A uniformly convergent hybrid scheme for singularly perturbed system of reaction-diffusion Robin type boundary-value pro
- PDF / 800,672 Bytes
- 25 Pages / 439.37 x 666.142 pts Page_size
- 31 Downloads / 274 Views
JAMC
C O M P U TAT I O N A L M AT H E M AT I C S
A uniformly convergent hybrid scheme for singularly perturbed system of reaction-diffusion Robin type boundary-value problems Pratibhamoy Das · Srinivasan Natesan
Received: 19 April 2012 / Published online: 1 November 2012 © Korean Society for Computational and Applied Mathematics 2012
Abstract This paper deals with the study on system of reaction diffusion differential equations for Robin or mixed type boundary value problems (MBVPs). A cubic spline approximation has been used to obtain the difference scheme for the system of MBVPs, on a piecewise uniform Shishkin mesh defined in the whole domain. It has been shown that our proposed scheme, i.e., central difference approximation for outer region with cubic spline approximation for inner region of boundary layers, leads to almost second order parameter uniform convergence whereas the standard method i.e., the forward-backward approximation for mixed boundary conditions with central difference approximation inside the domain leads to almost first order convergence on Shishkin mesh. Numerical results are provided to show the efficiency and accuracy of these methods. Keywords System of ordinary differential equations · Singularly perturbed problem · Boundary layers · Piecewise-uniform Shishkin mesh · Cubic spline · Uniform convergence Mathematics Subject Classification 65L10
1 Introduction Singularly perturbed problems arise in several branches of engineering and applied mathematics, including heat and mass transfer in chemical and nuclear engineering, linearized Navier-Stokes equation at high Reynolds number, control theory, etc. In
Dedicated to Professor N. Ramanujam on the occasion of his 60th birthday. P. Das · S. Natesan () Department of Mathematics, Indian Institute of Technology, Guwahati, 781 039, India e-mail: [email protected]
448
P. Das, S. Natesan
this article, we consider the following system of singularly perturbed Robin type reaction-diffusion problems, whose solution exhibits boundary layers: ⎧ d2 ⎪ 0 −ε1 dx L u 2 ⎪ 1 ⎪ L u(x) ≡ u(x) + B(x) u(x) = f(x), ≡ ⎪ d2 ⎪ L2 u ⎪ 0 −ε2 dx ⎨ 2 (1.1) x ∈ Ω = (0, 1), ⎪ ⎪ ⎪ ⎪ Ml u(0) ≡ αk uk (0) − βk uk (0) = Ak , ⎪ ⎪ ⎩ k Mrk u(1) ≡ γk uk (1) + δk uk (1) = Bk , k = 1, 2, where u(x) =
u1 (x) , u2 (x)
B(x) =
b11 (x) b21 (x)
b12 (x) , b22 (x)
f(x) =
f1 (x) . f2 (x)
For simplicity, define Ml u(0) = (Ml1 u(0), Ml2 u(0))T , and Mr u(1) = (Mr1 u(1), Mr2 u(1))T . Here ε1 , ε2 are two singular perturbation parameters, where 0 < ε1 , ε2 1. We consider, the functions bij (x) ∈ C3 (Ω) and fi (x) ∈ C2 (Ω) for i, j = 1, 2, in order to derive the derivative bounds of the solution, appeared in our error analysis. We shall also assume that αk , βk ≥ 0, αk + βk > 0, γk > 0, δk ≥ 0 for k = 1, 2. Under the above conditions, posed on the coefficients of the mixed boundaries, the system of MBVPs (1.1) admits a unique solution u(x) ∈ C4 (Ω) (see, e.g., [3]) on Ω = [0, 1]. In general, the solution u(x) exhibits boundary layers at
Data Loading...
