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Unusual bifurcation of a Neumann boundary value problem. (English) Zbl 07249289
Summary: Boundary layer behaviour of a family of second order nonlinear differential equations with Neumann boundary condition arising from an order-reduction of a pseudo-differential equation in fluid dynamics is analysed. The problem is considered with two perturbation parameters $$\mu, \delta$$. Using asymptotic and numerical methods it is shown that by perturbing $$\delta$$ the problem in the limit $$\mu \to 0$$ changes from a homogeneous problem with symmetric solutions (including outer solutions) of a simplified equation, to a non-homogeneous problem which, in general, does not have a non-zero outer limit solution (with boundary layers). By studying the bifurcation diagrams as $$\mu$$ and $$\delta$$ vary it is shown that the main branch of solutions will ‘tear’ for $$\mu = n^{-2}$$ where $$n$$ is an even integer. Blow-up regions, isolated islands of solution, and for $$\mu \ll 1$$ “exponentially small” regions, all occur: a structure that is difficult for conventional path following algorithms to find.
##### MSC:
 34D15 Singular perturbations of ordinary differential equations
##### Software:
Matlab; MATLAB ODE suite; ode113; Ode15s; ode23; ode23s; ode45
Full Text:
##### References:
 [1] Ai, S.; Hastings, S. P., A shooting approach to layers and chaos in a forced Duffing equation, J. Differ. Equ., 185, 2, 389-436 (2002) · Zbl 1025.34015 [2] Ai, S.; Chen, X.; Hastings, S. P., Layers and spikes in non-homogeneous bistable reaction-diffusion equations, Trans. Am. Math. Soc., 358, 07, 3169-3207 (2006) · Zbl 1087.35007 [3] Benjamin, T. B., A new kind of solitary wave, J. Fluid Mech., 245, 401-411 (1992) · Zbl 0779.76013 [4] Benjamin, T. B., Solitary and periodic waves of a new kind, Philos. Trans. R. Soc. Lond. A, Math. Phys. Eng. Sci., 354, 1713, 1775-1806 (1996) · Zbl 0862.76010 [5] Chen, X.; Sadhu, S., Uniform asymptotic expansions of solutions of an inhomogeneous equation, J. Differ. Equ., 253, 951-976 (2012) · Zbl 1365.34097 [6] Gradshteyn, I. S.; Ryzhik, I. M., Tables of Integrals, Series, and Products (1994), Academic Press: Academic Press New York · Zbl 0918.65002 [7] Montagu, E. L.; Norbury, J., A note on solitary and periodic waves of a new kind, Proc. R. Soc. A, Math. Phys. Eng. Sci., 454, 1975, 1831-1834 (1998) · Zbl 0933.76013 [8] Montagu, E. L.; Norbury, J., Bifurcation of positive solutions for a Neumann boundary value problem, ANZIAM J., 42, 324-340 (2001) · Zbl 0980.34018 [9] Montagu, E. L.; Norbury, J., Solution structure for nonautonomous nonlocal elliptic equations with Neumann boundary conditions, Integral Transforms Spec. Funct., 13, 461-470 (2002) · Zbl 1016.35028 [10] Shampine, L. F.; Reichelt, M. W., The MATLAB ODE suite, SIAM J. Sci. Comput., 18, 1, 1-22 (1997) · Zbl 0868.65040 [11] Torres, P. J., Some remarks on a Neumann boundary value problem arising in fluid dynamics, ANZIAM J., 45, 3, 327-332 (2004) · Zbl 1053.34019
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