JournalszaaVol. 18, No. 3pp. 611–624

The Generalized Riemann-Hilbert Boundary Value Problem for Non-Homogeneous Polyanalytic Differential Equation of Order nn in the Sobolev Space Wn,p(D)W_{n,p}(D)

  • Ali Seif Mshimba

    University of Dar es Salaam, Tanzania
The Generalized Riemann-Hilbert Boundary Value Problem for Non-Homogeneous Polyanalytic Differential Equation of Order $n$ in the Sobolev Space $W_{n,p}(D)$ cover
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Abstract

Given is a nonlinear non-homogeneous polyanalytic differential equation of order nn in a simply-connected domain DD in the complex plane. Initially we prove (under certain conditions) the existence of its general solution in Wn,p(D)W_{n,p}(D) by first transforming it into a system of integro-differential equations. Next we prove the solvability of a generalized Riemann-Hilbert problem for the differential equation. This is effected by first reducing the boundary value problem posed to a corresponding one for a polyanalytic function. The latter is then transformed into nn classical Riemann-Hilbert problems for holomorphic functions, whose solutions are known in the literature.

Cite this article

Ali Seif Mshimba, The Generalized Riemann-Hilbert Boundary Value Problem for Non-Homogeneous Polyanalytic Differential Equation of Order nn in the Sobolev Space Wn,p(D)W_{n,p}(D). Z. Anal. Anwend. 18 (1999), no. 3, pp. 611–624

DOI 10.4171/ZAA/901