Necessary and Sufficient Conditions for Associated Pairs in Quaternionic Analysis

Abstract

This paper deals with the initial value problem of the type

$$ \frac{{\partial w}} {{\partial t}} = L\left( {t,x,w,\frac{{\partial w}} {{\partial x_i }}} \right) $$

(1)

$$ w(0,x) = \phi (x) $$

(2)

where t is the time, L is a linear differential operator of first order in Quaternionic Analysis and ϕ is a regular function taking values in the Quaternionic Algebra. The necessary and sufficient conditions on the coefficients of the operator L under which L is associated to the generalized Cauchy-Riemann operator of the Quaternionic Analysis are proved.

This criterion makes it possible to construct the operator L for which the initial problem (1),(2) is solvable for an arbitrary initial regular function ϕ and the solution is also regular for each t.