No abstract

In this article, a parameter uniform Galerkin finite element method for solving singularly perturbed parabolic reaction diffusion problems with retarded argument is proposed. The solution of this class of problems exhibits parabolic boundary layers. The domain is discretized with a piecewise uniform mesh (Shishkin mesh) for spatial variable to capture the exponential behavior of the solution in the boundary layer region and backward‐Euler method on a equidistant mesh in the time direction. The method is proved to be unconditional stable and parameter uniform. The method is shown to be accurate of order O()N−1lnN2+Δt in maximum norm using Green's function approach. The convergence of the proposed method does not depend on the singular perturbation parameter.

In this article, a closed-form iterative analytic approximation to a class of nonlinear singularly perturbed parabolic partial differential equation is developed and analysed for convergence. We have considered both parabolic reaction diffusion and parabolic convection diffusion type of problems in this paper. The solution of this class of problem is polluted by a small dissipative parameter, due to which solution often shows boundary and interior layers. A sequence of approximate analytic solution for the above class of problems is constructed using Lagrange multiplier approach. Within a general frame work, the Lagrange multiplier is optimally obtained using variational theory and Liouville-Green’s transformation. The sequence of approximate analytic solutions so obtained is proved to converge the exact solution of the problem. To demonstrate the proposed method’s efficiency and accuracy, two linear and one nonlinear test problem have been taken into account.

In this article, a closed-form iterative analytic approximation to a class of nonlinear singularly perturbed parabolic partial differential equation is developed and analysed for convergence. We have considered both parabolic reaction diffusion and parabolic convection diffusion type of problems in this paper. The solution of this class of problem is polluted by a small dissipative parameter, due to which solution often shows boundary and interior layers. A sequence of approximate analytic solution for the above class of problems is constructed using Lagrange multiplier approach. Within a general frame work, the Lagrange multiplier is optimally obtained using variational theory and Liouville-Green's transformation. The sequence of approximate analytic solutions so obtained is proved to converge the exact solution of the problem. To demonstrate the proposed method's efficiency and accuracy, two linear and one nonlinear test problem have been taken into account.