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Volume 9
Issue 16
10.3390/math9161944
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Article

Solution of Inhomogeneous Fractional Differential Equations with Polynomial Coefficients in Terms of the Green’s Function, in Nonstandard Analysis

by
Tohru Morita
1,* and
Ken-ichi Sato
2
1
Graduate School of Information Sciences, Tohoku University, Sendai 980-8577, Japan
2
Kurume Library on Mathematics, Koriyama 963-8846, Japan
*
Author to whom correspondence should be addressed.
Mathematics 2021, 9(16), 1944; https://doi.org/10.3390/math9161944
Submission received: 13 July 2021 / Revised: 7 August 2021 / Accepted: 10 August 2021 / Published: 15 August 2021
Versions Notes

Abstract

Discussions are presented by Morita and Sato in Mathematics 2017; 5, 62: 1–24, on the problem of obtaining the particular solution of an inhomogeneous ordinary differential equation with polynomial coefficients in terms of the Green’s function, in the framework of distribution theory. In the present paper, a compact recipe in nonstandard analysis is presented, which is applicable to an inhomogeneous ordinary and also fractional differential equation with polynomial coefficients. The recipe consists of three theorems, each of which provides the particular solution of a differential equation for an inhomogeneous term, satisfying one of three conditions. The detailed derivation of the applications of these theorems is given for a simple fractional differential equation and an ordinary differential equation.
Keywords: Green’s function; fractional differential equations with polynomial coefficients; Kim and O’s differential equation; nonstandard analysis; distribution theory; operational calculus Green’s function; fractional differential equations with polynomial coefficients; Kim and O’s differential equation; nonstandard analysis; distribution theory; operational calculus

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MDPI and ACS Style

Morita, T.; Sato, K.-i. Solution of Inhomogeneous Fractional Differential Equations with Polynomial Coefficients in Terms of the Green’s Function, in Nonstandard Analysis. Mathematics 2021, 9, 1944. https://doi.org/10.3390/math9161944

AMA Style

Morita T, Sato K-i. Solution of Inhomogeneous Fractional Differential Equations with Polynomial Coefficients in Terms of the Green’s Function, in Nonstandard Analysis. Mathematics. 2021; 9(16):1944. https://doi.org/10.3390/math9161944

Chicago/Turabian Style

Morita, Tohru, and Ken-ichi Sato. 2021. "Solution of Inhomogeneous Fractional Differential Equations with Polynomial Coefficients in Terms of the Green’s Function, in Nonstandard Analysis" Mathematics 9, no. 16: 1944. https://doi.org/10.3390/math9161944

APA Style

Morita, T., & Sato, K.-i. (2021). Solution of Inhomogeneous Fractional Differential Equations with Polynomial Coefficients in Terms of the Green’s Function, in Nonstandard Analysis. Mathematics, 9(16), 1944. https://doi.org/10.3390/math9161944

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