Solution of Differential Equations with Polynomial Coefficients with the Aid of an Analytic Continuation of Laplace Transform
Abstract
:1. Introduction
1.1. Definition of the AC-Laplace Transform
1.2. Remarks on Recent Developments
2. AC-Laplace Transform
2.1. Recipe of Solving Differential Equation with Polynomial Coefficients
2.2. Term-by-Term Operators for and
3. Operational in the Spaces and
3.1. Operational Calculus in the Spaces
- (i)
- If and , then .
- (ii)
- If , then .
- (iii)
- The index law stated in the following lemma is valid.
3.1.1. Recipe of Solving Linear Differential Equation with Polynomial Coefficients
3.2. Operational Calculus in the Space
3.3. Distributions in the Space
3.4. Distributions in the Space
4. Solution of Modified Kummer’s DE
4.1. Solution Satisfying
4.2. Solution Satisfying
4.3. Solution by the Basic Method
4.4. Solution by the Modified Nishimoto’s Method
5. Solution of the Hypergeometric DE
5.1. Solution Satisfying
5.2. Solution Satisfying
5.3. Solution by the Basic Method
5.4. Solution by the Modified Nishimoto’s Method
6. Solution of Bessel’s DE
7. Solution of Hermite’s DE
8. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
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Morita, T.; Sato, K.-i. Solution of Differential Equations with Polynomial Coefficients with the Aid of an Analytic Continuation of Laplace Transform. Mathematics 2016, 4, 19. https://doi.org/10.3390/math4010019
Morita T, Sato K-i. Solution of Differential Equations with Polynomial Coefficients with the Aid of an Analytic Continuation of Laplace Transform. Mathematics. 2016; 4(1):19. https://doi.org/10.3390/math4010019
Chicago/Turabian StyleMorita, Tohru, and Ken-ichi Sato. 2016. "Solution of Differential Equations with Polynomial Coefficients with the Aid of an Analytic Continuation of Laplace Transform" Mathematics 4, no. 1: 19. https://doi.org/10.3390/math4010019
APA StyleMorita, T., & Sato, K.-i. (2016). Solution of Differential Equations with Polynomial Coefficients with the Aid of an Analytic Continuation of Laplace Transform. Mathematics, 4(1), 19. https://doi.org/10.3390/math4010019