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

Author

Listed:
  • 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)

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).

Suggested Citation

  • Seksan Jhanthanam & Kamsing Nonlaopon & Somsak Orankitjaroen, 2019. "Generalized Solutions of the Third-Order Cauchy-Euler Equation in the Space of Right-Sided Distributions via Laplace Transform," Mathematics, MDPI, vol. 7(4), pages 1-12, April.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:4:p:376-:d:225871
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