We present a new approach for expressing and solving boundary problems for linear ordinary differential equations in the language of differential algebras. Starting from an algebra with a derivation and integration operator, we construct an algebra of linear integro-differential operators that is expressive enough for specifying regular boundary problems with arbitrary Stieltjes boundary conditions as well as their solution operators. On the basis of these structures, we define a new multiplication on regular boundary problems in such a way that the resulting Green's operator is the reverse composition of the constituent Green's operators. We provide also a method for lifting any factorization of the underlying differential operator to the level of boundary problems. Since this method only needs the computation of initial value problems, it can be used as an effective alternative for computing Green's operators in the case where one knows how to factor the given differential operators.
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