Pell wavelet-optimization procedure for two classes of fractional partial differential equations with nonlocal boundary conditions

Studying initial value problems with nonlocal conditions is important because they have applications in physics and other areas of applied mathematics. This manuscript presents a hybrid scheme for solving two classes of fractional partial differential equations with nonlocal boundary conditions (N-BCs), namely fractional-order reaction–diffusion equations (F-RDEs), and fractional-order hyperbolic partial differential equations (FH-PDEs). We develop a new computational technique that employs Pell wavelet functions. To this end, we present a derivative pseudo-operational matrix and an extra pseudo-operational matrix for integral and Riemann–Liouville fractional integration and design the desired method with the help of optimization and collocation methods. The systems resulting from this technique are solved using the FindRoot package in Mathematica software. We also perform several numerical experiments to validate the accuracy and superiority of the suggested strategy.