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

Eugenia N.  Petropoulou; Panayiotis D.  Siafarikas

doi:10.3934/proc.2015.0923

Article Contents

2015, Volume 2015: 923-935. Doi: 10.3934/proc.2015.0923 open access

This issue Previous Article An in-host model of HIV incorporating latent infection and viral mutation Next Article Optimal design of sensors for a damped wave equation

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

Eugenia N. Petropoulou^1, and
Panayiotis D. Siafarikas²

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

Received: July 31, 2014

Revised: November 30, 2014

Published: October 2015

Abstract Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 35A01, 35A02; Secondary: 35B08, 35C10, 35G05, 35G20, 35Q53.

Citation:

Related Papers

Cited by

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), 47-78.
[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), 1167-1230.
[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), 419-434.
[4]	C. J. Earle and R. S. Hamilton, A fixed point theorem for holomorphic mappings in Global Analysis (Proc. Sympos. Pure Math., Vol.XVI, Berkeley, California 1968), Amer. Math. Soc., Providence R.I., (1970), 61-65.
[5]	L. C. Evans, Partial differential equations, $2^{nd}$ edition, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence R.I., 2010.
[6]	N. Hayashi and K. Kato, Global existence of small analytic solutions to Schrödinger equations with quadratic nonlinearity, Comm. Partial Differential Equations, 22 (1997), 773-798.
[7]	A. A. Himonas and G. Petronilho, Analytic well-posedness of periodic gKdV, J. Differential Equations, 253 (2012), 3101-3112.
[8]	E. K. Ifantis, Solution of the Schrödinger equation in the Hardy-Lebesgue space, J. Mathematical Phys., 12 (1971), 1961-1965.
[9]	E. K. Ifantis, Analytic solutions for nonlinear differential equations, J. Math. Anal. Appl., 124 (1987), 339-380.
[10]	E. K. Ifantis, Global analytic solutions of the radial nonlinear wave equation, J. Math. Anal. Appl., 124 (1987), 381-410.
[11]	J. Kajiwara, Holomorphic solutions of a partial differential equation of mixed type, Math. Balkanica, 2 (1972), 76-83.
[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), 85-95.
[13]	E. N. Petropoulou and P. D. Siafarikas, Analytic solutions of some non-linear ordinary differential equations, Dynam. Systems Appl. 13 (2004), 283-315.
[14]	E. N. Petropoulou and P. D. Siafarikas, Polynomial solutions of linear partial differential equations, Commun. Pure Appl. Anal. 8 (2009), 1053-1065.
[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), 279-296.
[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), 613-631.
[17]	E. N. Petropoulou and E. E. Tzirtzilakis, On the logistic equation in the complex plane, Numer. Funct. Anal. Optim. 34 (2013), 770-790.
[18]	I. G. Petrovsky, Lecture on partial differential equations. Translated from the Russian by A. Shenitzer, Intersicence Publishers Inc, New York, 1957.
[19]	A. Vourdas, Analytic representations in the unit disc and applications to phase states and squeezing, Phys. Rev. A 45 (1992), 1943-1950.
[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), 83-87.
[21]	G. Zampieri, Analytic solutions of P.D.E.'s. Ann. Univ. Ferrara-Sez. VII-Sc. Mat. XLV (1999), 365-372.