Abstract
In this study, the intelligent computational strength of neural networks (NNs) based on the backpropagated Levenberg-Marquardt (BLM) algorithm is utilized to investigate the numerical solution of nonlinear multiorder fractional differential equations (FDEs). The reference data set for the design of the BLM-NN algorithm for different examples of FDEs are generated by using the exact solutions. To obtain the numerical solutions, multiple operations based on training, validation, and testing on the reference data set are carried out by the design scheme for various orders of FDEs. The approximate solutions by the BLM-NN algorithm are compared with analytical solutions and performance based on mean square error (MSE), error histogram (EH), regression, and curve fitting. This further validates the accuracy, robustness, and efficiency of the proposed algorithm.
Highlights
Mathematicians have regarded the theory of fractional calculus as a branch of pure mathematics for nearly three centuries
Some highlighted features of the given study are illustrated as follows: (i) Novel applications of neuroheuristic techniques based on backpropagated Levenberg–Marquardt neural networks (BLM-NNs) are presented to obtain numerical solutions for different classes of nonlinear multi-order fractional differential equations
In order to illustrate the performance of the backpropagated Levenberg-Marquardt (BLM)-NN algorithm, we have considered various examples of nonlinear multiorder fractional differential equations
Summary
Mathematicians have regarded the theory of fractional calculus as a branch of pure mathematics for nearly three centuries. Computational Intelligence and Neuroscience based on semi-infinite interval to calculate the approximate solution for linear and nonlinear FDEs. Ahmadian [28, 29] applied the Jacobi operational matrix to study a class of linear fuzzy FDEs. e spectral approximation method is used by Li [30] to compute the fractional derivative and integral and presents the pseudo-spectral approximation technique for some classes of FDEs. Esmaeili [31] developed a numerical technique in which the properties of the Caputo derivative were used to reduce the fractional differential equation into a Volterra integral equation. Stochastic numerical techniques based on artificial intelligence have been developed to solve stiff nonlinear problems arising in various fields Such stochastic computing techniques use artificial neural networks to model approximate solutions. (i) Novel applications of neuroheuristic techniques based on backpropagated Levenberg–Marquardt neural networks (BLM-NNs) are presented to obtain numerical solutions for different classes of nonlinear multi-order fractional differential equations. It has simple and smooth implementation with exhaustive applicability and stability
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