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Existence of Unique Solutions to the Telegraph Equation in Binary Reproducing Kernel Hilbert Spaces

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Abstract

We demonstrate the existence of a unique solution to a nonhomogeneous telegraph initial/boundary value problem on the unit square in an appropriate binary reproducing kernel Hilbert space which depends on the smoothness of the driver. Examples are given to illustrate the numerical effectiveness of the reproducing kernel method when properly applied and the aberrations which can occur when no solution exists in the space.

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Acknowledgements

The authors thank Professor Jiří Šremr for clarifying discussions, patient corrections, and illustrative counterexamples connected with real functions on \(\Omega \) which are absolutely continuous in the sense of Carathéodory. Furthermore, the authors express their gratitude to Professor Tynisbek Kalmenov for pointing out the seminal importance of obtaining sufficient conditions on the driver guaranteeing a solution in \(W_2^{(3,3)}(\Omega )\) to (1.1)–(1.2)–(1.3). Finally, the authors are grateful to Mr. Jabar Hassan for suggesting the form of the extension of f to F in the proof of Theorem  4.1.

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Authors and Affiliations

  1. Art and Science Faculty, Department of Mathematics, Siirt University, 56100, Siirt, Turkey

    Ali Akgül

  2. Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO, 65409-0020, USA

    Ali Akgül & David Grow

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  1. Ali Akgül
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  2. David Grow
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Correspondence to David Grow.

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Akgül, A., Grow, D. Existence of Unique Solutions to the Telegraph Equation in Binary Reproducing Kernel Hilbert Spaces. Differ Equ Dyn Syst 28, 715–744 (2020). https://doi.org/10.1007/s12591-019-00453-3

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