  In this paper, we propose a procedure for constructing an infinite number of
families of solutions of given linear differential equations with partial
derivatives with constant coefficients. We use monogenic functions that are
defined on some sequences of commutative associative algebras over the field of
complex numbers. To achieve this goal, we first study the solutions of the
so-called characteristic equation on a given sequence of algebras. Further, we
investigate monogenic functions on the sequence of algebras and study their
relation with solutions of partial deferential equations. The proposed method
is used to construct solutions of some equations of mathematical physics. In
particular, for the three-dimensional Laplace equation and the wave equation,
for the equation of transverse oscillations of the elastic rod and the
conjugate equation, a generalized biharmonic equation and the two-dimensional
Helmholtz equation. We note that this method yields all analytic solutions of
the two-dimensional Laplace equation and the two-dimensional biharmonic
equation (Goursat formula).
