Framework for treating non-Linear multi-term fractional differential equations with reasonable spectrum of two-point boundary conditions

Najeeb Alam Khan; Samreen Ahmad; Najeeb Alam Khan; Samreen Ahmad

doi:10.3934/math.2019.4.1181

AIMS Mathematics

2019, Volume 4, Issue 4: 1181-1202. doi: 10.3934/math.2019.4.1181

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Framework for treating non-Linear multi-term fractional differential equations with reasonable spectrum of two-point boundary conditions

Najeeb Alam Khan ^,,
Samreen Ahmad

Department of Mathematics, University of Karachi, Karachi 75270, Pakistan

Received: 23 April 2019 Accepted: 13 August 2019 Published: 22 August 2019
MSC : 34A08

In this work, a novel bio-inspired meta-heuristic framework is presented for the assessment of linear and non-linear multi-term fractional differential equations (MFDEs) based on the idea of residual power series method (RPSM) in amalgamation with the bat algorithm (BATA) which, mimics the echolocation behavior of the foraging bats. The bat-inspired based methodology has been implemented to solve MFDEs with different possible variants of two point boundary conditions. The BATA is utilized in the recommended new residual optimization technique (NROT) for the minimization of the energy function attained by the spirit of RPSM. Moreover, to ratify the correctness and accuracy of the deliberated technique the results are evaluated by using two other meta-heuristic optimization techniques i.e., differential evolution algorithm (DEA) and the accelerated particle swarm optimization algorithm (APSOA) for the learning of unknown weights in the derived fitness function. The accuracy and competency of the NROT is validated by comparing the BATA computed results with the exact solution and the corresponding data acquired by the DEA and APSOA. Furthermore, detailed performance analysis is performed through statistical inference based on the large number of independent runs.
- residual method,
- bat algorithm,
- particle swarm optimization,
- differential evolution,
- fractional differential equation
Citation: Najeeb Alam Khan, Samreen Ahmad. Framework for treating non-Linear multi-term fractional differential equations with reasonable spectrum of two-point boundary conditions[J]. AIMS Mathematics, 2019, 4(4): 1181-1202. doi: 10.3934/math.2019.4.1181

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Abstract

In this work, a novel bio-inspired meta-heuristic framework is presented for the assessment of linear and non-linear multi-term fractional differential equations (MFDEs) based on the idea of residual power series method (RPSM) in amalgamation with the bat algorithm (BATA) which, mimics the echolocation behavior of the foraging bats. The bat-inspired based methodology has been implemented to solve MFDEs with different possible variants of two point boundary conditions. The BATA is utilized in the recommended new residual optimization technique (NROT) for the minimization of the energy function attained by the spirit of RPSM. Moreover, to ratify the correctness and accuracy of the deliberated technique the results are evaluated by using two other meta-heuristic optimization techniques i.e., differential evolution algorithm (DEA) and the accelerated particle swarm optimization algorithm (APSOA) for the learning of unknown weights in the derived fitness function. The accuracy and competency of the NROT is validated by comparing the BATA computed results with the exact solution and the corresponding data acquired by the DEA and APSOA. Furthermore, detailed performance analysis is performed through statistical inference based on the large number of independent runs.

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