Multi-Resolution Analysis and Approximate Solutions of Fractional Pantograph Differential Equations by Generalized Gegenbauer-Humbert Wavelet

Abstract

This study utilizes the generalized Gegenbauer-Humbert wavelet framework to approximate functions by decomposing them into localized basis functions at multiple resolutions. The wavelet basis is constructed using dilation and translation of the mother wavelet function formed from a combination of Gegenbauer polynomials modulated by Humbert functions. The involvement of Gegenbauer polynomials gives the wavelet high approximation order due to their spectral convergence properties. The Humbert function is frequently employed to handle singularities and non-local behaviors. The multiresolution analysis of this wavelet is investigated to ensure the reliability and efficiency of wavelets in mathematical computations. Furthermore, using moduli of continuity, convergence and error analyses are performed for solution functions belonging to Holder’s class. This article presents a numerical method for solving fractional pantograph differential equations using the generalized Gegenbauer-Humbert wavelet series. We compare our wavelet solutions with those obtained by Taylor and fractional order Legendre wavelet method (FLWM). Graphical representations are used to illustrate the accuracy of the successfully implemented wavelet collocation method. Therefore, the results reveal that the proposed technique may serve as a reliable tool for numerical simulations of mathematical models formulated as pantograph differential equations of both integer and fractional orders, under prescribed initial conditions.