Analytical solutions of fractional wave equation with memory effect using the fractional derivative with exponential kernel

Abstract Analytical solutions of the fractional wave equation via Caputo-Fabrizio fractional derivative are presented in this paper. For this analysis, three cases are considered, the classical, the damped and the damped with a source term defined by fractional wave equations. We show that these solutions are special cases of the time fractional equations with exponential law. Illustrative examples are presented.

[1]  Ayesha Sohail,et al.  Stability analysis for fractional‐order partial differential equations by means of space spectral time Adams‐Bashforth Moulton method , 2018 .

[2]  S. Arabia,et al.  Properties of a New Fractional Derivative without Singular Kernel , 2015 .

[3]  C. Dai,et al.  Vector optical soliton and periodic solutions of a coupled fractional nonlinear Schrödinger equation , 2020 .

[4]  Y. Chu,et al.  A new analyzing technique for nonlinear time fractional Cauchy reaction-diffusion model equations , 2020 .

[5]  Jinwei Gao,et al.  TEC power consumption in laser array packaging , 2017 .

[6]  Ali H. Nayfeh,et al.  Bifurcations and chaos in parametrically excited single-degree-of-freedom systems , 1990 .

[7]  A. Atangana,et al.  Numerical Simulation of Noninteger Order System in Subdiffusive, Diffusive, and Superdiffusive Scenarios , 2017 .

[8]  Jiwei Zhang,et al.  Superconvergence of Finite Element Approximations for the Fractional Diffusion-Wave Equation , 2017, J. Sci. Comput..

[9]  Z. Hammouch,et al.  Numerical simulation and stability analysis for the fractional-order dynamics of COVID-19 , 2020, Results in Physics.

[10]  Yuri Luchko,et al.  Multi-dimensional α-fractional diffusion-wave equation and some properties of its fundamental solution , 2017, Comput. Math. Appl..

[11]  Hossein Jafari,et al.  Numerical approach of Fokker-Planck equation with Caputo-Fabrizio fractional derivative using Ritz approximation , 2017, J. Comput. Appl. Math..

[12]  Dumitru Baleanu,et al.  Numerical approximation of higher-order time-fractional telegraph equation by using a combination of a geometric approach and method of line , 2016, J. Comput. Phys..

[13]  Dumitru Baleanu,et al.  Two analytical methods for time-fractional nonlinear coupled Boussinesq–Burger’s equations arise in propagation of shallow water waves , 2016 .

[14]  M. Caputo,et al.  A new Definition of Fractional Derivative without Singular Kernel , 2015 .

[15]  Bosko S. Jovanovic,et al.  Finite Difference Approximation of Fractional Wave Equation with Concentrated Capacity , 2017, Comput. Methods Appl. Math..

[16]  R. Ellahi,et al.  Extracting new solitary wave solutions of Benny–Luke equation and Phi-4 equation of fractional order by using (G′/G)-expansion method , 2017 .

[17]  D. Baleanu,et al.  A Variety of Novel Exact Solutions for Different Models With the Conformable Derivative in Shallow Water , 2020, Frontiers in Physics.

[18]  Michele Caputo,et al.  Applications of New Time and Spatial Fractional Derivatives with Exponential Kernels , 2016 .

[19]  R. Ellahi,et al.  Exact traveling wave solutions of fractional order Boussinesq-like equations by applying Exp-function method , 2018 .

[20]  A. Atangana,et al.  A Note on Fractional Order Derivatives and Table of Fractional Derivatives of Some Special Functions , 2013 .

[21]  B. Samet,et al.  Numerical solution for generalized nonlinear fractional integro-differential equations with linear functional arguments using Chebyshev series , 2020 .

[22]  S. Abdel-Khalek,et al.  Optical solitons of space-time fractional Fokas–Lenells equation with two versatile integration architectures , 2020, Advances in Difference Equations.

[23]  Devendra Kumar,et al.  Analysis of a New Fractional Model for Damped Bergers’ Equation , 2017 .

[24]  I. Podlubny.  Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications , 1999 .

[25]  Ji-Huan He,et al.  Periodic property of the time-fractional Kundu–Mukherjee–Naskar equation , 2020 .

[26]  M. Osman,et al.  Optical soliton solutions of perturbing time-fractional nonlinear Schrödinger equations , 2020 .

[27]  Kai Zhang,et al.  Exact solutions to the space–time fractional shallow water wave equation via the complete discrimination system for polynomial method , 2021 .

[28]  Vasily E. Tarasov,et al.  Economic interpretation of fractional derivatives , 2017, 1712.09575.

[29]  Dumitru Baleanu,et al.  On Analytical Solutions of the Fractional Differential Equation with Uncertainty: Application to the Basset Problem , 2015, Entropy.

[30]  Syed Zubair,et al.  A new computing approach for power signal modeling using fractional adaptive algorithms. , 2017, ISA transactions.

[31]  Devendra Kumar,et al.  A new analysis for fractional model of regularized long‐wave equation arising in ion acoustic plasma waves , 2017 .

[32]  R. Ellahi,et al.  Correction to: Extracting new solitary wave solutions of Benny–Luke equation and Phi-4 equation of fractional order by using (G′/G)-expansion method , 2017 .

[33]  Hossein Jafari,et al.  Analysis of Riccati Differential Equations within a New Fractional Derivative without Singular Kernel , 2017, Fundamenta Informaticae.

[34]  O. Arqub,et al.  A Numerical Algorithm for the Solutions of ABC Singular Lane–Emden Type Models Arising in Astrophysics Using Reproducing Kernel Discretization Method , 2020, Mathematics.

[35]  M. Osman,et al.  Novel optical solitons to the perturbed Gerdjikov–Ivanov equation with truncated M-fractional conformable derivative , 2020 .

[36]  Bahaaudin M. Raffah,et al.  Optical soliton solutions for a space-time fractional perturbed nonlinear Schrödinger equation arising in quantum physics , 2020 .

[37]  Trifce Sandev,et al.  Fractional wave equation with a frictional memory kernel of Mittag-Leffler type , 2012, Appl. Math. Comput..

[38]  Badr Saad T. Alkahtani,et al.  Analysis of the Keller-Segel Model with a Fractional Derivative without Singular Kernel , 2015, Entropy.