Extension and decomposition method for differential and integro-differential equations

Abstract

A direct method for finding exact solutions of differential or Fredholm integro-differential equations with nonlocal boundary conditions is proposed. We investigate the abstract equations of the form Bu = Au - gF(Au) = f and B1u = A²u - qF(Au) - gF(A²u) = f with abstract nonlocal boundary conditions Φ(u) = NΨ(Au) and Φ(u) = DF(Au) + NΨ(A²u), respectively, where q, g are vectors, D, N are matrices, F, Φ, Ψ are vector-functions. In this paper: 1. we investigate the correctness of the equation Bu = f and find its exact solution, 2. we investigate the correctness of the equation B1u = f and find its exact solution, 3. we find the conditions under which the operator B1 has the decomposition B1 = B², i.e. B1 is a quadratic operator, and then we investigate the correctness of the equation B²u = f1 and find its exact solution.