Dependence of the unique solution of a periodic boundary value problem on the parameter

作者：Bellout, Hamid; Kong, Qingkai^*; Wang, Min

来源：Applied Mathematics and Computation, 2011, 217(19): 7838-7844.

DOI：10.1016/j.amc.2011.02.110

• Dependence of solution on parameter
• Uniqueness of positive solution
• Periodic boundary value problem
• Implicit function theorem

In this paper, we study the dependence of the unique positive solution of the periodic boundary value problem @@@ -u '' + rho(2)u = w(t)u(s), 0 < s < 1 and rho not equal 0, @@@ u(0) = u(1), u'(0) = u'(1) @@@ on the parameter rho as rho -> 0. We show that this solution u( t; rho) is infinitely differentiable in rho and has singularity in the order of rho (alpha) with alpha = 2/(1 s) as rho -> 0. Furthermore, the graph of u( t; rho) has fluctuation in t only of the order O(rho (alpha+2)). This will help us to determine the location of the solution in numerical computations to study the behavior of the solution when rho is sufficiently close to 0.

• 出版日期2011-6-1