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Positive solutions of nonlinear Dirichlet BVPs in ODEs with time and space singularities

Irena Rachůnková1*, Alexander Spielauer2, Svatoslav Staněk1 and Ewa B Weinmüller2

Author Affiliations

1 Department of Mathematical Analysis, Faculty of Science, Palacký University, 17. listopadu 12, Olomouc, CZ-771 46, Czech Republic

2 Department of Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8-10, Wien, A-1040, Austria

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Boundary Value Problems 2013, 2013:6  doi:10.1186/1687-2770-2013-6

Published: 16 January 2013

Abstract

In this paper, we discuss the existence of positive solutions to the singular Dirichlet boundary value problems (BVPs) for ordinary differential equations (ODEs) of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M2">View MathML</a>. The nonlinearity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M3">View MathML</a> may be singular for the space variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M4">View MathML</a> and/or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M5">View MathML</a>. Moreover, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M6">View MathML</a>, the differential operator on the left-hand side of the differential equation is singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M7">View MathML</a>. Sufficient conditions for the existence of positive solutions of the above BVPs are formulated and asymptotic properties of solutions are specified. The theory is illustrated by numerical experiments computed using the open domain MATLAB code bvpsuite, based on polynomial collocation.

MSC: 34B18, 34B16, 34A12.

Keywords:
singular ordinary differential equation of the second order; time singularities; space singularities; positive solutions; existence of solutions; polynomial collocation