Numerical techniques for two-parameter elastic foundation using integro-partial differential equations

This article presents numerical techniques for solving two-parameter singularly perturbed differential equations, which include a Fredholm integral term. Such problems arise in shell structures interacting with two-parameter elastic foundations. The proposed approach employs a developed exponentially fitted operator for the spatial component, the composite trapezoidal rule for the integral component on a uniform grid, and the backward Euler method for the temporal component to approximate the solution. The method achieves a first-order convergence rate when ϵ ≪ μ 2 , and a second-order rate when μ 2 ≪ ϵ in the spatial direction and first-order convergence in the temporal direction. Numerical findings are presented to demonstrate the theoretical framework of the proposed technique.