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Abstract. We investigate the solvability of the Dirichlet boundary value problemwhere λ is a nonnegative parameter. We discuss the existence of multiple positive solutions and show that for certain values of λ, there also exist solutions that vanish on a subinterval [ρ, 1 − ρ] ⊂ (0, 1), the so-called dead core solutions. In order to illustrate the theoretical findings, we present computational results for g(u) = 1/ √ u, computed using the collocation method implemented in bvpsuite, a new version of the standard MATLAB code sbvp1.0.

Transparent boundary conditions for the 2D Schrödinger equationWe consider the 2D time-dependent Schrödinger equation on the infinite stripeΩ = R × (0, Y ):with a given real-valued potential V (x, y, t). For numerical simulations we have to restrict (1) to a finite domain, e.g. Ω = (0, X) × (0, Y ), and we assume supp ψ I ⊂ Ω, ψ I ∈ L 2 (Ω). Also, we assume that V is constant on the exterior domain:for the partial Fourier seriesψ m (x, t) = k∈N ψ(x, y, t) sin kπy Y of ψ w.r.t. y. η denotes the unit outward normal vector at the artificial boundaries at x = 0, x = X. √ ∂ t is the fractional time derivative of order 1 2 with the Fourier symbol √ −iω,Discretizing (1) with the standard Crank-Nicolson finite difference scheme, performing a discrete sine-transformation in y, and using the Z-transformation in discrete time leads to the sine-transformed discrete TBCs (DTBCs) for each modeψ m :at the right boundary x J = X for the discrete wave functionψ n j,m ∼ψ m (x j , t n ), x j = j∆x, j ∈ Z, t n = n∆t, n ∈ N, m ∈ N (cf.[2], [4]) (and for the left boundary x = 0 analogously). The convolution coefficients s (n) j,m are given in [2]. The discretization of the Schrödinger equation (1) along with the DTBCs (2) is solved by time stepping. The convolution length in (2) therefore grows with every time step. This makes it impossible to simplify (or accelerate) the numerical evaluation of these time convolutions by using a fast Fourier transform (FFT). Hence, the evaluation of the convolution sum up to time level n requires a memory of the order O(Kn) and numerical costs of the order O(Kn 2 ), where K denotes the number of discretization points in y-direction. The latter can easily surpass the costs of solving the PDE in the interior domain which only grows linearly in n. As a remedy it is possible to approximate the convolution coefficients by a sum-of-exponentials (say L terms) and to calculate the convolution sum recursively, as shown in [2]. We write s (n) j,m ≈s (n) j,m = ⎧ ⎨ ⎩ s (n) j,m : n = 0, . . . , r − 1 L l=1 b j,m,l q −n j,m,l : n ≥ r ; j = 0, J; m ∈ N (3) for some parameters L, r ∈ N. Typical values to obtain good results are L = 10 -20, r = 1, 2. Furthermore, the non-local part C n ( ψ j,m ) := n−r k=1s (n−k) j,m ψ k