# American Institute of Mathematical Sciences

2015, 2015(special): 923-935. doi: 10.3934/proc.2015.0923

## A functional-analytic technique for the study of analytic solutions of PDEs

 1 Department of Civil Engineering, University of Patras, 26500 Patras, Greece 2 Department of Mathematics, University of Patras, 26500 Patras, Greece

Received  August 2014 Revised  December 2014 Published  November 2015

A functional-analytic method is used to study the existence and the uniqueness of bounded, analytic and entire complex solutions of partial differential equations. As a benchmark problem, this method is applied to the nonlinear Benjamin--Bona--Mahony equation and the associated to this, linear equation. The predicted solutions are in power series form and two concrete examples are given for specific initial conditions.
Citation: Eugenia N. Petropoulou, Panayiotis D. Siafarikas. A functional-analytic technique for the study of analytic solutions of PDEs. Conference Publications, 2015, 2015 (special) : 923-935. doi: 10.3934/proc.2015.0923
##### References:
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##### References:
 [1] T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems,, Philos. Trans. Roy. Soc. London Ser. A, 272 (1972), 47.   Google Scholar [2] G. Caciotta and F. Nicoló, Local and global analytic solutions for a class of characteristic problems of the Einstein vacuum equations in the "double null foliation gauge",, Ann. Henri Poincare, 13 (2012), 1167.   Google Scholar [3] G. M. Coclite, F. Gargano and V. Sciacca, Analytic solutions and singularity formation for the Peakon $b$-family equations,, Acta Appl. Math., 122 (2012), 419.   Google Scholar [4] C. J. Earle and R. S. Hamilton, A fixed point theorem for holomorphic mappings, in Global Analysis (Proc. Sympos. Pure Math., (1970), 61.   Google Scholar [5] L. C. Evans, Partial differential equations,, $2^{nd}$ edition, (2010).   Google Scholar [6] N. Hayashi and K. Kato, Global existence of small analytic solutions to Schrödinger equations with quadratic nonlinearity,, Comm. Partial Differential Equations, 22 (1997), 773.   Google Scholar [7] A. A. Himonas and G. Petronilho, Analytic well-posedness of periodic gKdV,, J. Differential Equations, 253 (2012), 3101.   Google Scholar [8] E. K. Ifantis, Solution of the Schrödinger equation in the Hardy-Lebesgue space,, J. Mathematical Phys., 12 (1971), 1961.   Google Scholar [9] E. K. Ifantis, Analytic solutions for nonlinear differential equations,, J. Math. Anal. Appl., 124 (1987), 339.   Google Scholar [10] E. K. Ifantis, Global analytic solutions of the radial nonlinear wave equation,, J. Math. Anal. Appl., 124 (1987), 381.   Google Scholar [11] J. Kajiwara, Holomorphic solutions of a partial differential equation of mixed type,, Math. Balkanica, 2 (1972), 76.   Google Scholar [12] T. Kusano and S. Oharu, Bounded entire solutions of second order semilinear elliptic equations with application to a parabolic initial value problem,, Indiana Univ. Math. J., 34 (1985), 85.   Google Scholar [13] E. N. Petropoulou and P. D. Siafarikas, Analytic solutions of some non-linear ordinary differential equations,, Dynam. Systems Appl. 13 (2004), 13 (2004), 283.   Google Scholar [14] E. N. Petropoulou and P. D. Siafarikas, Polynomial solutions of linear partial differential equations,, Commun. Pure Appl. Anal. 8 (2009), 8 (2009), 1053.   Google Scholar [15] E. N. Petropoulou, P. D. Siafarikas and E. E. Tzirtzilakis, A "discretization" technique for the solution of ODEs,, J. Math. Anal. Appl. 331 (2007), 331 (2007), 279.   Google Scholar [16] E. N. Petropoulou, P. D. Siafarikas and E. E. Tzirtzilakis, A "discretization" technique for the solution of ODEs II,, Numer. Funct. Anal. Optim. 30 (2009), 30 (2009), 613.   Google Scholar [17] E. N. Petropoulou and E. E. Tzirtzilakis, On the logistic equation in the complex plane,, Numer. Funct. Anal. Optim. 34 (2013), 34 (2013), 770.   Google Scholar [18] I. G. Petrovsky, Lecture on partial differential equations., Translated from the Russian by A. Shenitzer, (1957).   Google Scholar [19] A. Vourdas, Analytic representations in the unit disc and applications to phase states and squeezing,, Phys. Rev. A 45 (1992), 45 (1992), 1943.   Google Scholar [20] G. Zampieri, A sufficient condition for existence of real analytic solutions of P.D.E. with constant coefficients, in open sets of $\mathbbR^{2}$,, Rend. Sem. Mat. Univ. Padova 63 (1980), 63 (1980), 83.   Google Scholar [21] G. Zampieri, Analytic solutions of P.D.E.'s., Ann. Univ. Ferrara-Sez. VII-Sc. Mat. XLV (1999), XLV (1999), 365.   Google Scholar
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