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Representation of solutions of second order one-dimensional model hyperbolic equations

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Abstract

We consider second order weakly hyperbolic operators. Some representation formulas are known (see [16]) for the fundamental solution of the initial value problem for the Tricomi-type equation

$$\partial _t^2u - {t^{2\ell }}\partial _x^2u = f\left( {t,x} \right)$$

. In this paper, we solve the initial value problem for

$$\partial _t^2u - {x^{2k}}\partial _x^2u = f\left( {t,x} \right)\;and\;\partial _t^2u - {e^{2kx}}\partial _x^2u = f\left( {t,x} \right)$$

. Furthermore, we solve the mixed initial boundary value problem for

$$\partial _t^2u - {t^{2\ell }}{x^{2k}}\partial _x^2u = 0$$

.

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Authors and Affiliations

  1. Department of Mathematics, University of Texas-Pan American, Edinburg, Texas, 78541-2999, USA

    Anahit Galstian

  2. Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki, 305-8571, Japan

    Tamotu Kinoshita

Authors
  1. Anahit Galstian
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  2. Tamotu Kinoshita
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Correspondence to Anahit Galstian.

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Galstian, A., Kinoshita, T. Representation of solutions of second order one-dimensional model hyperbolic equations. JAMA 130, 355–374 (2016). https://doi.org/10.1007/s11854-016-0040-x

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  • DOI: https://doi.org/10.1007/s11854-016-0040-x

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