Numerical investigation of \(\Psi \)-fractional differential equations using wavelets neural networks

This work is appropriated to the numerical approximation of \(\varPsi \) $\Psi$ -fractional differential equations and \(\varPsi \) $\Psi$ -fractional integro-differential equations. First, the under study problems are converted into the equivalent second kind Volterra integral equations. Then, a numerical learning technique namely fractional-Genocchi wavelets neural networks is introduced for solving the obtained second kind Volterra integral equations. The fractional-Genocchi wavelets, and the \(\sinh \) $sinh$ function are utilized as activation functions of hidden and output layers of the network, respectively. The fractional-Genocchi wavelets neural networks and Gauss–Legendre integration rule are used to transform the Volterra equations into the associated optimization problems. Eventually, using an optimization technique, some weights are utilized so that the approximating functions satisfy the initial conditions. The analysis of convergence of the established approach is investigated in the Sobolev space. Some illustrative examples are considered to display the accuracy and efficiency of the established scheme.