# Differential-symbol method of constructing the quasipolynomial solutions of a two-point problem for a partial differential equation

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## Abstract

We studied the solvability of a problem with local inhomogeneous conditions two-point in time for a homogeneous differential equation which is second-order in time and has generally the infinite order in spatial variables in the case where the set of zeros of the characteristic determinant of the problem is not empty and does not coincide with ℂ^{s}*:* The existence of a solution of the problem under the condition that the right-hand sides of the two-point conditions are quasipolynomials is proved. A differential-symbol method of constructing a solution of the problem is proposed.

## Keywords

Quasipolynomial solutions differential-symbol method characteristic determinant of the problem two-point conditions## Preview

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## References

- 1.Ch. J. Vallée-Poussin, “Sur l’equation differentielle lineaire du second ordre. Determination d’une integrale par deux valeurs assignees. Extension aux equations d’ordre
*n*,”*J. Math. de Pura et Appl.*,**9**, No. 8, 125–144 (1929).zbMATHGoogle Scholar - 2.M. Picone, “Sui valori eccezionali di un parametro do cui dipend un equazione differentiale lineare ordinaria del secondo ordine,”
*Ann. Scuola Norm. Sup. Pisa*,**11**, 1–144 (1909).Google Scholar - 3.Ya. D. Tamarkin,
*On Some General Problems of the Theory of Ordinary Differential Equations and on the Expansion of Arbitrary Functions in Series*[in Russian], M.P. Frolova’s Typolith., Petrograd, 1917.Google Scholar - 4.B. I. Ptashnyk, “A Vallée-Poussin-type problem for hyperbolic equations with constant coefficients,”
*Dokl. AN UkrRSR*, No. 10, 1254–1257 (1966).Google Scholar - 5.B. I. Ptashnik,
*Ill-Posed Boundary Problems for Partial Differential Equations*[in Russian], Naukova Dumka, Kiev, 1984.Google Scholar - 6.B. I. Ptashnyk, I. Ya. Kmit’, V. S. Il’kiv, and V. M. Polishchuk,
*Nonlocal Boundary-Value Problems for Partial Differential Equations*[in Ukrainian], Naukova Dumka, Kiev, 2002.Google Scholar - 7.B. I. Ptashnyk and M. M. Symotyuk, “A multipoint problem with multiple nodes for partial differential equations with constant coefficients,”
*Ukr. Mat. Zh*.,**55**, No. 3, 400–413 (2003).CrossRefzbMATHGoogle Scholar - 8.T. Kiguradze, “The Vallée-Poussin problem for higher order nonlinear hyperbolic equations,”
*Comput. Math. Applic.*,**59**, Issue 2, 994-1002 (2010).Google Scholar - 9.V. M. Borok, “Classes of uniqueness of a solution of a boundary-value problem in the infinite layer,”
*Dokl. AN SSSR*,**183**, No. 5, 995–998 (1968).MathSciNetGoogle Scholar - 10.V. M. Borok and M. A. Perel’man, “On the classes of uniqueness of a solution of a multipoint boundary-value problem in the infinite layer,”
*Izv. Vyssh. Ucheb. Zav. Mat.*,**8**, 29–34 (1973).Google Scholar - 11.I. L. Vilents’, “Classes of uniqueness of the solution of a general boundary-value problem in a layer for systems of linear partial differential equations,”
*Dokl. AN UkrRSR, Ser. A*,**3**, 195–197 (1974).Google Scholar - 12.L. V. Fardigola, “Well-posed problems in a layer with differential operators in a boundary condition,”
*Ukr. Mat. Zh.*,**44**, No. 8, 1083–1090 (1992).Google Scholar - 13.L. V. Fardigola, “Nonlocal two-point boundary-value problems in a layer with differential operators in a boundary condition,”
*Ukr. Mat. Zh.*,**47**, No. 8, 1122–1128 (1995).MathSciNetCrossRefzbMATHGoogle Scholar - 14.W. K. Hayman and Z. G. Shanidze, “Polynomial solutions of partial differential equations,”
*Meth. Applic. Analysis*,**6**, No. 1, 97–108 (1999).MathSciNetCrossRefzbMATHGoogle Scholar - 15.G. N. Hile and A. Stanoyevitch, “Heat polynomial analogues for equations with higher order time derivatives,”
*J. Math. Anal. Appl.*,**295**, 595–610 (2004).MathSciNetCrossRefzbMATHGoogle Scholar - 16.P. Pedersen, “A basis for polynomial solutions for systems of linear constant coefficient PDE’s,”
*Adv. Math.*,**117**, 157–163 (1996).MathSciNetCrossRefzbMATHGoogle Scholar - 17.O. Malanchuk and Z. Nytrebych, “Homogeneous two-point problem for PDE of the second order in time variable and infinite order in spatial variables,”
*Open Math.*, No. 15, 101–110 (2017).Google Scholar - 18.Z. Nytrebych, O. Malanchuk, V. Il’kiv, and P. Pukach, “Homogeneous problem with two-point in time conditions for some equations of mathematical physics,”
*Azerb. J. of Math*.,**7**, No. 2, 176–191 (2017).zbMATHGoogle Scholar - 19.Z. Nytrebych, V. Il’kiv, P. Pukach, and O. Malanchuk, “On nontrivial solutions of homogeneous Dirichlet problem for partial differential equations in a layer,”
*Krag. J. of Math.*,**42**, No. 2, 193–207 (2018).MathSciNetCrossRefGoogle Scholar - 20.Z. Nytrebych, O. Malanchuk, V. Il’kiv, and P. Pukach, “On the solvability of two-point in time problem for PDE,”
*Italian J. of Pure and Appl. Math.*, No. 38, 715–726 (2017).Google Scholar - 21.P. I. Kalenyuk and Z. M. Nytrebych,
*A Generalized Scheme of Separation of Variables. The Differential-Symbol Method*[in Ukrainian], Univ. “L’vivs’ka Politekhnika”, Lviv, 2002.Google Scholar - 22.P. I. Kalenyuk, I. V. Kohut, and Z. M. Nytrebych, “Problem with integral condition for partial differential equation of the first order with respect to time,”
*J. Math. Sci.*,**181**, Issue 3, 293–304 (2012).Google Scholar - 23.Z. M. Nitrebich, “An operator method of solving the Cauchy problem for a homogeneous system of partial differential equations,”
*J. Math. Sci.*,**81**, Issue 6, 3034–3038 (1996).Google Scholar - 24.Z. Nytrebych and O. Malanchuk, “The differential-symbol method of solving the two-point problem with respect to time for a partial differential equation,”
*J. Math. Sci.*,**224**, Issue 4, 541–554 (2017).Google Scholar - 25.A. Kampf, D. M. Jackson, and A. H. Morales, “New Dirac delta function based methods with applications to perturbative expansions in quantum field theory,”
*J. Phys. Math. Theor.*,**47**, No. 41, 415204 (2014).Google Scholar - 26.A. Kampf, D. M. Jackson, and A. H. Morales, “How to (path) integrate by differentiating,”
*J. Phys.: Conf. Ser.*,**626**, No. 626012015 (2015).Google Scholar

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