Finite Series of Distributional Solutions for Certain Linear Differential Equations

• Main Authors: Nipon Waiyaworn, Kamsing Nonlaopon, Somsak Orankitjaroen
• Format: Article
• Language: English
• Published: MDPI AG 2020-10-01
• Series: Axioms
• Subjects: Dirac delta function; distributional solution; Laplace transform; power series solution
• Online Access: https://www.mdpi.com/2075-1680/9/4/116

In this paper, we present the distributional solutions of the modified spherical Bessel differential equations <inline-formula><math display="inline"><semantics><mrow><msup><mi>t</mi><mn>2</mn></msup><msup><mi>y</mi><mrow><mo>′</mo><mo>′</mo></mrow></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>+</mo><mn>2</mn><mi>t</mi><msup><mi>y</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>−</mo><mrow><mo>[</mo><msup><mi>t</mi><mn>2</mn></msup><mo>+</mo><mi>ν</mi><mrow><mo>(</mo><mi>ν</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>]</mo></mrow><mi>y</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> and the linear differential equations of the forms <inline-formula><math display="inline"><semantics><mrow><msup><mi>t</mi><mn>2</mn></msup><msup><mi>y</mi><mrow><mo>′</mo><mo>′</mo></mrow></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>+</mo><mn>3</mn><mi>t</mi><msup><mi>y</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><msup><mi>t</mi><mn>2</mn></msup><mo>+</mo><msup><mi>ν</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mi>y</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math display="inline"><semantics><mrow><mi>ν</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi><mo>∪</mo><mo>{</mo><mn>0</mn><mo>}</mo></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><mi>t</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>. 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 <inline-formula><math display="inline"><semantics><mi>ν</mi></semantics></math></inline-formula>. The results of several examples are also presented.