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

Boundary value problems

Alexander Spielauer¹  Svatoslav Stank²  Ewa B Weinmller³  Irena Rachnkov³ 

[1] University, Olomouc, Czech Republic;Department of Analysis and Scientific Computing, Vienna University of Technology, Wien, Austria;Department of Mathematical Analysis, Faculty of Science, Palacký

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

DOI  :  10.1186/1687-2770-2013-6

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

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 formu″(t)+atu′(t)−at2u(t)=f(t,u(t),u′(t)),u(0)=0,u(T)=0,wherea∈(−1,0). The nonlinearityf(t,x,y)may be singular for the space variablesx=0and/ory=0. Moreover, sincea≠0, the differential operator on the left-hand side of the differential equation is singular att=0. 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.

【 授权许可】

CC BY