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Lie symmetry, convergence analysis, explicit solutions, and conservation laws for the time-fractional modified Benjamin-Bona-Mahony equation

Al-deiakeh Rawaya, Alquran Marwan, Ali Mohammed, Qureshi Sania, Momani Shaher, Malkawi Abed Al-Rahman

Journal of Applied Mathematics and Computational Mechanics

2024

Vol. 23, nr 1

19-31

Lie symmetry analysis is considercd as one of the most powerful techniques that has been used for analyzing and extracting various types of solulions to partial differential equations. Conservation laws reflect important aspects of the behavior and pcoperties of physical systems. This paper focuses on the investigation of the (1+1)-dimensional time-fractional modified Benjamin-Bona-Mahony equation (mBBM) incorporating Riemann-Louville derivatives (RLD). Through the application of Lie symmetry analysis, ihe study cxplores similarity reductions and transforms the problem into a nonlinear ordinary differential equation with fractional order. A power series solution is obtained using the Erdelyi-Kober fractional operator, and the convergence of the solutions is analyzed. Furthemore, novel conservation laws for the time-fractional mBBM equation are established. The findings of the current work contribute to a deeper understanding of the dynamics of this fractional evolution equation and provide valuable insights into its behavior.

Synchronization of FitzHugh-Nagumo reaction-diffusion systems via one-dimensional linear control law

Ouannas Adel, Mesdoui Fatiha, Momani Shaher, Batiha Iqbal, Grassi Giuseppe

Archives of Control Sciences

2021

Vol. 31, No. 2

333--345

The Fitzhugh-Nagumo model (FN model), which is successfully employed in modeling the function of the so-called membrane potential, exhibits various formations in neuronal networks and rich complex dynamics. This work deals with the problem of control and synchronization of the FN reaction-diffusion model. The proposed control law in this study is designed to be uni-dimensional and linear law for the purpose of reducing the cost of implementation. In order to analytically prove this assertion, Lyapunov’s second method is utilized and illustrated numerically in one- and/or two-spatial dimensions.