American Institute of Mathematical Sciences

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

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

Eugenia N. Petropoulou 1, and  Panayiotis D. Siafarikas 2,

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.
Keywords: Benjamin--Bona--Mahony equation., series solutions, Bounded solutions, entire solutions, analytic solutions.
Mathematics Subject Classification: Primary: 35A01, 35A02; Secondary: 35B08, 35C10, 35G05, 35G20, 35Q5.
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:
[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. 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), 1167-1230. 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), 419-434. Google Scholar

[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. Google Scholar

[5]

L. C. Evans, Partial differential equations, $2^{nd}$ edition, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence R.I., 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), 773-798. Google Scholar

[7]

A. A. Himonas and G. Petronilho, Analytic well-posedness of periodic gKdV, J. Differential Equations, 253 (2012), 3101-3112. Google Scholar

[8]

E. K. Ifantis, Solution of the Schrödinger equation in the Hardy-Lebesgue space, J. Mathematical Phys., 12 (1971), 1961-1965. Google Scholar

[9]

E. K. Ifantis, Analytic solutions for nonlinear differential equations, J. Math. Anal. Appl., 124 (1987), 339-380. Google Scholar

[10]

E. K. Ifantis, Global analytic solutions of the radial nonlinear wave equation, J. Math. Anal. Appl., 124 (1987), 381-410. Google Scholar

[11]

J. Kajiwara, Holomorphic solutions of a partial differential equation of mixed type, Math. Balkanica, 2 (1972), 76-83. 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), 85-95. Google Scholar

[13]

E. N. Petropoulou and P. D. Siafarikas, Analytic solutions of some non-linear ordinary differential equations, Dynam. Systems Appl. 13 (2004), 283-315. Google Scholar

[14]

E. N. Petropoulou and P. D. Siafarikas, Polynomial solutions of linear partial differential equations, Commun. Pure Appl. Anal. 8 (2009), 1053-1065. 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), 279-296. 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), 613-631. Google Scholar

[17]

E. N. Petropoulou and E. E. Tzirtzilakis, On the logistic equation in the complex plane, Numer. Funct. Anal. Optim. 34 (2013), 770-790. Google Scholar

[18]

I. G. Petrovsky, Lecture on partial differential equations. Translated from the Russian by A. Shenitzer, Intersicence Publishers Inc, New York, 1957. Google Scholar

[19]

A. Vourdas, Analytic representations in the unit disc and applications to phase states and squeezing, Phys. Rev. A 45 (1992), 1943-1950. 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), 83-87. Google Scholar

[21]

G. Zampieri, Analytic solutions of P.D.E.'s. Ann. Univ. Ferrara-Sez. VII-Sc. Mat. XLV (1999), 365-372. Google Scholar

show all references

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. 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), 1167-1230. 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), 419-434. Google Scholar

[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. Google Scholar

[5]

L. C. Evans, Partial differential equations, $2^{nd}$ edition, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence R.I., 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), 773-798. Google Scholar

[7]

A. A. Himonas and G. Petronilho, Analytic well-posedness of periodic gKdV, J. Differential Equations, 253 (2012), 3101-3112. Google Scholar

[8]

E. K. Ifantis, Solution of the Schrödinger equation in the Hardy-Lebesgue space, J. Mathematical Phys., 12 (1971), 1961-1965. Google Scholar

[9]

E. K. Ifantis, Analytic solutions for nonlinear differential equations, J. Math. Anal. Appl., 124 (1987), 339-380. Google Scholar

[10]

E. K. Ifantis, Global analytic solutions of the radial nonlinear wave equation, J. Math. Anal. Appl., 124 (1987), 381-410. Google Scholar

[11]

J. Kajiwara, Holomorphic solutions of a partial differential equation of mixed type, Math. Balkanica, 2 (1972), 76-83. 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), 85-95. Google Scholar

[13]

E. N. Petropoulou and P. D. Siafarikas, Analytic solutions of some non-linear ordinary differential equations, Dynam. Systems Appl. 13 (2004), 283-315. Google Scholar

[14]

E. N. Petropoulou and P. D. Siafarikas, Polynomial solutions of linear partial differential equations, Commun. Pure Appl. Anal. 8 (2009), 1053-1065. 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), 279-296. 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), 613-631. Google Scholar

[17]

E. N. Petropoulou and E. E. Tzirtzilakis, On the logistic equation in the complex plane, Numer. Funct. Anal. Optim. 34 (2013), 770-790. Google Scholar

[18]

I. G. Petrovsky, Lecture on partial differential equations. Translated from the Russian by A. Shenitzer, Intersicence Publishers Inc, New York, 1957. Google Scholar

[19]

A. Vourdas, Analytic representations in the unit disc and applications to phase states and squeezing, Phys. Rev. A 45 (1992), 1943-1950. 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), 83-87. Google Scholar

[21]

G. Zampieri, Analytic solutions of P.D.E.'s. Ann. Univ. Ferrara-Sez. VII-Sc. Mat. XLV (1999), 365-372. Google Scholar

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