Two numerical algorithms for finding solutions of multiparameter semipositone Dirichlet problems
Introduction
In this paper we will use a variational technique to obtain approximation solutions to our PDE equations:where Ω is a bounded domain of RN with the boundary ∂Ω ∈ C2, and Δ is the Laplacian of u, λ > 0 and μ ∈ R are parameters, f and g are Caratheodory functions on Ω × (0, ∞) such thatfor some 1 ⩽ q < 2 and constants a1, a2 ⩾ 0,for some a3, t1 ⩾ 0, and g is bounded on bounded sets. We make no assumptions about the signs of f(x, 0) and g(x, 0) and hence allow the semipositone case λf(x, 0) + μg(x, 0) < 0.
Caldwell et al. [2] studied the evolution of solution curves of (1) as λ, μ > 0, vary for the ODE case n = 1 and Caldwell et al. [1] studied the existence and nonexistence of classical solutions for the PDE case n ⩾ 2. In [4] the authors seek weak solutions in the general quasilinear case p-laplacian for 1 < p < ∞, n ⩾ 1 that we provide a sketch of proof for the semilinear case p = 2. Theorem 1.1 There is a λ0 > 0 such that (1) has two solutions for λ ⩾ λ0, and μ small in the following cases: g is subcritical and p-sublinear: ∣g(x, t)∣ ⩽ a4tr−1 + a5 1 ⩽ r < p∗ and a4, a5 ⩾ 0 and 0 < θG(x, t) ⩽ tg(x, t), t ⩾ t2 for some θ > p and t2 > 0, n > p and . Denoted bythe critical Sobolev exponent.
Section snippets
Existence results
We begin by constructing a positive subsolution. Let λ1 > 0 and 0 < φ1 ⩽ 1 be the first Dirichlet eigenvalue of −Δ on Ω and the corresponding eigenfunction, respectively. By (1.2), (1.3), f ⩾ −a6 for some a6 > 0. Let Lemma 2.1 u is a subsolution of (1) for λ sufficiently large and ∣μ∣ small. Proof We haveSince φ1 = 0 and ∇φ1 ≠ 0 on ∂Ω, in some neighborhood Ω′ ⊂ Ω of ∂Ω the right-hand side of (2.2) is ⩽−a7 and henceOn Ω⧹Ω′, φ1 ⩾ a8
Numerical algorithm
The subject of partial differential equations, PDE, has many applications in real life. In particular nonlinear elliptic PDE have been used in many physical problems such as fluid dynamics, chemical reactions, and steady state solutions of reaction diffusion equations (see [5]).
When studying a nonlinear PDE, one might be interested in finding solutions which satisfy some boundary value problem, BVP.
It is essential to approximate the solution of partial differential equations numerically in
Numerical results
We consider problem (2) with q = 1.5, r = 3 and let Ω = [0, 1] × [0, 1], we want to obtain a numerical solution for the problem
We present just some of values of u(x, y) in following tables for varying λ:x y 0.2 0.4 0.6 0.8 Approximation of u for λ = 10 0.2 −0.2554 + 0.2080i −0.3493 + 0.3078i −0.3493 + 0.3078i −0.2554 + 0.2080i 0.4 −0.3493 + 0.3078i −0.4889 + 0.4601i −0.4889 + 0.4601i −0.3493 + 0.3078i 0.6 −0.3493 + 0.3078i −0.4889 + 0.4601i −0.4889 + 0.4601i −0.3493 + 0.3078i 0.8 −0.2554 + 0.2080i −0.3493 + 0.3078i −0.3493 +
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