Numerical simulation of $(k, \psi )$-Hilfer fractional integro-differential equations using airfoil wavelets

This study successfully introduces a new class of fractional integro-differential equations using the \((k, \psi )\) -Hilfer fractional derivative (HFD). A hybrid computational technique for solving these equations is developed for the first time, by introducing the airfoil wavelets as an appropriate set of wavelet functions. To do this, the \((k, \psi )\) -Hilfer fractional integro-differential equations (HFIDEs) are reformulated as an equivalent fractional integral equation (FIE) by utilizing the properties of the \((k, \psi )\) -Riemann-Liouville fractional (RLF) integral and the \((k, \psi )\) -HFD. The unknown function is approximated by applying an activation function on a finite expansion of airfoil wavelets and unknown coefficients. The solution to the problem at hand is then found by solving a system whose components are algebraic equations by employing the Gauss-Legendre numerical integration (GLNI) and collocation scheme. Furthermore, the convergence of the proposed method is shown in Hilbert space. A series of computational examples are provided to illustrate the accuracy and effectiveness of the mentioned strategy.