Extension and decomposition method for differential and integro-differential equations

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 $B_1u = A^2u - qF(Au) - gF(A^2u) = f$ with abstract nonlocal boundary conditions $\Phi(u) = N\Psi(Au)$ and $\Phi(u) = N\Psi(Au)$, $\Phi(Au) = DF(Au) + N\Psi(A^2u)$, respectively, where $q$, $g$ are vectors, $D$, $N$ are matrices, $F$, $\Phi$, $\Psi$ are vector-functions. In this paper:

• we investigate the correctness of the equation $Bu = f$ and find its exact solution,
• we investigate the correctness of the equation $B_1u = f$ and find its exact solution,
• we find the conditions under which the operator $B_1$ has the decomposition $B_1=B^2$, i.e. $B_1$ is a quadratic operator, and then we investigate the correctness of the equation $B^2u = f_1$ and find its exact solution.