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

Seksan Jhanthanam, Kamsing Nonlaopon and Somsak Orankitjaroen
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Seksan Jhanthanam: Department of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand
Kamsing Nonlaopon: Department of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand
Somsak Orankitjaroen: Department of Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand

Mathematics, 2019, vol. 7, issue 4, 1-12

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).

Keywords: Cauchy-Euler equation; Dirac delta function; distributional solutions; Laplace transform; weak solutions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
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