Neural network method for solving fractional diffusion equations

Abstract In this paper, neural networks based on Legendre polynomials are established to solve space and time fractional diffusion equations. The error functions containing adjustable parameters (the weights) for the training sets are constructed. The range of learning rate is analyzed to ensure that the error decreases with respect to training times. Several numerical examples including numerical results and graphs are illustrated. The results show that more training can achieve high precision.

[1]  Yang Liu,et al.  A mixed finite element method for a time-fractional fourth-order partial differential equation , 2014, Appl. Math. Comput..

[2]  Santanu Saha Ray,et al.  An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method , 2005, Appl. Math. Comput..

[3]  Fawang Liu,et al.  A Crank-Nicolson ADI Spectral Method for a Two-Dimensional Riesz Space Fractional Nonlinear Reaction-Diffusion Equation , 2014, SIAM J. Numer. Anal..

[4]  M. Meerschaert,et al.  Finite difference approximations for fractional advection-dispersion flow equations , 2004 .

[5]  A. H. Hadian-Rasanan,et al.  A single layer fractional orthogonal neural network for solving various types of Lane–Emden equation , 2020 .

[6]  Yang Zhang,et al.  A finite difference method for fractional partial differential equation , 2009, Appl. Math. Comput..

[7]  Fawang Liu,et al.  Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation , 2007, Appl. Math. Comput..

[8]  F. Quirós,et al.  A P ] 1 4 Ja n 20 10 A fractional porous medium equation by , 2010 .

[9]  Santanu Saha Ray,et al.  A new approach for the application of Adomian decomposition method for the solution of fractional space diffusion equation with insulated ends , 2008, Appl. Math. Comput..

[10]  Jingtang Ma,et al.  High-order finite element methods for time-fractional partial differential equations , 2011, J. Comput. Appl. Math..

[11]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[12]  Dumitru Baleanu,et al.  On artificial neural networks approach with new cost functions , 2018, Appl. Math. Comput..

[13]  Yangquan Chen,et al.  Matrix approach to discrete fractional calculus II: Partial fractional differential equations , 2008, J. Comput. Phys..

[14]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[15]  Manoj Kumar,et al.  Multilayer perceptrons and radial basis function neural network methods for the solution of differential equations: A survey , 2011, Comput. Math. Appl..

[16]  Tofigh Allahviranloo,et al.  SOLVING FUZZY FRACTIONAL DIFFERENTIAL EQUATIONS BY FUZZY LAPLACE TRANSFORMS , 2012 .

[17]  Theodore E. Simos,et al.  Neural network solution of pantograph type differential equations , 2020, Mathematical Methods in the Applied Sciences.

[18]  Francesco Mainardi,et al.  The Wright functions as solutions of the time-fractional diffusion equation , 2003, Appl. Math. Comput..

[19]  Snehashish Chakraverty,et al.  Artificial Neural Network Based Solution of Fractional Vibration Model , 2020 .

[20]  Sohrab Effati,et al.  A Neural Network Approach for Solving a Class of Fractional Optimal Control Problems , 2017, Neural Processing Letters.

[21]  Dimitrios I. Fotiadis,et al.  Artificial neural networks for solving ordinary and partial differential equations , 1997, IEEE Trans. Neural Networks.

[22]  M. T. Cicero.  FRACTIONAL CALCULUS AND WAVES IN LINEAR VISCOELASTICITY , 2012 .

[23]  B. Mandelbrot,et al.  Fractional Brownian Motions, Fractional Noises and Applications , 1968 .

[24]  S. Effati,et al.  Solving differential equations of fractional order using an optimization technique based on training artificial neural network , 2017, Appl. Math. Comput..

[25]  T. E. Simos,et al.  Neural Network Solution of Single-Delay Differential Equations , 2019, Mediterranean Journal of Mathematics.

[26]  Fawang Liu,et al.  Galerkin finite element approximation of symmetric space-fractional partial differential equations , 2010, Appl. Math. Comput..

[27]  Dumitru Baleanu,et al.  Artificial neural network approach for a class of fractional ordinary differential equation , 2016, Neural Computing and Applications.

[28]  Weihua Deng,et al.  Finite Element Method for the Space and Time Fractional Fokker-Planck Equation , 2008, SIAM J. Numer. Anal..

[29]  Ahmad Jafarian,et al.  A new artificial neural network structure for solving high-order linear fractional differential equations , 2018, Int. J. Comput. Math..

[30]  Jie Shen,et al.  Spectral Methods: Algorithms, Analysis and Applications , 2011 .

[31]  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..

[32]  D. Benson,et al.  Application of a fractional advection‐dispersion equation , 2000 .

[33]  M. A. Manzar,et al.  An efficient computational intelligence approach for solving fractional order Riccati equations using ANN and SQP , 2015 .