Finite Series of Distributional Solutions for Certain Linear Differential Equations

Waiyaworn, Nipon; Nonlaopon, Kamsing; Orankitjaroen, Somsak

doi:10.3390/axioms9040116

Open AccessArticle

Finite Series of Distributional Solutions for Certain Linear Differential Equations

by

Nipon Waiyaworn

¹,

Kamsing Nonlaopon

^1,*

and

Somsak Orankitjaroen

²

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.

Axioms 2020, 9(4), 116; https://doi.org/10.3390/axioms9040116

Submission received: 31 August 2020 / Revised: 30 September 2020 / Accepted: 2 October 2020 / Published: 13 October 2020

(This article belongs to the Special Issue Nonlinear Differential Equations and Dynamical Systems: Theory and Applications)

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Abstract

In this paper, we present the distributional solutions of the modified spherical Bessel differential equations

t^{2} y^{″} (t) + 2 t y^{'} (t) - [t^{2} + ν (ν + 1)] y (t) = 0

and the linear differential equations of the forms

t^{2} y^{″} (t) + 3 t y^{'} (t) - (t^{2} + ν^{2} - 1) y (t) = 0,

where

ν \in N \cup {0}

and

t \in R

. We find that the distributional solutions, in the form of a finite series of the Dirac delta function and its derivatives, depend on the values of

ν

. The results of several examples are also presented.

Keywords: Dirac delta function; distributional solution; Laplace transform; power series solution

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MDPI and ACS Style

Waiyaworn, N.; Nonlaopon, K.; Orankitjaroen, S. Finite Series of Distributional Solutions for Certain Linear Differential Equations. Axioms 2020, 9, 116. https://doi.org/10.3390/axioms9040116

AMA Style

Waiyaworn N, Nonlaopon K, Orankitjaroen S. Finite Series of Distributional Solutions for Certain Linear Differential Equations. Axioms. 2020; 9(4):116. https://doi.org/10.3390/axioms9040116

Chicago/Turabian Style

Waiyaworn, Nipon, Kamsing Nonlaopon, and Somsak Orankitjaroen. 2020. "Finite Series of Distributional Solutions for Certain Linear Differential Equations" Axioms 9, no. 4: 116. https://doi.org/10.3390/axioms9040116

APA Style

Waiyaworn, N., Nonlaopon, K., & Orankitjaroen, S. (2020). Finite Series of Distributional Solutions for Certain Linear Differential Equations. Axioms, 9(4), 116. https://doi.org/10.3390/axioms9040116

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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Finite Series of Distributional Solutions for Certain Linear Differential Equations

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