On the generalized Elzaki transform and its applications to fractional differential equations

Aysun Yurttas Gunes; Halim Benali; Mohammed Said Souid; Aysun Yurttas Gunes; Halim Benali; Mohammed Said Souid

doi:10.3934/math.2025593

AIMS Mathematics

2025, Volume 10, Issue 6: 13231-13250. doi: 10.3934/math.2025593

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Research article

On the generalized Elzaki transform and its applications to fractional differential equations

1.
Department of Mathematics, Faculty of Arts and Science, Bursa Uludag University, Bursa, Turkey
2.
Department of Mathematics, Faculty of Mathematics and Computer Sciences, University of Tiaret, Algeria
3.
Department of Mathematics, Saveetha School of Engineering, SIMATS, India
4.
Department of Economic Sciences, University of Tiaret, Algeria

Received: 17 March 2025 Revised: 25 May 2025 Accepted: 28 May 2025 Published: 09 June 2025
MSC : 26A33, 34A08, 65L05

In this paper, we introduce a modified Elzaki transform and its generalization, namely the Elzaki transform, and its own convolution theorem is given. These generalization are given by composition with a monotonic increasing function $ \varrho $ having a continuous derivative. A revised version of the Elzaki transform that is broader in scope and applicable over a wider range is developed, and some of its fundamental properties are given. This modified transform is performed to find solutions of certain non homogeneous linear $ \varrho $ Riemann–Liouville, $ \varrho $ Caputo fractional differential equations. Using comparision graphs, one can determine the effectiveness of the solutions. This research opens up new avenues for future research and has the potential to make a significant impact on the field of mathematics and its applications.
- Elzaki transform,
- convolution,
- generalized fractional operators,
- fractional integrals,
- graph modeling
Citation: Aysun Yurttas Gunes, Halim Benali, Mohammed Said Souid. On the generalized Elzaki transform and its applications to fractional differential equations[J]. AIMS Mathematics, 2025, 10(6): 13231-13250. doi: 10.3934/math.2025593

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Abstract

In this paper, we introduce a modified Elzaki transform and its generalization, namely the Elzaki transform, and its own convolution theorem is given. These generalization are given by composition with a monotonic increasing function $ \varrho $ having a continuous derivative. A revised version of the Elzaki transform that is broader in scope and applicable over a wider range is developed, and some of its fundamental properties are given. This modified transform is performed to find solutions of certain non homogeneous linear $ \varrho $ Riemann–Liouville, $ \varrho $ Caputo fractional differential equations. Using comparision graphs, one can determine the effectiveness of the solutions. This research opens up new avenues for future research and has the potential to make a significant impact on the field of mathematics and its applications.

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