Initial value problems with regular initial functions in quaternionic analysis

Abstract

In the present article, we consider the initial value problems (IVPs) of type

where t is the time variable and L is a linear partial differential operator of the first order acting with respect to space like variables and taking value in the quaternion algebra. We formulate conditions on the coefficients of the operator L under which L is associated with the Cauchy–Riemann operator in quaternionic analysis. Hence we can construct all linear operators for which the IVP with an arbitrary regular function ϕ is always solvable.

Keywords:

• quaternionic analysis
• associated operator
• Cauchy–Riemann operator
• initial value problem

AMS Subject Classifications::

• 35F10
• 30G35
• 35A10