Analysis and numerical solution of singularly perturbed partial differential equations with large spatial delays and integral boundary conditions: applications in chemical and catalytic systems

This work proposes a hybrid numerical strategy to effectively solve the singularly perturbed partial differential equations (SPPDEs) with integral boundary conditions and substantial spatial delays. For the time discretization, the Crank-Nicolson scheme was chosen because of its stability and second-order precision. In order to maximize accuracy in the vicinity of layers coming from the tiny perturbation parameter and delay parameter, the computational implementation will be carried out using a non-uniform Shishkin-type mesh for spatial discretization using cubic spline interpolation. The approach is tested numerically to verify its robustness and efficiency with respect to integral boundary conditions and delayed feedback. Applications to reaction-diffusion systems, catalytic reactions in porous media, and transport-reaction dynamics in tubular reactors are presented to illustrate the effectiveness of the proposed approach.