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Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case

## An efficient numerical method for fractional model of allelopathic stimulatory phytoplankton species with Mittag-Leffler law

 1 Department of Engineering Science, Kermanshah University of Technology, Kermanshah, Iran, Department of Mathematics, Faculty of Engineering and Natural Sciences, Bahçeşehir University, 34349 Istanbul, Turkey 2 Department of Mathematics, University of Rajasthan, Jaipur-302004, Rajasthan, India 3 Department of Mathematics, JECRC University, Jaipur-303905, Rajasthan, India

* Corresponding author: jagdevsinghrathore@gmail.com

Received  December 2019 Revised  March 2020 Published  September 2020

The principal aim of the present article is to study a mathematical pattern of interacting phytoplankton species. The considered model involves a fractional derivative which enjoys a nonlocal and nonsingular kernel. We first show that the problem has a solution, then the proof of the uniqueness is included by means of the fixed point theory. The numerical solution of the mathematical model is also obtained by employing an efficient numerical scheme. From numerical simulations, one can see that this is a very efficient method and provides precise and outstanding results.

Citation: Behzad Ghanbari, Devendra Kumar, Jagdev Singh. An efficient numerical method for fractional model of allelopathic stimulatory phytoplankton species with Mittag-Leffler law. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020428
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##### References:
Influence of $\rho$ on response behavior of solutions when $\gamma = 0.001$.
Influence of $\rho$ on response behavior of solutions when $\gamma = 0.002$.
Influence of $\rho$ on response behavior of solutions when $\gamma = 0.003$.
Influence of $\rho$ on response behavior of solutions when $\gamma = 0.005$.
Influence of $\rho$ on response behavior of solutions when $\gamma = 0.05$.
Influence of $\rho$ on response behavior of solutions when $\gamma = 0.01$.
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