Parameter-uniformly convergent numerical scheme for singularly perturbed delay parabolic differential equation via extended B-spline collocation

Abstract

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This paper presents a parameter-uniform numerical method to solve the time dependent singularly perturbed delay parabolic convection-diffusion problems. The solution to these problems displays a parabolic boundary layer if the perturbation parameter approaches zero. The retarded argument of the delay term made to coincide with a mesh point and the resulting singularly perturbed delay parabolic convection-diffusion problem is approximated using the implicit Euler method in temporal direction and extended cubic B-spline collocation in spatial orientation by introducing artificial viscosity both on uniform mesh. The proposed method is shown to be parameter uniform convergent, unconditionally stable, and linear order of accuracy. Furthermore, the obtained numerical results agreed with the theoretical results.

Keywords

• singularly perturbed delay differential equations
• extended cubic B-spline collocation scheme
• implicit Euler method
• artificial viscosity
• parabolic convection-diffusion
• blending function