Asymptotic Solution for Three-Dimensional Reaction–Diffusion–Advection Equation with Periodic Boundary Conditions

Abstract: In this study, we investigate the dynamics of moving fronts in three-dimensional spaces,which form as a result of in-situ combustion during oil production. This phenomenon is alsoobserved in other contexts, such as various autowave models and the propagation of acousticwaves. Our analysis involves a singularly perturbed reaction–diffusion–advection typeinitial–boundary value problem of a general form. We employ methods from asymptotic theory todevelop an approximate smooth solution with an internal layer. Using local coordinates, we focuson the transition layer, where the solution undergoes rapid changes. Once the location of thetransition layer is established, we can describe the solution across the full domain of the problem.Numerical examples are provided, demonstrating the high accuracy of the asymptotic method inpredicting the behaviors of moving fronts.