A new approach to the generalized Touchard wavelet approximation of fractional integro-differential equations with weakly singular kernels: Moduli of continuity and convergence

In this work, we introduce the generalized Touchard wavelet to solve fractional integro-differential equations with weakly singular kernels. These equations are effective in modeling various physical phenomena. In this approach, the unknown function is approximated by a truncated series of generalized Touchard wavelets. This method focuses primarily on reducing such problems to solving systems of algebraic equations. We conduct an inquiry into the convergence and error analysis of the solution functions. Moreover, we obtain estimates of the moduli of continuity of functions belonging to Hölder's class. In addition, to demonstrate the immutability and precision of the proposed method, numerical results are presented in graphical and tabular form. We perform a comparative analysis of the generalized Touchard wavelet solution against those obtained using different wavelets. The numerical findings reveal that the solutions are sufficiently accurate, even when the number of collocation points is small. The error results are consistent with the convergence analysis of the method.