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Expedite homotopy perturbation method based on metaheuristic technique mimicked by the flashing behavior of fireflies

1 Department of Mathematics, University of Karachi, Karachi 75270, Pakistan
2 Department of Electrical and Computer Engineering, COMSATS University Islamabad, Attock Campus, Attock, Pakistan

Special Issues: Initial and Boundary Value Problems for Differential Equations

In this work, an endeavor is made for assessing the solutions of nonlinear fractional differential equations, by taking into account the derivative and integral operator in the Caputo sense. The proposed technique is developed by the merger of classical and modern ideas of mathematical analysis. The approximate solution of the fractional differential equation (FDE) accomplished by the careful and profitable implementation of the homotopy perturbation method is fast track by using a powerful and proficient metaheuritic technique, which mimics the flashing pattern and behavior of the fireflies. Accuracy and accelerated convergence are the main attributes of the proposed technique, which are attained by using the firefly algorithm (FA) for the optimization of the fitness function constructed in the mean square sense. Numerical experiments are performed to illustrate the worth mentioning performance of the design methodology in term of accuracy, convergence and competency. The solutions found by the deliberated scheme are far superior to the former results emphasized in the literature.
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Keywords Riccati equation; fractional derivative; firefly algorithm; homotopy perturbation method; Adam bashforth method

Citation: Najeeb Alam Khan, Samreen Ahmed, Tooba Hameed, Muhammad Asif Zahoor Raja. Expedite homotopy perturbation method based on metaheuristic technique mimicked by the flashing behavior of fireflies. AIMS Mathematics, 2019, 4(4): 1114-1132. doi: 10.3934/math.2019.4.1114

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