximation to order δ(ε) satisfying the modified Van Dyke principle (MVDP) E1E0y(x, ε) = E0E1y(x, ε). This is the main idea underlying the method of matched asymptotic expansions (MMAE)
Summary
Nonlinear problems have always been more attractive than linear ones for scientists. The main reason for this is that almost all natural phenomena in nature lead us to nonlinear models to describe them. Perturbed problems occurs frequently in electrical systems, celestial mechanics, particle physics, quantum mechanics, (semi/super) conductor systems, fluid mechanics, thermal processes and in chemical/biochemical reactions (Kumar 2011) These problems are characterized by the presence of a very small positive parameter 0 < ε ≪ 1 that multiplies the highest order derivative term in the differential equation. A number of excellent books were published such as O’Malley (1974), Bender and Orszag (1978), Kevorkian and Cole (1981), Eckhaus (1973), Eckhaus (1979), Lagerstrom (1988), Hinch (1991), Van Dyke (1975), Johnson (2006), Verhulst (2006), Holmes (1995) and Roos et al (1996) Thanks to these great books and the other works, today we have certain traditional asymptotic methods. We study on an efficient asymptotic method called SCEM that generates uniformly valid approximations (UVA) to the solution of singularly perturbed nonlinear boundary value problems.
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