Analysis on determining the solution of fourth-order fuzzy initial value problem with Laplace operator

Muhammad Akram; Tayyaba Ihsan; Tofigh Allahviranloo; Mohammed M. Ali Al-Shamiri; Muhammad Akram; Tayyaba Ihsan; Tofigh Allahviranloo; Mohammed M. Ali Al-Shamiri

doi:10.3934/mbe.2022554

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 12: 11868-11902. doi: 10.3934/mbe.2022554

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Analysis on determining the solution of fourth-order fuzzy initial value problem with Laplace operator

1.
Department of Mathematics, University of the Punjab, New Campus, Lahore, Pakistan
2.
Faculty of Engineering and Natural Sciences, Istinye University, Istanbul, Turkey
3.
Department of Mathematics, Faculty of Science and Arts, Muhayl Asser, King Khalid University, K.S.A
4.
Department of Mathematics and Computer, Faculty of Science, Ibb University, Ibb, Yemen

Academic Editor: Jesús Martín Vaquero

Received: 04 June 2022 Revised: 04 August 2022 Accepted: 11 August 2022 Published: 17 August 2022

This study presents a new analytical method to extract the fuzzy solution of the fuzzy initial value problem (FIVP) of fourth-order fuzzy ordinary differential equations (FODEs) using the Laplace operator under the strongly generalized Hukuhara differentiability (SGH-differentiability). To this end, firstly the fourth-order derivative of the fuzzy valued function (FVF) according to the type of the SGH-differentiability is introduced, and then the relationships between the fourth-order derivative of the FVF and its Laplace transform are established. Furthermore, considering the types of differentiabilities and switching points, some fundamental theorems related to the Laplace transform of the fourth-order derivative of the FVF are stated and proved in detail and a method to solve FIVP by the fuzzy Laplace transform is presented in detail. An application of our proposed method in Resistance-Inductance circuit (RL circuit) is presented. Finally, FIVP's solution is graphically analyzed to visualize and support theoretical results.
- fuzzy number,
- SGH-differentiability,
- fuzzy initial-value problem,
- fuzzy Laplace transform,
- Mittag-Leffler function
Citation: Muhammad Akram, Tayyaba Ihsan, Tofigh Allahviranloo, Mohammed M. Ali Al-Shamiri. Analysis on determining the solution of fourth-order fuzzy initial value problem with Laplace operator[J]. Mathematical Biosciences and Engineering, 2022, 19(12): 11868-11902. doi: 10.3934/mbe.2022554

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Abstract

This study presents a new analytical method to extract the fuzzy solution of the fuzzy initial value problem (FIVP) of fourth-order fuzzy ordinary differential equations (FODEs) using the Laplace operator under the strongly generalized Hukuhara differentiability (SGH-differentiability). To this end, firstly the fourth-order derivative of the fuzzy valued function (FVF) according to the type of the SGH-differentiability is introduced, and then the relationships between the fourth-order derivative of the FVF and its Laplace transform are established. Furthermore, considering the types of differentiabilities and switching points, some fundamental theorems related to the Laplace transform of the fourth-order derivative of the FVF are stated and proved in detail and a method to solve FIVP by the fuzzy Laplace transform is presented in detail. An application of our proposed method in Resistance-Inductance circuit (RL circuit) is presented. Finally, FIVP's solution is graphically analyzed to visualize and support theoretical results.

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