On the boundedness of the solution set for the $ \psi $-Caputo fractional pantograph equation with a measure of non-compactness via simulation analysis

Reny George; Fahad Al-shammari; Mehran Ghaderi; Shahram Rezapour; Reny George; Fahad Al-shammari; Mehran Ghaderi; Shahram Rezapour

doi:10.3934/math.20231025

AIMS Mathematics

2023, Volume 8, Issue 9: 20125-20142. doi: 10.3934/math.20231025

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On the boundedness of the solution set for the $ \psi $-Caputo fractional pantograph equation with a measure of non-compactness via simulation analysis

1.
Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
2.
Institute of Research and Development, Duy Tan University, Da Nang 550000, Vietnam
3.
Department of Mathematics, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul 02447, Republic of Korea
4.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

Received: 28 February 2023 Revised: 11 May 2023 Accepted: 24 May 2023 Published: 19 June 2023
MSC : 34A08, 34A12

A large number of physical phenomena can be described and modeled by differential equations. One of these famous models is related to the pantograph, which has been investigated in the history of mathematics and physics with different approaches. Optimizing the parameters involved in the pantograph is very important due to the task of converting the type of electric current in the relevant circuit. For this reason, it is very important to use fractional operators in its modeling. In this work, we will investigate the existence of the solution for the fractional pantograph equation by using a new $ \psi $-Caputo operator. The novelty of this work, in addition to the $ \psi $-Caputo fractional operator, is the use of topological degree theory and numerical results from simulations. Techniques in fixed point theory and the use of inequalities will also help to prove the main results. Finally, we provide two examples with some graphical and numerical simulations to make our results more objective. Our data indicate that the boundedness of the solution set for the desired problem depends on the choice of the $ \psi(\kappa) $ function.
- pantograph equation,
- fixed-point theory,
- topological degree theory,
- $ \psi $-Caputo derivative,
- measure of non-compactness
Citation: Reny George, Fahad Al-shammari, Mehran Ghaderi, Shahram Rezapour. On the boundedness of the solution set for the $ \psi $-Caputo fractional pantograph equation with a measure of non-compactness via simulation analysis[J]. AIMS Mathematics, 2023, 8(9): 20125-20142. doi: 10.3934/math.20231025

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Abstract

A large number of physical phenomena can be described and modeled by differential equations. One of these famous models is related to the pantograph, which has been investigated in the history of mathematics and physics with different approaches. Optimizing the parameters involved in the pantograph is very important due to the task of converting the type of electric current in the relevant circuit. For this reason, it is very important to use fractional operators in its modeling. In this work, we will investigate the existence of the solution for the fractional pantograph equation by using a new $ \psi $-Caputo operator. The novelty of this work, in addition to the $ \psi $-Caputo fractional operator, is the use of topological degree theory and numerical results from simulations. Techniques in fixed point theory and the use of inequalities will also help to prove the main results. Finally, we provide two examples with some graphical and numerical simulations to make our results more objective. Our data indicate that the boundedness of the solution set for the desired problem depends on the choice of the $ \psi(\kappa) $ function.

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