Solutions to Polynomial Generalized Bers–Vekua Equations in Clifford Analysis

摘要

In this paper, we mainly study polynomial generalized Vekua-type equation p(D)w = 0and polynomial generalized Bers–Vekua equation p(D)w = 0defined in Ω∩R~(n+1) where D and D mean generalized Vekua-type operator and generalized Bers–Vekua operator, respectively. Using Clifford algebra, we obtain the Fischer-type decomposition theorems for the solutions to these equations including (D - λ)~k w =0, -D - λ)~k w= 0 (k ∈ N)with complex parameter λ as special cases,which derive the Almansi-type decomposition theorems for iterated generalized Bers–Vekua equation and polynomial generalized Cauchy–Riemann equation defined in Ω∩R~(n+1). Making use of the decomposition theorems we give the solutions to polynomial generalized Bers–Vekua equation defined in Ω∩R~(n+1)under some conditions. Furthermore we discuss inhomogeneous polynomial generalized Bers–Vekua equationp(D)w = v defined in Ω∩R~(n+1),and develop the structure of the solutions to inhomogeneous polynomial generalized Bers–Vekua equation p(D)w = v defined in Ω∩R~(n+1).