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Generalized Solutions of the Third-Order Cauchy-Euler Equation in the Space of Right-Sided Distributions via Laplace Transform

1
Department of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand
2
Department of Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(4), 376; https://doi.org/10.3390/math7040376
Received: 3 April 2019 / Revised: 12 April 2019 / Accepted: 17 April 2019 / Published: 25 April 2019
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Abstract

Using the Laplace transform technique, we investigate the generalized solutions of the third-order Cauchy-Euler equation of the form t 3 y ( t ) + a t 2 y ( t ) + b y ( t ) + c y ( t ) = 0 , where a , b , and c Z and t R . We find that the types of solutions in the space of right-sided distributions, either distributional solutions or weak solutions, depend on the values of a, b, and c. At the end of the paper, we give some examples showing the types of solutions. Our work improves the result of Kananthai (Distribution solutions of the third order Euler equation. Southeast Asian Bull. Math. 1999, 23, 627–631). View Full-Text
Keywords: Cauchy-Euler equation; Dirac delta function; distributional solutions; Laplace transform; weak solutions Cauchy-Euler equation; Dirac delta function; distributional solutions; Laplace transform; weak solutions
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Jhanthanam, S.; Nonlaopon, K.; Orankitjaroen, S. Generalized Solutions of the Third-Order Cauchy-Euler Equation in the Space of Right-Sided Distributions via Laplace Transform. Mathematics 2019, 7, 376.

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