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Conditions for the Existence of Solutions of a Periodic Boundary-Value Problem for an Inhomogeneous Linear Hyperbolic Equation of the Second Order. I

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Abstract

We consider the periodic boundary-value problem u tt − u xx = g(x, t), u(0, t) = u(π, t) = 0, u(x, t + ω) = u(x, t). By representing a solution of this problem in the form u(x, t) = u 0(x, t) + ũ(x, t), where u 0(x, t) is a solution of the corresponding homogeneous problem and ũ(x, t) is the exact solution of the inhomogeneous equation such that ũ(x, t + ω) u x = ũ(x, t), we obtain conditions for the solvability of the inhomogeneous periodic boundary-value problem for certain values of the period ω. We show that the relation obtained for a solution includes known results established earlier.

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REFERENCES

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Authors and Affiliations

  1. Institute of Mathematics, Ukrainian Academy of Sciences, Kyiv, Ukraine

    Yu. A. Mitropol'skii

  2. Ternopil' Academy of National Economy, Ternopil', Ukraine

    S. H. Khoma-Mohyl's'ka

Authors
  1. Yu. A. Mitropol'skii
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  2. S. H. Khoma-Mohyl's'ka
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Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 7, pp. 912–921, July, 2005.

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Mitropol'skii, Y.A., Khoma-Mohyl's'ka, S.H. Conditions for the Existence of Solutions of a Periodic Boundary-Value Problem for an Inhomogeneous Linear Hyperbolic Equation of the Second Order. I. Ukr Math J 57, 1077–1088 (2005). https://doi.org/10.1007/s11253-005-0249-9

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  • DOI: https://doi.org/10.1007/s11253-005-0249-9

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