Nonlocal Boundary Value Problems

The theory of the nonlocal linear boundary value problems is still on the level of examples. Any attempt to encompass them by a unified scheme sticks upon the lack of general methods. Here we are to outline an algebraic approach to linear nonlocal boundary value problems. It is based on the notion of convolution of linear operator and on operational calculus on it. Our operators are right inverses of the differentiation operator and its square. These right inverse operators are determined by the boundary value conditions of the problems we are to deal with. The idea of the algebraic approach consists in algebraization of the problem by reducing it to a single linear algebraic equation of the first degree in a corresponding commutative ring, containing as subrings: the function space, the multipliers ring of the convolution algebra and the number field. Thus we reduce all the consideration into a single algebraic system: the ring of the multiplier fractions. As applications it is possible to be considered the following nonlocal BVPs: 1) Nonlocal Cauchy problems; 2) Dezin BVP; 3) Samarskii – Ionkin problem; 4) Beilin problem; 5) Bitsadze – Samarskii problem. Following our algebraic approach we obtain for all of them explicit representations of the solutions. These representations may be considered as extensions of the classical Duhamel principle. They can be used for numerical calculation of the solutions using quadrature formulae.