Effect of fractional temporal variation on the vibration of waves on elastic substrates with spatial non-homogeneity

Ahmed SM Alzaidi; Ali M Mubaraki; Rahmatullah Ibrahim Nuruddeen; Ahmed SM Alzaidi; Ali M Mubaraki; Rahmatullah Ibrahim Nuruddeen

doi:10.3934/math.2022757

AIMS Mathematics

2022, Volume 7, Issue 8: 13746-13762. doi: 10.3934/math.2022757

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Effect of fractional temporal variation on the vibration of waves on elastic substrates with spatial non-homogeneity

1.
Department of Mathematics and Statistics, Collage of Science, Taif University, P. O. Box 11099, Taif 21944, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Federal University Dutse, P. O. Box 7156 Dutse, Jigawa State, Nigeria

Received: 06 March 2022 Revised: 02 May 2022 Accepted: 04 May 2022 Published: 23 May 2022
MSC : 73D05, 26A33

The current manuscript examines the effect of the fractional temporal variation on the vibration of waves on non-homogeneous elastic substrates by applying the Laplace integral transform and the asymptotic approach. Four different non-homogeneities, including linear and exponential forms, are considered and scrutinized. In the end, it is reported that the fractional temporal variation significantly affects the respective vibrational fields greatly as the vibrations increase with a decrease in the fractional-order $\mu$. Besides, the two approaches employed for the cylindrical substrates are also shown to be in good agreement for very small non-homogeneity parameter $\alpha$. More so, the present study is set to play a vital role in the fields of material science, and non-homogenization processes to state a few.
- Laplace transform,
- non-homogeneous substrates,
- elastic waves,
- fractional order
Citation: Ahmed SM Alzaidi, Ali M Mubaraki, Rahmatullah Ibrahim Nuruddeen. Effect of fractional temporal variation on the vibration of waves on elastic substrates with spatial non-homogeneity[J]. AIMS Mathematics, 2022, 7(8): 13746-13762. doi: 10.3934/math.2022757

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Abstract

The current manuscript examines the effect of the fractional temporal variation on the vibration of waves on non-homogeneous elastic substrates by applying the Laplace integral transform and the asymptotic approach. Four different non-homogeneities, including linear and exponential forms, are considered and scrutinized. In the end, it is reported that the fractional temporal variation significantly affects the respective vibrational fields greatly as the vibrations increase with a decrease in the fractional-order $\mu$. Besides, the two approaches employed for the cylindrical substrates are also shown to be in good agreement for very small non-homogeneity parameter $\alpha$. More so, the present study is set to play a vital role in the fields of material science, and non-homogenization processes to state a few.

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