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Zeitschrift für Analysis und ihre Anwendungen


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Volume 20, Issue 2, 2001, pp. 513–524
DOI: 10.4171/ZAA/1028

Published online: 2001-06-30

The Generalized Riemann Problem of Linear Conjugation for Non-Homogeneous Polyanalytic Equations of Order $n$ in $W_{n,p}(D)$

Ali Seif Mshimba[1]

(1) University of Dar es Salaam, Tanzania

We consider a non-homogeneous polyanalytic partial differential equation of order $n$ in a simply-connected domain $D$ with smooth boundary $\partial D$ in the complex plane $\mathbb C$. Initially we transform the given equation into an equivalent system of integro-differential equations and then find the general solution of the former in $W_{n,p} (D)$. Next we pose and prove the solvability of a generalized Riemann problem of linear conjugation to the differential equation. This is effected by first reducing the Riemann problem to a corresponding one for a polyanalytic function. The latter is solved by first transforming it into $n$ classical Riemann problems of linear conjugation for $n$ holomorphic functions expressed in terms of the analytic functions which define the polyanalytic function. The solution of the classical Riemann problem is available in the literature.

Keywords: Polyanalytic functions, generalized Cauchy-Pompeiu integral operators of higher order, Riemann problem of linear conjugation

Mshimba Ali Seif: The Generalized Riemann Problem of Linear Conjugation for Non-Homogeneous Polyanalytic Equations of Order $n$ in $W_{n,p}(D)$. Z. Anal. Anwend. 20 (2001), 513-524. doi: 10.4171/ZAA/1028