Two numerical algorithms for finding solutions of multiparameter semipositone Dirichlet problems

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Abstract

In this paper, we investigate numerically positive solutions of the equation −Δu = λf(x, u) + μg(x, u) with Dirichlet boundary condition in a bounded domain Ω for λ > 0, μ  R using sub-super solution algorithm and mountain pass algorithm. We will show that in which range of λ, this problem achieves numerical solutions.

Introduction

In this paper we will use a variational technique to obtain approximation solutions to our PDE equations:-Δu=λf(x,u)+μg(x,u)inΩu>0inΩu=0inΩ,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 that|f(x,t)|a1tq-1+a2,for some 1  q < 2 and constants a1, a2  0,f(x,t)a3,tt1for 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 C1,α(Ω¯) solutions for λ  λ0, and μ small in the following cases:

  • (i)

    g is subcritical and p-sublinear:g(x, t)  a4tr1 + 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,

  • (ii)

    n > p and g(x,t)=tp-1.

    Denoted byp=np/(n-p),n>p,n<pthe 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. Let1<β<2,a7>λ1a6βa1,cλ=λa6+1a7,u̲=cλφ1β.

Lemma 2.1

u is a subsolution of (1) for λ sufficiently large andμsmall.

Proof

We have-Δφ1β=βλ1φ1β-(β-1)|φ1|2φ11-(β-1).Since φ1 = 0 and ∇φ1  0 on ∂Ω, in some neighborhood Ω  Ω of ∂Ω the right-hand side of (2.2) is ⩽−a7 and hence-Δu̲-cλa7=-(λa6+1)λf(x,u̲)-1.On ΩΩ′, 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 problemΔu=λ(u12-1)+μu2inΩu>0inΩu=0inΩ.

We present just some of values of u(x, y) in following tables for varying λ:

xy
0.20.40.60.8
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 + 

References (5)

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    Positive solution for classes of multiparameter boundary value problems

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    (2002)
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