Electron. J. Diff. Equ., Vol. 2015 (2015), No. 201, pp. 1-18.

Analytic solutions of a class of nonlinear partial differential equations

Eugenia N. Petropoulou

We study a class of nonlinear partial differential equations, which can be connected with wave-type equations and Laplace-type equations, by using a functional-analytic technique. We establish primarily the existence and uniqueness of bounded solutions in the two-dimensional Hardy-Lebesque space of analytic functions with independent variables lying in the open unit disc. However these results can be modified to expand the domain of definition. The proofs have a constructive character enabling the determination of concrete and easily verifiable conditions, and the determination of the coefficients appearing in the power series solution. Illustrative examples are given related to the sine-Gordon equation, the Klein-Gordon equation, and to equations with nonlinear terms of algebraic, exponential and logistic type.

Submitted April 29, 2015. Published August 4, 2015.
Math Subject Classifications: 35A01, 35A02, 35B99, 35C10, 35J60, 35L70.
Key Words: Analytic solution; series solution; bounded solution; wave-type PDE; Laplace-type PDE; PDE with mixed derivatives; sine-Gordon; Klein-Gordon.

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Eugenia N. Petropoulou
Department of Civil Engineering
University of Patras
26500 Patras, Greece
email: jenpetr@upatras.gr

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