An integrated Laplace transform and accelerated Adomian decomposition approach for solving time-fractional nonlinear partial differential equations

Mohamed A. Ramadan; Mahmoud A. A. Abd El-Latif; Mohammed Z. Alqarni; Mohamed A. Ramadan; Mahmoud A. A. Abd El-Latif; Mohammed Z. Alqarni

doi:10.3934/era.2025201

Electronic Research Archive

2025, Volume 33, Issue 7: 4398-4434. doi: 10.3934/era.2025201

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Research article

An integrated Laplace transform and accelerated Adomian decomposition approach for solving time-fractional nonlinear partial differential equations

1.
Mathematics and Computer Science Department, Faculty of Science, Menoufia University, Egypt
2.
Mathematics Department, Faculty of Education, Ain Shams University, Egypt
3.
Department of Mathematics, Faculty of Science, King Khalid University, Abha 61471, Saudi Arabia

Received: 16 May 2025 Revised: 13 July 2025 Accepted: 21 July 2025 Published: 05 August 2025

This study focuses on the efficient and accurate solution of time-fractional nonlinear partial differential equations (PDEs), which arise in various scientific and engineering applications but are often challenging to solve due to their complexity. We propose a novel method that integrates the Laplace transform with the accelerated Adomian decomposition method (AADM), forming the Laplace transform accelerated Adomian decomposition method (LAADM). The key innovation of this approach lies in its ability to handle the fractional time derivative expressed in the Caputo sense, while enhancing convergence speed and reducing computational effort. The methodology is systematically formulated, and several numerical experiments are conducted to validate its performance. The results demonstrate that LAADM provides highly accurate solutions and exhibits superior efficiency when compared to traditional solution techniques for fractional PDEs.
- fractional Laplace transform,
- nonlinear partial fractional differential equations,
- Laplace transform accelerated Adomian,
- convergence,
- error analysis uniqueness,
- accuracy
Citation: Mohamed A. Ramadan, Mahmoud A. A. Abd El-Latif, Mohammed Z. Alqarni. An integrated Laplace transform and accelerated Adomian decomposition approach for solving time-fractional nonlinear partial differential equations[J]. Electronic Research Archive, 2025, 33(7): 4398-4434. doi: 10.3934/era.2025201

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Abstract

This study focuses on the efficient and accurate solution of time-fractional nonlinear partial differential equations (PDEs), which arise in various scientific and engineering applications but are often challenging to solve due to their complexity. We propose a novel method that integrates the Laplace transform with the accelerated Adomian decomposition method (AADM), forming the Laplace transform accelerated Adomian decomposition method (LAADM). The key innovation of this approach lies in its ability to handle the fractional time derivative expressed in the Caputo sense, while enhancing convergence speed and reducing computational effort. The methodology is systematically formulated, and several numerical experiments are conducted to validate its performance. The results demonstrate that LAADM provides highly accurate solutions and exhibits superior efficiency when compared to traditional solution techniques for fractional PDEs.

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