A UNIFIED APPROACH TO SINGULAR PROBLEMS ARISING IN THE MEMBRANE THEORY

作者：Rachunkova Irena^*; Pulverer Gernot; Weinmueller Ewa B

来源：Applications of Mathematics, 2010, 55(1): 47-75.

DOI：10.1007/s10492-010-0002-z

• singular mixed boundary value problem
• positive solution
• shallow membrane
• collocation method
• lower and upper functions

We consider the singular boundary value problem
(t(n) u'(t))' + t(n) f(t, u(t)) = 0, lim(t -> 0+) t(n) u'(t) = 0, a(0)u(1) + a(1)u'(1-) = A,
where f(t, x) is a given continuous function defined on the set (0, 1] x ( 0, infinity) which can have a time singularity at t = 0 and a space singularity at x = 0. Moreover, n is an element of N, n >= 2, and a(0), a(1), A are real constants such that a(0) is an element of (0, infinity), whereas a(1), A is an element of [ 0, infinity). The main aim of this paper is to discuss the existence of solutions to the above problem and apply the general results to cover certain classes of singular problems arising in the theory of shallow membrane caps, where we are especially interested in characterizing positive solutions. We illustrate the analytical findings by numerical simulations based on polynomial collocation.

• 出版日期2010-2