Residual Power Series Technique for Simulating Fractional Bagley–Torvik Problems Emerging in Applied Physics

Numerical simulation of physical issues is often performed by nonlinear modeling, which typically involves solving a set of concurrent fractional differential equations through effective approximate methods. In this paper, an analytic-numeric simulation technique, called residual power series (RPS), is proposed in obtaining the numerical solution a class of fractional Bagley–Torvik problems (FBTP) arising in a Newtonian fluid. This approach optimizes the solutions by minimizing the residual error functions that can be directly applied to generate fractional PS with a rapidly convergent rate. The RPS description is presented in detail to approximate the solution of FBTPs by highlighting all the steps necessary to implement the algorithm in addressing some test problems. The results indicate that the RPS algorithm is reliable and suitable in solving a wide range of fractional differential equations applying in physics and engineering.

[1]  A. H. Bhrawy.  A Jacobi spectral collocation method for solving multi-dimensional nonlinear fractional sub-diffusion equations , 2015, Numerical Algorithms.

[2]  Şuayip Yüzbaşı,et al.  Numerical solution of the Bagley–Torvik equation by the Bessel collocation method , 2013 .

[3]  M. Al‐Smadi,et al.  Numerical Solutions of Fractional Systems of Two-Point BVPs by Using the Iterative Reproducing Kernel Algorithm , 2018, Ukrainian Mathematical Journal.

[4]  M. Shamsi,et al.  A pseudo-spectral scheme for the approximate solution of a family of fractional differential equations , 2011 .

[5]  Santanu Saha Ray,et al.  Analytical solution of the Bagley Torvik equation by Adomian decomposition method , 2005, Appl. Math. Comput..

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

[7]  Mohammed Al-Smadi,et al.  Computational algorithm for solving fredholm time-fractional partial integrodifferential equations of dirichlet functions type with error estimates , 2019, Appl. Math. Comput..

[8]  Ian Grout,et al.  Remote Laboratories as a Means to Widen Participation in STEM Education , 2017 .

[9]  Mohammed Al-Smadi,et al.  Construction of fractional power series solutions to fractional stiff system using residual functions algorithm , 2019, Advances in Difference Equations.

[10]  Abu Arqub,et al.  Series Solution of Fuzzy Differential Equations under Strongly Generalized Differentiability , 2013 .

[11]  Mohammed Al-Smadi,et al.  An Analytical Numerical Method for Solving Fuzzy Fractional Volterra Integro-Differential Equations , 2019, Symmetry.

[12]  Mustafa Gülsu,et al.  Numerical solution the fractional Bagley–Torvik equation arising in fluid mechanics , 2017, Int. J. Comput. Math..

[13]  Omar Abu Arqub,et al.  Solutions of Bagley–Torvik and Painlevé equations of fractional order using iterative reproducing kernel algorithm with error estimates , 2018, Neural Computing and Applications.

[14]  Vidhya Saraswathy Krishnasamy,et al.  The Numerical Solution of the Bagley–Torvik Equation With Fractional Taylor Method , 2016 .

[15]  Mohammed Al-Smadi,et al.  Two computational approaches for solving a fractional obstacle system in Hilbert space , 2019, Advances in Difference Equations.

[16]  Dumitru Baleanu,et al.  Two Efficient Generalized Laguerre Spectral Algorithms for Fractional Initial Value Problems , 2013 .

[17]  S. Momani,et al.  Numerical Multistep Approach for Solving Fractional Partial Differential Equations , 2017 .

[18]  Asghar Ghorbani,et al.  Application of He's Variational Iteration Method to Solve Semidifferential Equations of th Order , 2008 .

[19]  Inmaculada Maíz,et al.  Factors for academic success in the integration of MOOCs in the university classroom , 2017 .

[20]  Enrique Navarro-Asencio,et al.  Validación de constructo de un instrumento para medir la competencia digital docente de los profesores (CDD) , 2018 .

[21]  Syed Tauseef Mohyud-Din,et al.  A fractional-order Legendre collocation method for solving the Bagley-Torvik equations , 2016, Advances in Difference Equations.

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

[23]  Aytac Arikoglu,et al.  Solution of fractional differential equations by using differential transform method , 2007 .

[24]  Habibolla Latifizadeh.  APPLICATION OF HOMOTOPY ANALYSIS TRANSFORM METHOD TO FRACTIONAL BIOLOGICAL POPULATION MODEL , 2013 .

[25]  Shuanghua Luo,et al.  Local Polynomial Smoother for Solving Bagley-Torvik Fractional Differential Equations , 2016 .

[26]  Shahrokh Esmaeili.  THE NUMERICAL SOLUTION OF THE BAGLEY-TORVIK EQUATION BY EXPONENTIAL INTEGRATORS , 2017 .

[27]  J. Tourón,et al.  Construct validation of a questionnaire to measure teachers’ digital competence (TDC) , 2018 .

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

[29]  Mohammed Al-Smadi,et al.  Toward computational algorithm for time-fractional Fokker–Planck models , 2019, Advances in Mechanical Engineering.

[30]  Mike Borowczak,et al.  Integrated STEM: Focus on Informal Education and Community Collaboration through Engineering. , 2018 .

[31]  Mohammed Al-Smadi,et al.  Advanced Analytical Treatment of Fractional Logistic Equations Based on Residual Error Functions , 2019, International Journal of Differential Equations.

[32]  Ali H. Bhrawy,et al.  The operational matrix of fractional integration for shifted Chebyshev polynomials , 2013, Appl. Math. Lett..

[33]  Y. H. Youssri,et al.  A new operational matrix of Caputo fractional derivatives of Fermat polynomials: an application for solving the Bagley-Torvik equation , 2017, Advances in Difference Equations.

[34]  M. Al‐Smadi.  Simplified iterative reproducing kernel method for handling time-fractional BVPs with error estimation , 2017, Ain Shams Engineering Journal.

[35]  E. A. Rawashdeh,et al.  Numerical solution of semidifferential equations by collocation method , 2006, Appl. Math. Comput..

[36]  M. Al‐Smadi,et al.  Numerical algorithm for solving time‐fractional partial integrodifferential equations subject to initial and Dirichlet boundary conditions , 2018 .