Electron. J. Diff. Eqns., Vol. 2003(2003), No. 33, pp. 1-25.

Analytic solution to a class of integro-differential equations

Xuming Xie

Abstract:
In this paper, we consider the integro-differential equation
$$ \epsilon^2 y''(x)+L(x){\cal H}(y)=N(\epsilon,x,y,{\cal H}(y)), $$
where ${cal H}(y)[x]=\frac{1}{\pi}(P)\int_{-\infty}^{\infty} \frac{y(t)}{t-x}dt$ is the Hilbert transform. The existence and uniqueness of analytic solution in appropriately chosen space is proved. Our method consists of extending the equation to an appropriately chosen region in the complex plane, then use the Contraction Mapping Theorem.

Submitted August 13, 2002. Published March 28, 2003.
Math Subject Classifications: 34A20, 45E05.
Key Words: Analytic solution, singular integro-differential equation.

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Xuming Xie
Department of Mathematical Sciences
University of Delaware
501 Ewing Hall
Newark, DE 19716, USA
email: xie@math.udel.edu

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