Finite Series of Distributional Solutions for Certain Linear Differential Equations
Abstract
:1. Introduction
2. Preliminaries
- (i)
- The locally integrable function is a distribution generated by the locally integrable function . Then we define , where Ω is the support of and .
- (ii)
- The Dirac delta function is a distribution defined by and the support of is .
- (i)
- ;
- (ii)
- .
- (i)
- for all ;
- (ii)
- There exists a real number c such that is absolutely integrable over .
- (i)
- is infinitely differentiable—i.e., ;
- (ii)
- , as well as its derivatives of all orders, vanish at infinity faster than the reciprocal of any polynomial which is expressed by the inequalitywhere is a constant depending on , and . Then is called a test function in the space S.
- (i)
- for ;
- (ii)
- for every null sequence .We shall let denote the set of all distributions of slow growth.
- (i)
- is a right-sided distribution, that is, .
- (ii)
- There exists a real number c such that is a tempered distribution.
- (i)
- , ;
- (ii)
- , ;
- (iii)
- , ;
- (iv)
- , ;
- (v)
- , .
3. Main Results
4. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Waiyaworn, N.; Nonlaopon, K.; Orankitjaroen, S. Finite Series of Distributional Solutions for Certain Linear Differential Equations. Axioms 2020, 9, 116. https://doi.org/10.3390/axioms9040116
Waiyaworn N, Nonlaopon K, Orankitjaroen S. Finite Series of Distributional Solutions for Certain Linear Differential Equations. Axioms. 2020; 9(4):116. https://doi.org/10.3390/axioms9040116
Chicago/Turabian StyleWaiyaworn, Nipon, Kamsing Nonlaopon, and Somsak Orankitjaroen. 2020. "Finite Series of Distributional Solutions for Certain Linear Differential Equations" Axioms 9, no. 4: 116. https://doi.org/10.3390/axioms9040116
APA StyleWaiyaworn, N., Nonlaopon, K., & Orankitjaroen, S. (2020). Finite Series of Distributional Solutions for Certain Linear Differential Equations. Axioms, 9(4), 116. https://doi.org/10.3390/axioms9040116