Abstract
This paper deals with numerical treatment of nonstationary singularly perturbed delay convection-diffusion problems. The solution of the considered problem exhibits boundary layer on the right side of the spatial domain. To approximate the term with the delay, Taylor’s series approximation is used. The resulting time-dependent singularly perturbed convection-diffusion problems are solved using Crank-Nicolson method for temporal discretization and hybrid method for spatial discretization. The hybrid method is designed using mid-point upwind in regular region with central finite difference in boundary layer region on piecewise uniform Shishkin mesh. Numerical examples are used to validate the theoretical findings and analysis of the proposed scheme. The present method gives accurate and nonoscillatory solutions in regular and boundary layer regions of the solution domain. The stability and the uniform convergence of the scheme are proved. The scheme converges uniformly with almost second-order rate of convergence.
Highlights
Perturbed differential equations are differential equations in which the highest-order derivative term is multiplied by a small perturbation parameter ε
Perturbed Delay Differential Equations (SPDDEs) are differential equations in which the highestorder derivative term is multiplied by small perturbation parameter ε and involves at least one delay term
Ramesh and Kadalbajo [5] first treated the delays using Taylor’s series approximation and developed numerical scheme using upwind for time derivative discretization with upwind and mid-point upwind for the spatial derivatives discretization on Shishkin mesh
Summary
Perturbed differential equations are differential equations in which the highest-order derivative term is multiplied by a small perturbation parameter ε. While using the standard numerical methods, a International Journal of Differential Equations large number of mesh points are required, which is not practical due to round-off error, computer processing ability, and computer memory issue To handle this drawback, recently, different authors have developed numerical schemes for solving singularly perturbed parabolic problems having delay on the spatial variable. Ramesh and Kadalbajo [5] first treated the delays using Taylor’s series approximation and developed numerical scheme using upwind for time derivative discretization with upwind and mid-point upwind for the spatial derivatives discretization on Shishkin mesh. E proposed scheme consists of Crank-Nicolson method for temporal discretization with hybrid FDM (mid-point upwind in outer layer and central difference in boundary layer) on Shishkin mesh for spatial discretization. Symbol C(in some cases indexed) denotes positive constant independent of ε and N. e norm ‖.‖ represents the maximum norm
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
