A Study on Spatially Loaded Time‐Fractional Diffusion Equation: Well‐Posedness, Discretization, and Simulation

ABSTRACT Loaded differential equations are crucial for representing occurrences having multiple pointwise effects on the overall state. This article presents an analytical as well as numerical study on the spatially loaded Caputo‐type time‐fractional diffusion equation with initial and non‐homogeneous Dirichlet boundary conditions. The well‐posedness of the system is established using the Galerkin approximation method, and some a priori estimates of the state variable are derived. The numerical discretization of the problem is achieved via the finite difference method, which uses the well‐known method for Caputo derivative approximation. This process yields a spatially loaded implicit finite difference scheme, which reduces to a parametric system of linear equations. There, the superposition property of systems of linear algebraic equations is used to obtain a parametric representation of solutions using auxiliary linear systems of special parametric structures. In subsequent analysis, the unique solvability of the proposed scheme is proved. Also, the discrete energy method is implemented to derive unconditional stability and convergence of the proposed scheme for non‐positive load coefficients. In the end, several numerical experiments are conducted over a few test examples to validate the accuracy and efficiency of the proposed scheme.