Browsing by Author "Wang, Shin-Hwa"

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Classification and Evolution of Bifurcation Curves for a Dirichlet-Neumann Boundary Value Problem and its Application
(2019-04-01)
Kuo, Da-Chang
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Wang, Shin-Hwa
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Liang, Yu-Hao
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應用數學系
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Department of Applied Mathematics
We study the classification and evolution of bifurcation curves of positive solutions for the one-dimensional Dirichlet-Neumann boundary value problem {u ''(x) + lambda f (u) = 0, 0 < x < 1, u(0) = 0, u' (1) = -c < 0, where lambda > 0 is a bifurcation parameter and c > 0 is an evolution parameter. We mainly prove that, under some suitable assumptions on f, there exists c(1) > 0, such that, on the (lambda,parallel to u parallel to(infinity))-plane, (i) when 0 < c < c(1), the bifurcation curve is S-shaped; (ii) when c >= c(1), the bifurcation curve is subset of-shaped. Our results can be applied to the one-dimensional perturbed Gelfand equation with f(u) = exp (au/a+u) for a >= 4.37.
Classification and evolution of bifurcation curves for the one-dimensional perturbed Gelfand equation with mixed boundary conditions
(2016-06-15)
Liang, Yu-Hao
;
Wang, Shin-Hwa
;
應用數學系
;
Department of Applied Mathematics
We study the classification and evolution of bifurcation curves of positive solutions for the onedimensional perturbed Gelfand equation with mixed boundary conditions given by {u"(x) + lambda exp(au/a+u) = 0, 0 < x < 1, u(0) = 0, u\' (1) = -c < 0. We prove that, for positive a <= a(0) (approximate to 0.501) and c > 0, the bifurcation curve is strictly increasing on the (A, II ulloo)-plane, and there exists a positive Xo such that the problem has no positive solution for 0 < lambda < lambda(0) and exactly one positive solution for lambda > lambda(0). While for a >= a(1) (approximate to 4.107), there exists c(1) (= c(1) (a)) > 1.057 such that, on the (A, 11u1100) -plane, (i) when 0 < c < cl, the bifurcation curve is S-shaped, and the problem has at least three positive solutions for some range of positive A; (ii) when c > cl, the bifurcation curve is c-shaped and the problem has at least two positive solutions for some range of positive A. (C) 2016 Elsevier Inc. All rights reserved.
Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity
(2012-06-15)
Tzeng, Chih-Chun
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Hung, Kuo-Chih
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Wang, Shin-Hwa
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應用數學系
;
Department of Applied Mathematics
We study the global bifurcation and exact multiplicity of positive solutions of {u'' (chi) + lambda f(epsilon)(u) = 0, -1 < chi < 1, u(-1) = u(1) = 0, f epsilon(u) = -epsilon u(3) + sigma u(2) - kappa u + rho, where lambda, epsilon > 0 are two bifurcation parameters, and sigma, rho > 0, 0 < kappa <= root sigma rho are constants. We prove the global bifurcation of bifurcation curves for varying epsilon > 0. More precisely, there exists (epsilon) over tilde > 0 such that, on the (lambda, parallel to u parallel to infinity)-plane, the bifurcation curve is S-shaped for 0 < epsilon < (epsilon) over tilde and is monotone increasing for epsilon >= (epsilon) over tilde. Thus we are able to determine the exact number of positive solutions by the values of epsilon and lambda. Our results extend those of Hung and Wang (K.-C. Hung, S.-H. Wang, Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity and their applications, Trans. Amer. Math. Soc., in press) from kappa <= 0 to kappa <= root sigma rho. (C) 2012 Elsevier Inc. All rights reserved.