A new fractional dynamic cobweb model based on nonsingular kernel derivatives. (April 2021) - Salahshour, Soheil; Ahmadian, Ali; Allahviranloo, Tofigh
Highlights: To propose a new cobweb model based on nonsingular kernel derivatives. To find the analytical solutions of the new model based on Laplace transform method. To make a comprehensive analysis on the advantages and disadvantages of employing different types of fractional derivative for cobweb phenomenon. To investigate of effects of choosing different normalized functions on the behaviors of solutions. To make a comparative study on new model with other existing ones considering the convergence rate to the equilibrium value under various conditions Abstract: This study proposes a new dynamic differential-based cobweb model that is built under nonsingular kernel fractional derivative. The general solution of this new model is obtained using Laplace transform method. Besides, the existence and uniqueness of solutions are investigated. A comprehensive analysis is carried out to study asymptotic behaviours of solutions compared with the conformable and Caputo-type cobweb models. In fact, we have a deep observation to the convergence or divergence rate of solutions under different values of important parameters in such models to find out the effects of varying these parameters on the trends of solutions over different times interval. The theoretical foundations of this letter is supported by solving a number of examples in the last part where the new model performs better in terms of convergence rate to the equilibrium value. In fact, this report demonstrates the effectiveness of proposed cobweb model in the different times intervals where in some conditions the solutions of other models are completely divergent. Besides, we exemplify the proficiency of the new model compared with the corresponding cobweb model based on integer differential equations in which the relevant solution is divergent whereas the presented model has convergent solution in the same period time.