Hypercomplex method for solving linear PDEs

Abstract

Algebraic-analytic approach to constructing solutions for given partial differential equations were investigated in many papers. In particular, in papers [1-14]. It involves solving two problems. Problem (P 1) is to describe all the sets of vectors \( e_1, e_2, \ldots, e_d \), which satisfy the characteristic equation (or specify the procedure by which they can be found). And the problem (P 2) is to describe all the components of monogenic (i.e., continuous and differentiable in sense Gateaux) functions. In particular, for the equation (4) we must describe the components of the function \( \Phi(\zeta) = \exp \zeta \). Note that in the papers [15, 16] a constructive description of all analytic functions with values is obtained in an arbitrary finite-dimensional commutative associative algebra over the field \(\mathbb{C}\). The Theorem 5.1 of the paper [17] states that it is enough to limit the study of monogenic functions in algebras with the basis of \( \{1, \eta_1, \eta_2, \ldots, \eta_{n-1} \} \), where \( \eta_1, \eta_2, \ldots, \eta_{n-1} \) are nilpotents. In addition, in [18] it is showed that in each algebra with a basis of the form \( \{1, \eta_1, \eta_2, \ldots, \eta_{n-1} \} \) the equation (3) has solutions. That is, the problems (P 1) and (P 2) are completely solved on the classes of commutative associative algebras with the basis \( \{1, \eta_1, \eta_2, \ldots, \eta_{n-1}\} \). It is worth noting that in a finite-dimensional algebra a decomposition of monogenic functions has a finite number of components, and therefore, it generates a finite number of solutions of a given partial differential equations. 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).