  We define a class of pseudo-differential operators in a completely new way,
which is called the abstract operators and expounded systematically the theory
of abstract operators. By combining abstract operators with the Laplace
transform, we can apply the Laplace transform to any $n+1$ dimensional linear
higher-order partial differential equations $P(\partial_x,\partial_t)u=f(x,t)$
directly, without using the Fourier transform. By making introduction of
abstract operators $G(\partial_x,t):=\mathcal{L}^{-1}[1/P(\partial_x,s)]$, the
analytic solutions of initial value problems are expressed in these abstract
operators, including the multiple inhomogeneous wave equation associated with
the shifted Laplace-Beltrami operator on real hyperbolic spaces. By writing
abstract operators in this class into integral forms, the solutions in operator
form are represented into integral forms. Thus the analytic solutions of Cauchy
problem for the multiple wave equation on $\mathbb{R}^n$ can be represented in
the integrations of some given functions, without using the traditional Fourier
transform technique. As a further application, we study the solvability of
initial-boundary value problem for the linear higher-order partial differential
equations and deduce new distinguishable method associated with the
second-order linear self-adjoint elliptic operators.
