Numerical solution of Bagley–Torvik equations using Legendre artificial neural network method

In this article, we have used the Legendre artificial neural network to find the solution of the Bagley–Torvik equation, which is a fractional-order ordinary differential equation. Caputo fractional derivative has been considered throughout the presented work to handle the fractional order differential equation. The training of optimal weights of the network has been carried out using a simulated annealing optimization technique. Here we have presented three examples to exhibit the precision and relevance of the proposed technique with comparison to the other numerical methods with error analysis. The proposed technique is an easy, highly efficient, and robust technique for finding the approximate solution of fractional-order ordinary differential equations.

[1]  Arvet Pedas,et al.  On the convergence of spline collocation methods for solving fractional differential equations , 2011, J. Comput. Appl. Math..

[2]  Ali Ugur Ozturk,et al.  Numerical analysis on corrosion resistance of mild steel structures , 2013, Engineering with Computers.

[3]  E. Castillo,et al.  Working with differential, functional and difference equations using functional networks , 1999 .

[4]  R. Bagley,et al.  On the Appearance of the Fractional Derivative in the Behavior of Real Materials , 1984 .

[5]  Stefania Tomasiello,et al.  New sinusoidal basis functions and a neural network approach to solve nonlinear Volterra–Fredholm integral equations , 2019, Neural Computing and Applications.

[6]  P. Borne,et al.  Lyapunov analysis of sliding motions: Application to bounded control , 1996 .

[7]  Snehashish Chakraverty,et al.  Chebyshev Neural Network based model for solving Lane-Emden type equations , 2014, Appl. Math. Comput..

[8]  I. Podlubny.  Fractional differential equations , 1998 .

[9]  Azmat Ullah Khan Niazi,et al.  Analytical solution of the generalized Bagley–Torvik equation , 2019, Advances in Difference Equations.

[10]  Aydin Kurnaz,et al.  The solution of the Bagley-Torvik equation with the generalized Taylor collocation method , 2010, J. Frankl. Inst..

[11]  J. Piqueira,et al.  Analytical Solution of General Bagley-Torvik Equation , 2015 .

[12]  Manoj Kumar,et al.  Numerical Solution of Lane–Emden Type Equations Using Multilayer Perceptron Neural Network Method , 2019, International Journal of Applied and Computational Mathematics.

[13]  Najeeb Alam Khan,et al.  Numerical Simulation Using Artificial Neural Network on Fractional Differential Equations , 2016 .

[14]  Igor Podlubny,et al.  Matrix approach to discretization of fractional derivatives and to solution of fractional differential equations and their systems , 2009, 2009 IEEE Conference on Emerging Technologies & Factory Automation.

[16]  Z. Hammouch,et al.  Approximate analytical solutions to the Bagley-Torvik equation by the Fractional Iteration Method , 2012 .

[17]  Stefania Tomasiello,et al.  A functional network to predict fresh and hardened properties of self‐compacting concretes , 2011 .

[18]  Snehashish Chakraverty,et al.  Comparison of Artificial Neural Network Architecture in Solving Ordinary Differential Equations , 2013, Adv. Artif. Neural Syst..

[19]  Rasit Köker.  A neuro-simulated annealing approach to the inverse kinematics solution of redundant robotic manipulators , 2012, Engineering with Computers.

[20]  I. L. El-Kalla,et al.  Analytical and numerical solutions of multi-term nonlinear fractional orders differential equations , 2010 .

[21]  Marina Popolizio,et al.  Numerical Solution of Multiterm Fractional Differential Equations Using the Matrix Mittag–Leffler Functions , 2018 .

[22]  Muhammed I. Syam,et al.  A Numerical Solution of Fractional Lienard’s Equation by Using the Residual Power Series Method , 2017 .

[23]  M. Bansal,et al.  ANALYTICAL SOLUTION OF BAGLEY TORVIK EQUATION BY GENERALIZE DIFFERENTIAL TRANSFORM , 2016 .

[24]  Witold Pedrycz,et al.  A Granular Functional Network with delay: Some dynamical properties and application to the sign prediction in social networks , 2018, Neurocomputing.

[25]  Raja Muhammad Asif Zahoor,et al.  Design of unsupervised fractional neural network model optimized with interior point algorithm for solving Bagley-Torvik equation , 2017, Math. Comput. Simul..

[26]  N. Ford,et al.  Numerical Solution of the Bagley-Torvik Equation , 2002, BIT Numerical Mathematics.

[27]  Junaid Ali Khan,et al.  Solution of Fractional Order System of Bagley-Torvik Equation Using Evolutionary Computational Intelligence , 2011 .

[28]  Mathieu Colin.  Stability of Standing waves for a Quasilinear Schrodinger Equation in Space Dimension 2 , 2003 .

[29]  Snehashish Chakraverty,et al.  Single layer Chebyshev neural network model with regression-based weights for solving nonlinear ordinary differential equations , 2020, Evolutionary Intelligence.

[30]  Peter J. Torvik,et al.  Fractional calculus-a di erent approach to the analysis of viscoelastically damped structures , 1983 .

[31]  M. Caputo.  Linear Models of Dissipation whose Q is almost Frequency Independent-II , 1967 .

[32]  Zhengyi Lu,et al.  Analytical solution of the linear fractional differential equation by Adomian decomposition method , 2008 .