Predicting the solution of fractional order differential equations with Artificial Neural Network

• Published in: Partial differential equations in applied mathematics : a spin-off of Applied Mathematics Letters Vol. 10; p. 100690
• Main Authors: Khan, A.M., Gaur, Sanjay, Suthar, D.L.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.06.2024; Elsevier
• Subjects: 34K37; 41A10; 68T07; Approximation using quadrature method; Back propagation algorithm; Caputo fractional derivative operator; Unsupervised learning; Approximation using quadrature method; Caputo fractional derivative operator; 41A10; 34K37; 68T07; Back propagation algorithm; Unsupervised learning
• ISSN: 2666-8181; 2666-8181
• DOI: 10.1016/j.padiff.2024.100690

The present paper aims to propose an approximation method of Caputo fractional operator using discretization based on quadrature theory to minimize the error function for an Artificial Neural Network (ANN) with higher convergence rate. In the proposed work authors have verified the suitability of multilayer feed forward ANN architecture to get an estimated solution of fractional order differential equations. The back propagation algorithm and an unsupervised learning, was imposed in order to minimize the error function including the optimizing the network parameters such as weights of synapses and the biases. The algorithm uses truncated power series to replace the unknown function in the fractional differential equations. The novelty of the present work is an approximation of a fractional operator using discretization based on quadrature theory to construct an error function to provide an appropriate estimated solution of the nonlinear fractional differential equations. Four illustrative examples with different orders of nonlinear fractional differential equations are unraveled to approve the validity of the model along with demonstration of effectiveness and fast convergence of the proposed method.