Numerical solution to loaded difference scheme for time-fractional diffusion equation with temporal loads

This paper investigates the temporally loaded time-fractional diffusion equation with initial and Dirichlet-type boundary conditions. To begin with, a solution form is established using the method of eigenfunction expansions, and its existence and uniqueness are examined along with some apriori estimates. Thereafter, a finite difference approximation is performed using the so-called L1 method for the Caputo fractional derivative, resulting in a loaded difference scheme. The superposition property of systems of linear algebraic equations is applied to solve the loaded difference scheme by appointing an appropriate solution representation. The unique solvability of the proposed scheme is set up. The stability and convergence of the proposed difference scheme are analysed by the discrete energy method with an order of accuracy \(\mathcal {O}(\tau ^{2-\alpha }+h^2)\) $\mathcal{O}({\tau }^{2-\alpha }+h^2)$ . Numerical results via two test problems are presented to validate the theoretical findings of the proposed scheme by observing the errors.