Positive Solutions for Boundary Value Problems of Second Order Difference Equations and Their Computation

Authors

Jun Ji, Kennesaw State UniversityFollow
Bo Yang, Kennesaw State UniversityFollow

Department

Mathematics

Document Type

Article

Publication Date

7-15-2010

Abstract

We consider the following two classes of second order boundary value problems for difference equation: Δ(r_i―1Δy_i―1) ― b_iy_¡ + λa_¡y_¡ = 0, 1 ≤ i ≤ n, y₀ ― τy₁ = y_n+1 ― δy_n = 0 with δ, τ ∈ [0, 1] and Δ(r_i―1Δy_i―1) ― b_iy_i + λa_iy_i = 0, 1 ≤ i ≤ n, y₀ = αY_n, Y_n+1 = βy₁ with α, β ∈ [0, 1]. We establish the existence of positive solutions to both problems. A solver with linear computational complexity for almost tridiagonal linear systems is developed by exploring the special structure of linear system of equations. Based on fast solvers for linear systems, effective algorithms for the computation of positive solutions will be proposed.