Abstract
In this article, a numerical solution is proposed for singularly perturbed delay parabolic reaction-diffusion problem with mixed-type boundary conditions. The problem is discretized by the implicit Euler method on uniform mesh in time and extended cubic B-spline collocation method on a Shishkin mesh in space. The parameter-uniform convergence of the method is given, and it is shown to be ε -uniformly convergent of O Δ t + N − 2 ln 2 N , where Δ t and N denote the step size in time and number of mesh intervals in space, respectively. The proposed method gives accurate results by choosing suitable value of the free parameter λ . Some numerical results are carried out to support the theory.
Highlights
IntroductionMany researchers have proposed different numerical methods to solve singularly perturbed time delay parabolic reaction-diffusion equations; for instance, see [3,4,5,6,7,8]
Let Ω = ð0, 1Þ, D = Ω × ð0, T, and Γ = Γl ∪ Γb ∪ Γr, where Γl and Γr are the left and right sides of the rectangular domain D corresponding to x = 0 and x = 1, respectively, and Γb is the base of the domain and given by Γb=1⁄20, 1 × 1⁄2−τ, 0
To see the applicability and efficiency of the proposed method, an example is considered from the literature
Summary
Many researchers have proposed different numerical methods to solve singularly perturbed time delay parabolic reaction-diffusion equations; for instance, see [3,4,5,6,7,8]. The authors in [11] studied singularly perturbed time delay parabolic reaction-diffusion equation subject to mixed-type boundary conditions, yet to the best of our knowledge, no further study has been done. We apply the extended form of cubic B-spline collocation method for singularly perturbed time delay parabolic reaction-diffusion problem subject to mixed boundary conditions. The authors in [17, 18] studied singularly perturbed semilinear differential equation of reaction-diffusion and convection-diffusion type, respectively, using cubic B-spline collocation method on a piecewise Shishkin mesh.
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