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- Title
Solution of Inhomogeneous Differential Equations with Polynomial Coefficients in Terms of the Green's Function, in Nonstandard Analysis.
- Authors
Morita, Tohru
- Abstract
Discussions are presented by Morita and Sato on the problem of obtaining the particular solution of an inhomogeneous differential equation with polynomial coefficients in terms of the Green's function. In a paper, the problem is treated in distribution theory, and in another paper, the formulation is given on the basis of nonstandard analysis, where fractional derivative of degree, which is a complex number added by an infinitesimal number, is used. In the present paper, a simple recipe based on nonstandard analysis, which is closely related with distribution theory, is presented, where in place of Heaviside's step function H (t) and Dirac's delta function δ (t) in distribution theory, functions H ϵ (t) : = 1 Γ (1 + ϵ) t ϵ H (t) and δ ϵ (t) : = d d t H ϵ (t) = 1 Γ (ϵ) t ϵ − 1 H (t) for a positive infinitesimal number ϵ , are used. As an example, it is applied to Kummer's differential equation.
- Subjects
FRACTIONAL differential equations; GREEN'S functions; FRACTIONAL calculus; HYPERGEOMETRIC functions; RIEMANN integral
- Publication
AppliedMath, 2022, Vol 2, Issue 3, p379
- ISSN
2673-9909
- Publication type
Article
- DOI
10.3390/appliedmath2030022