Comparative analysis for fractional nonlinear Sturm-Liouville equations with singular and non-singular kernels

Ahu Ercan; Ahu Ercan

doi:10.3934/math.2022736

AIMS Mathematics

2022, Volume 7, Issue 7: 13325-13343. doi: 10.3934/math.2022736

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Comparative analysis for fractional nonlinear Sturm-Liouville equations with singular and non-singular kernels

Ahu Ercan ^,

Department of Mathematics, Science Faculty, Firat University, Elazig 23119, Turkey

Received: 09 February 2022 Revised: 11 April 2022 Accepted: 21 April 2022 Published: 16 May 2022
MSC : 34A08, 34A45, 34L30

This article presents the Laplace-Adomian decomposition method (LADM), which produces a fast convergence series solution, for two types of nonlinear fractional Sturm-Liouville (SL) problems. The fractional derivatives are defined in the Caputo, conformable, Caputo-Fabrizio in the sense of Caputo (CFC), Caputo type Atangana-Baleanu (ABC) senses. With the help of this method, approximate solutions of the investigated problems were obtained. The solutions generated from the Caputo and ABC derivatives are represented by the Mittag-Leffler function, which is intrinsic to fractional derivatives, and the solution obtained using the conformable and CFC derivatives generate the hyperbolic sine and cosine functions. Thus, we derive some novel solutions for fractional-order versions of nonlinear SL equations. The fractional calculus provides more data than classical calculus and has been widely used in mathematical modeling with memory effect. Finally, we analyzed and compared these novel solutions of the considered problems by graphs under different values of $ p $, $ \lambda $ and different orders of $ \alpha $.
- nonlinear Sturm-Liouville equation,
- Laplace-Adomian decomposition method,
- Caputo,
- conformable,
- Caputo-Fabrizio,
- Atangana-Baleanu fractional derivatives
Citation: Ahu Ercan. Comparative analysis for fractional nonlinear Sturm-Liouville equations with singular and non-singular kernels[J]. AIMS Mathematics, 2022, 7(7): 13325-13343. doi: 10.3934/math.2022736

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Abstract

This article presents the Laplace-Adomian decomposition method (LADM), which produces a fast convergence series solution, for two types of nonlinear fractional Sturm-Liouville (SL) problems. The fractional derivatives are defined in the Caputo, conformable, Caputo-Fabrizio in the sense of Caputo (CFC), Caputo type Atangana-Baleanu (ABC) senses. With the help of this method, approximate solutions of the investigated problems were obtained. The solutions generated from the Caputo and ABC derivatives are represented by the Mittag-Leffler function, which is intrinsic to fractional derivatives, and the solution obtained using the conformable and CFC derivatives generate the hyperbolic sine and cosine functions. Thus, we derive some novel solutions for fractional-order versions of nonlinear SL equations. The fractional calculus provides more data than classical calculus and has been widely used in mathematical modeling with memory effect. Finally, we analyzed and compared these novel solutions of the considered problems by graphs under different values of $ p $, $ \lambda $ and different orders of $ \alpha $.

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