Abstract
In this paper, we investigate the singular Sturm-Liouville problem , , , where is a nonnegative parameter, , , and . We discuss the existence of multiple positive solutions and show that for certain values of , there also exist solutions that vanish on a subinterval , the so-called dead core solutions. The theoretical findings are illustrated by computational experiments for and for some model problems from the class of singular differential equations discussed in Agarwal et al. (2007). For the numerical simulation, the collocation method implemented in our MATLAB code bvpsuite has been applied.
Highlights
In the theory of diffusion and reaction see, e.g., 1, the reaction-diffusion phenomena are described by the equationΔv φ2h x, v, 1.1 where x ∈ Ω ⊂ RN
Baxley and Gersdorff 2 discussed problem 1.3, where f and h were continuous and h was allowed to be unbounded for u → 0. They proved the existence of positive solutions and dead core solutions vanishing on a subinterval 0, t0, 0 < t0 < 1 of problem 1.3, and covered the case of the function h approximated by some regular function hκ
The numerical treatment of problems 1.9a - 1.9b and 1.5a - 1.5b based on the collocation method is discussed in Section 3, where for different values of λ, we study positive, pseudo dead core, and dead core solutions of problem 1.9a - 1.9b and positive solutions of problem 1.5a - 1.5b
Summary
In the theory of diffusion and reaction see, e.g., 1 , the reaction-diffusion phenomena are described by the equation. If u 0 0, u > 0 on 0, T , φ u ∈ ACloc 0, T , u satisfies 1.4b and 1.4a holds a.e. on 0, T , u is called a pseudo dead core solution of problem 1.4a - 1.4b. A function u ∈ C2 0, 1 is a positive solution of problem 1.6a - 1.6b if u satisfies the boundary conditions 1.6b , u > 0 on 0, 1 and 1.6a holds for t ∈ 0, 1. A function u : 0, 1 → 0, ∞ is called a dead core solution of problem 1.6a - 1.6b if there exists a point ρ ∈ 0, 1 such that u t 0 for t ∈ 0, ρ , u ∈ C1 0, 1 ∩ C2 ρ, 1 , u satisfies 1.6b and 1.6a holds for t ∈ ρ, 1. The numerical treatment of problems 1.9a - 1.9b and 1.5a - 1.5b based on the collocation method is discussed in Section 3, where for different values of λ, we study positive, pseudo dead core, and dead core solutions of problem 1.9a - 1.9b and positive solutions of problem 1.5a - 1.5b
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