A new modern scheme for solving fractal–fractional differential equations based on deep feedforward neural network with multiple hidden layer

Abstract

The recent development of knowledge in fractional calculus introduced an advanced superior operator known as fractal–fractional derivative (FFD). This operator combines memory effect and self-similar property that give better accurate representation of real world problems through fractal–fractional differential equations (FFDEs). However, the existence of fresh and modern numerical technique on solving FFDEs is still scarce. Originally invented for machine learning technique, artificial neural network (ANN) is cutting-edge scheme that have shown promising result in solving the fractional differential equations (FDEs). Thus, this research aims to extend the application of ANN to solve FFDE with power law kernel in Caputo sense (FFDEPC) by develop a vectorized algorithm based on deep feedforward neural network that consists of multiple hidden layer (DFNN-2H) with Adam optimization. During the initial stage of the method development, the basic framework on solving FFDEs is designed. To minimize the burden of computational time, the vectorized algorithm is constructed at the next stage for method to be performed efficiently. Several example have been tested to demonstrate the applicability and efficiency of the method. Comparison on exact solutions and some previous published method indicate that the proposed scheme have give good accuracy and low computational time.