The discrete convolution for fractional cosine-sine series and its application in convolution equations

Rongbo Wang; Qiang Feng; Jinyi Ji; Rongbo Wang; Qiang Feng; Jinyi Ji

doi:10.3934/math.2024130

AIMS Mathematics

2024, Volume 9, Issue 2: 2641-2656. doi: 10.3934/math.2024130

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Research article

The discrete convolution for fractional cosine-sine series and its application in convolution equations

School of Mathematics and Computer Science, Yan'an University, Yan'an 716000, China

Received: 01 November 2023 Revised: 13 December 2023 Accepted: 18 December 2023 Published: 27 December 2023
MSC : 44A35, 45E10, 47A30

The fractional sine series (FRSS) and the fractional cosine series (FRCS) were defined. Three types of discrete convolution operations for FRCS and FRSS were introduced, along with a detailed investigation into their corresponding convolution theorems. The interrelationship between these convolution operations was also discussed. Additionally, as an application of the presented results, two forms of discrete convolution equations based on the proposed convolution theorems were examined, resulting in explicit solutions for these equations. Furthermore, numerical simulations were provided to demonstrate that our proposed solution can be easily implemented with low computational complexity.
- fractional sine series (FRSS),
- fractional cosine series (FRCS),
- discrete convolution,
- convolution equations
Citation: Rongbo Wang, Qiang Feng, Jinyi Ji. The discrete convolution for fractional cosine-sine series and its application in convolution equations[J]. AIMS Mathematics, 2024, 9(2): 2641-2656. doi: 10.3934/math.2024130

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Abstract

The fractional sine series (FRSS) and the fractional cosine series (FRCS) were defined. Three types of discrete convolution operations for FRCS and FRSS were introduced, along with a detailed investigation into their corresponding convolution theorems. The interrelationship between these convolution operations was also discussed. Additionally, as an application of the presented results, two forms of discrete convolution equations based on the proposed convolution theorems were examined, resulting in explicit solutions for these equations. Furthermore, numerical simulations were provided to demonstrate that our proposed solution can be easily implemented with low computational complexity.

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