  We first strictly expressed the basic notions and research methods of
abstract operators, which systematically expounded the main results of abstract
operator theory. By combining abstract operators with the Laplace transform, we
can easily apply this Laplace transform to n+1 dimensional partial differential
equations. Further, all the analytic solutions to an initial value problem of
an arbitrary order linear partial differential equation are expressed in these
abstract operators. By writing abstract operators in this class into integral
forms, the solutions in operator form are represented into integral forms. We
thus solved the important problem of representing the solutions of linear
higher-order partial differential equations into the integrations of some given
functions. By introduction of abstract operators on Hilbert space, we further
discuss the solvability of initial-boundary value problem for the linear
higher-order partial differential equations.
