Solution of Inhomogeneous Differential Equations with Polynomial Coefficients in Terms of the Green’s Function, in Nonstandard Analysis
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 i...
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MDPI AG
2022-07-01
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| Online Access: | https://www.mdpi.com/2673-9909/2/3/22 |
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| author | Tohru Morita |
| author_facet | Tohru Morita |
| author_sort | Tohru Morita |
| collection | DOAJ |
| description | 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> and Dirac’s delta function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>δ</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> in distribution theory, functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mi>ϵ</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>:</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mi mathvariant="sans-serif">Γ</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi>ϵ</mi><mo>)</mo></mrow></mfrac><msup><mi>t</mi><mi>ϵ</mi></msup><mi>H</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>δ</mi><mi>ϵ</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>:</mo><mo>=</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><msub><mi>H</mi><mi>ϵ</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mrow><mi mathvariant="sans-serif">Γ</mi><mo>(</mo><mi>ϵ</mi><mo>)</mo></mrow></mfrac><msup><mi>t</mi><mrow><mi>ϵ</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>H</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for a positive infinitesimal number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϵ</mi></semantics></math></inline-formula>, are used. As an example, it is applied to Kummer’s differential equation. |
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| institution | Directory Open Access Journal |
| issn | 2673-9909 |
| language | English |
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| publishDate | 2022-07-01 |
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| spelling | doaj.art-6318ad8a3d3345f7bb837a812703a81a2023-11-24T07:33:05ZengMDPI AGAppliedMath2673-99092022-07-012337939210.3390/appliedmath2030022Solution of Inhomogeneous Differential Equations with Polynomial Coefficients in Terms of the Green’s Function, in Nonstandard AnalysisTohru Morita0Graduate School of Information Sciences, Tohoku University, Sendai 980-8577, JapanDiscussions 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> and Dirac’s delta function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>δ</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> in distribution theory, functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mi>ϵ</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>:</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mi mathvariant="sans-serif">Γ</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi>ϵ</mi><mo>)</mo></mrow></mfrac><msup><mi>t</mi><mi>ϵ</mi></msup><mi>H</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>δ</mi><mi>ϵ</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>:</mo><mo>=</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><msub><mi>H</mi><mi>ϵ</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mrow><mi mathvariant="sans-serif">Γ</mi><mo>(</mo><mi>ϵ</mi><mo>)</mo></mrow></mfrac><msup><mi>t</mi><mrow><mi>ϵ</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>H</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for a positive infinitesimal number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϵ</mi></semantics></math></inline-formula>, are used. As an example, it is applied to Kummer’s differential equation.https://www.mdpi.com/2673-9909/2/3/22Green’s functiondifferential equations with polynomial coefficientsnonstandard analysisdistribution theory |
| spellingShingle | Tohru Morita Solution of Inhomogeneous Differential Equations with Polynomial Coefficients in Terms of the Green’s Function, in Nonstandard Analysis AppliedMath Green’s function differential equations with polynomial coefficients nonstandard analysis distribution theory |
| title | Solution of Inhomogeneous Differential Equations with Polynomial Coefficients in Terms of the Green’s Function, in Nonstandard Analysis |
| title_full | Solution of Inhomogeneous Differential Equations with Polynomial Coefficients in Terms of the Green’s Function, in Nonstandard Analysis |
| title_fullStr | Solution of Inhomogeneous Differential Equations with Polynomial Coefficients in Terms of the Green’s Function, in Nonstandard Analysis |
| title_full_unstemmed | Solution of Inhomogeneous Differential Equations with Polynomial Coefficients in Terms of the Green’s Function, in Nonstandard Analysis |
| title_short | Solution of Inhomogeneous Differential Equations with Polynomial Coefficients in Terms of the Green’s Function, in Nonstandard Analysis |
| title_sort | solution of inhomogeneous differential equations with polynomial coefficients in terms of the green s function in nonstandard analysis |
| topic | Green’s function differential equations with polynomial coefficients nonstandard analysis distribution theory |
| url | https://www.mdpi.com/2673-9909/2/3/22 |
| work_keys_str_mv | AT tohrumorita solutionofinhomogeneousdifferentialequationswithpolynomialcoefficientsintermsofthegreensfunctioninnonstandardanalysis |