IJPAM: Volume 102, No. 1 (2015)

MODIFIED RIEMANN-HILBERT BOUNDARY VALUE
PROBLEM FOR NONLINEAR COMPLEX PARTIAL
DIFFERENTIAL EQUATION

N. Taghizadeh$^1$, H. Hoseini$^2$
$^{1,2}$Department of Mathematics
Faculty of Mathematical Sciences
University of Guilan
P.O. Box 1914, Rasht, IRAN


Abstract. In this paper we discuss on the existence and uniqueness solution of the modified Riemann-Hilbert boundary value problem in the form:

\begin{displaymath} \frac{\partial w}{\partial \bar{z}}=F(z,w,\frac{\partial w}{\partial z}),\quad z\in{D}, \end{displaymath} (1)


\begin{displaymath} Re(a+ib)w=g+\varphi \quad \text{on}\quad \partial D \end{displaymath} (2)

in the $C_{1,\alpha}(\bar{D})$, wherer $a$, $b$, $\varphi$ and $g$ are given Holder continuously differentiable real-valued function of a real parameter $t$ on $\partial{D}$. We shall assume that $ a^2+b^2=1$ everywhere on $\partial{D}$, and $\varphi$ is identically zero if $\chi \geq 0$ ($\chi$ is the index of the Riemann-Hilbert problem) and for $\chi < 0$

\begin{displaymath}\varphi (z)=\sum_{k=\chi+1}^{-\chi-1} h_k \omega^k(z),\quad\quad z\in \bar{D},\end{displaymath}

the coefficients $h_k$ are restricted to

\begin{displaymath}h_{-k}=\bar{h_k}\quad\quad \vert k \vert \leq -\chi-1,\end{displaymath}

and $\omega $ is the conformal map from $D$ into unit disc $\mathcal{D}$.

Received: January 9, 2014

AMS Subject Classification: 35D05, 35J55, 35j67

Key Words and Phrases: boundary value problem, Banach space, fixed point theorem, complex differential equation, holomorphic function

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DOI: 10.12732/ijpam.v102i1.1 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2015
Volume: 102
Issue: 1
Pages: 1 - 7


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