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

This study successfully introduces a new class of fractional integro-differential equations using the \((k, ψ )\) -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 \((k, ψ )\) -Hilfer fractional integro-differential equations (HFIDEs) are reformulated as an equivalent fractional integral equation (FIE) by utilizing the properties of the \((k, ψ )\) -Riemann-Liouville fractional (RLF) integral and the \((k, ψ )\) -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.