Multiple Solutions of a Generalized Singular Perturbed Bratu Problem

Bakri, Taoufik; Kuznetsov, Yu. А.; Verhulst, Ferdinand; Doedel, Eusebius J.

doi:10.1142/s0218127412500952

Cited by 11 publications

(13 citation statements)

References 12 publications

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“…where, additionally, we assume that D(u) > 0, D ′ (u) ≥ 0 and g ′ (u) > 0. The results are generalizations of the ones obtained in [7]. Therein, the authors consider the special case D = 1, g(u) = 2u and f (u) = e u .…”

Section: Theoretical Observationssupporting

confidence: 60%

“…Next, we note that the equation has a stable slow manifold M with an O(ϵ 2 ) approximation by the manifold M 0 described by u 0 (x) + ϵu 1 (x). This observation can be found by applying Fenichel's geometric perturbation theory (for more details, see [7,135]). This implies that, away from the boundary layer at x = x L , we find near x = x R :…”

Section: Theoremmentioning

confidence: 96%

“…Proof (we use elements from [7] u ′ (x 0 ) = 0 and also u ′′ (x 0 ) = −λf (x 0 ). For λ > 0, we find u ′′ (x 0 ) < 0, since f (x 0 ) > 0.…”

Section: Lemmamentioning

confidence: 99%

“…We assume that there exists a unique solution for model (3.1). Note, however, that multiple solutions may exist for specific choices of the functions f and g (see Chapter 2, [148] and [7]). Furthermore, we may expect boundary layers in the solution as well.…”

Section: Convection-diffusion-reaction Modelmentioning

confidence: 99%

“…In Table 3.1 we illustrate this for the special case with α = β = 0, x L = 0, x R = 1, f (u) = e u , g(u) = 2u and D = 1 (as in [7]). Then the result in Theorem 2 predicts that u ′ 0 (1) = −1/2 and u ′ 1 (1) = −1/8 and u ′ (1) = −1/2 − ϵ/8 + O(ϵ 2 ).…”

Section: Theoremmentioning

confidence: 99%

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Numerical Methods for Nonlinear Elliptic Boundary Value Problems with Parameter Dependence

Iqbal¹

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Chapter 1 1.3 Nonstandard finite difference schemes The majority of differential equation models (both linear and nonlinear), which correspond to dynamical systems stemming from engineering and natural sciences, can't be solved exactly in terms of elementary functions. As a consequence, a variety of numerical methods has been constructed to find an approximate solution. One of the traditional techniques in this area, is the finite difference method [78, 11, 52, 128, 73]. Unfortunately, numerical instabilities may arise in these finite difference approximations. Normally, numerical instabilities can be recognized as • Rule 1. The order of the discrete derivatives should be equal to the orders of the corresponding derivatives of the differential equations. This rule demonstrates that spurious solutions may exist, if the order of the discrete derivatives is larger than that of the derivatives that appear in the differential equations. The following example shows the consequences, if Rule 1 is violated. Consider the decay differential equation: du dt = −u. (1.5) A central finite difference approximation for ODE (1.5) gives: u n+1 + 2∆t u n − u n−1 = 0. It has the general solution [89]: u n = C 1 (r +) n + C 2 (r −) n , with r ± = −∆t ± √ 1 + (∆t) 2 , (1.6) where C 1 and C 2 are arbitrary constants. The r ± are functions of ∆t and the following result holds for large ∆t : 1.4 Singularly perturbed boundary value problems with boundary layers It is well-known that many phenomena in fluid dynamics, biology, engineering, physics and many more can be described by boundary value problems (BVPs) associated with different types ∆x j+1 − ∆x j ∆x j ∆x j+1 , c = ∆x j ∆x j+1 (∆x j + ∆x j+1). Figure 1.5: Sequence of grids with spacing h (fine grid), 2h, 4h (coarsest grid) of the set of grid points Ω h , Ω 2h and Ω 4h , respectively. direct solution: = () − No Post-smooth: relax 2 times on fine grid = MGV (* , ℎ , 2) if coarsest grid Figure 1.6: Multigrid V-cycle flow diagram This chapter is based on the article published as [148]:

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Section: Theoretical Observationssupporting

confidence: 60%

Section: Theoremmentioning

confidence: 96%

“…Proof (we use elements from [7] u ′ (x 0 ) = 0 and also u ′′ (x 0 ) = −λf (x 0 ). For λ > 0, we find u ′′ (x 0 ) < 0, since f (x 0 ) > 0.…”

Section: Lemmamentioning

confidence: 99%

Section: Convection-diffusion-reaction Modelmentioning

confidence: 99%

Section: Theoremmentioning

confidence: 99%