Abstract
A general approach is presented for proving existence and uniqueness of solutions to the singular boundary value problem y ″ ( x ) + m x y ′ ( x ) = f ( x , y ( x ) ) , x ∈ ( 0 , 1 ] , y ′ ( 0 ) = 0 , A y ( 1 ) + B y ′ ( 1 ) = C , A > 0 , B , C ⩾ 0 . The proof is constructive in nature, and could be used for numerical generation of the solution. The only restriction placed on f ( x , y ) is that it not be a singular function of the independent variable x ; singularities in y are easily avoided. Solutions are found in finite regions where ∂ f / ∂ y ⩾ 0 , using an integral equation whose Green’s function contains an adjustable parameter that secures convergence of the Picard iterative sequence. Methods based on the theory are developed and applied to a set of problems that have appeared previously in published works.
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