Solution of initial value problems for first order systems using matrix notation

Abstract

Depending on the structure of the underlying Clifford Algebra, the generalized Cauchy—Riemann equation of Clifford Analysis turns out to be elliptic, parabolic or hyperbolic (See Mrs. E. Obolashvili's paper [13] and the book [14] of the same author where one can find further references to related topics; cf. also the books [1] of H. Begehr and R. P. Gilbert, [6] of R. P. Gilbert And J. L. Buchanan, [2] of F. Brackx, R. Delanghe and F. Sommen, 141 of R. Delanghe, F. Sommen and V. Soucek, [7] of K. GUrlebeck and W. Spröbig, and [ll] of V. V. Kravchenko and M. V. Shapiro. Concerning an elementary approach to Clifford Analysis see the paper [16] in the Proceedings [17])

In the hyperbolic case the real-valued components of a solution of ∂u= 0 are solutions of the wave equation, and thus initial value problems for ∂u= 0 can be reduced to initial value problems for the wave equation. In the already quoted book [14] an explicit representation of the solution of the wave equation is used for solving explicitly the initial value problem for ∂u= 0 in the hyperbolic case. Further, with use of an analogous explicit representation of the initial value problem for the Klein-Gordon equation, in the same book [14] the initial value problem for the differential equation ∂u=+ūh = 0 for h-regular functions (where h is constant and is a certain conjugate to u) is solved explicitly.

Using a matrix notation, the present paper generalizes the results of [14] concerning the hyperbolic case.

Keywords:

• Reduction to second order equations
• explicit solution of systems with constant coefficients
• hyperbolic systems in Clifford analysis

AMS:

• 35L40
• 35L45