Approximation of a system of semilinear singularly perturbed parabolic reaction-diffusion equations on a vertical strip

Second-Order Robust Numerical Method for a Partially Singularly Perturbed Time-Dependent Reaction–Diffusion System

“…SPPs often occur in different applied fields such as control systems or fluid dynamics, among others [1][2][3]. In particular, systems of Singularly Perturbed Partial Differential Equations (SPPDEs) frequently arise in the modeling of heat and mass transfer processes whenever the diffusion coefficients and the thermal conductivity are very small [4]. In [2,3], some mathematical models are developed for generators and their controls in control systems using singular perturbation techniques.…”

Pointwise error estimates for a system of two singularly perturbed time‐dependent semilinear reaction–diffusion equations

“…A linear system of parabolic reaction–diffusion equations is considered in Gracia and Lisbona 32 and Gracia et al 33 . A semilinear system of two singularly perturbed parabolic reaction–diffusion equations on a vertical strip is considered in GI Shishkin and LP Shishkin, 34,35 where the highest order derivatives having divergent form are multiplied by a small perturbation parameter ε 2 . In the solution, the presence of parabolic boundary layers of width is observed in the neighborhood of the strip boundary when ε → 0.…”

“…In the solution, the presence of parabolic boundary layers of width is observed in the neighborhood of the strip boundary when ε → 0. In GI Shishkin and LP Shishkin, 34 the condensing mesh method and the classical finite difference method are applied for the construction of ε ‐uniformly convergent difference scheme that converges ε ‐uniformly at the rate of , where are number of mesh intervals on x 1 ‐axis, t ‐axis respectively, N 2 is the number of mesh intervals per unit length on the x 2 ‐axis and . Whereas in GI Shishkin and LP Shishkin, 35 the integro‐interpolation method is used to construct a conservative nonlinear finite difference scheme that converges ε ‐uniformly at the rate of , where N 1 and N 0 are number of mesh intervals on x 1 ‐axis and t ‐axis, respectively, and N 2 is the number of mesh intervals per unit length on the x 2 ‐axis.…”

Conservative Numerical Method for a System of Semilinear Singularly Perturbed Parabolic Reaction‐diffusion Equations

“…When ε tends to zero, the parabolic boundary layer with the typical width ε appears in a neighbourhood of the strip boundary. A similar problem first was considered in [14], where the condensing grid method and classical difference approximations of the boundary value problem were applied for the construction of ε-uniformly convergent difference schemes. In this paper, using the integro-interpolational method, a conservative nonlinear finite difference scheme is constructed that converges ε-uniformly at the rate O N −2 1…”