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) = / with abstract nonlocal boundary conditions Φ(u) = N Ψ(Au) and Φ(u) = N Ψ(Au); Φ(Au) = DF(Au) + N Ψ(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, anil then we investigate the correctness of the equation B2u = f1 and find its exact solntion. © 2019 L.N. Gumilyov Eurasian National University.