Staněk, Svatoslav; Pulverer, Gernot; Weinmüller, Ewa B. Analysis and numerical simulation of positive and dead-core solutions of singular two-point boundary value problems. (English) Zbl 1152.34320 Comput. Math. Appl. 56, No. 7, 1820-1837 (2008). Summary: We investigate the solvability of the Dirichlet boundary value problem\[ u^{\prime\prime}(t)=\lambda g(u(t)),\qquad \lambda \geq 0,\quad u(0)=1,\quad u(1)=1 \] where \(\lambda \) is a nonnegative parameter. We discuss the existence of multiple positive solutions and show that for certain values of \(\lambda \), there also exist solutions that vanish on a subinterval \([\rho ,1 - \rho ]\subset (0,1)\), the so-called dead-core solutions. In order to illustrate the theoretical findings, we present computational results for \(g(u)=1/\sqrt u\), computed using the collocation method implemented in bvpsuite, a new version of the standard MATLAB code sbvp1.0. Cited in 7 Documents MSC: 34B15 Nonlinear boundary value problems for ordinary differential equations 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations Keywords:singular Dirichlet boundary value problem; positive solution; dead-core solution; pseudo-dead-core solution; existence; uniqueness; dead-core; multiplicity; collocation methods Software:bvpsuite; Matlab; Sbvp PDFBibTeX XMLCite \textit{S. Staněk} et al., Comput. Math. Appl. 56, No. 7, 1820--1837 (2008; Zbl 1152.34320) Full Text: DOI References: [1] Aris, R., The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts (1975), Clarendon Press: Clarendon Press Oxford · Zbl 0315.76051 [2] Bobisud, L. E., Asymptotic dead-cores for reaction-diffusion equations, J. Math. Anal. Appl., 147, 249-262 (1990) · Zbl 0706.34052 [3] Bobisud, L. E.; Royalty, W. 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