The numerical simulation of convection delayed dominated diffusion equation

In this paper, we propose a fitted numerical method for solving convection delayed dominated diffusion equation. A fitting factor is introduced and the model equation is discretized by cubic spline method. The error analysis is analyzed for the consider problem. The numerical examples are solved using the present method and compared the result with the exact solution.


Introduction
Consider the following convection delayed dominated diffusion equation ''( ) ( ) '( ) ( ) ( ) 0 y x a x y x b x y x H G on [0,1] : (1) subject to the interval conditions ( ) ( ) y x x (2) where 0 1 H is a perturbation parameter and G is a small shifting parameter of order H .It is also assumed that ( ), ( ), ( ) a x b x x M are smooth functions and J is a constant.These convection diffusion delayed types with dominated convection term problems play an important role in the mathematical modelling of various practical phenomena in the engineering and environmental sciences, for examples include high Reynold's number flow in fluid dynamics, heat transport problems with large Pecklet number, modelling the problems in mathematical biology and semi-conductor devices etc.It is challenging to develop efficient numerical methods for solving convection diffusion with dominated convection term due to the existence of boundary layers.Standard discretization methods for solving such kind of problems are unstable and fails to give accurate results when perturbation parameter H is small.Therefore, it is challenging to develop suitable numerical methods to these problems, whose accuracy does not depend on the parameter H . Lange and Miura [1] initiated the singular perturbation analysis of boundary value problems for differential difference equations with small shifts.The numerical study of second order singularly perturbed delay differential equations has been given in [2][3] and references therein.
In this paper, we present an exponentially fitted method on uniform mesh based on cubic spline method for the convection delayed dominated diffusion equation.The layout of the paper is organized as follows: Continuation of the problem is presented in next section and follows description of the method.In Section 3, the error analysis of the method is discussed.Section 4 ends with the Numerical results.

Continuous problem
An application of Taylor series in (1) yields ICAET 2016 - The fitting factor V U is to be determined in such a way that the solution of Eq.( 6) converges uniformly to the solution Eq.(3 Therefore by the maximum principle [5], we obtain which gives the required estimate.

Let
The spline function x x and the differential equation

S x y and 0
W ! is termed as cubic spline in compression.Following Aziz and Khan [6], we obtain the tridiagonal system 10) and using the following three point approximations for first order derivatives We get the following difference scheme Where 11) gives a system of 1 N equations with 1 N unknowns.These 1 N equations together with the Eq.( 7) are sufficient to solve the system by using Thomas algorithm.

Determination of fitting factor
Taking the limit as 0 h o in Eq.( 11), we obtain Substituting Eq.( 5) in Eq.( 12) and then simplifying, we get the variable fitting factor as follows is a constant fitting factor for left end boundary layer.ICAET 2016 -

Error analysis
Putting exact solution in (14), we get for any choice of 1 O and 2 O whose sum is 1 2 .
Subtracting Eq.( 14) from Eq.( 15) and substituting ( ) , , 1 j j j e y x y j i i r , we get ' ' ' ) ( ) , , , i i x x K K K K Using these expansions and Eq.( 21), we have Where Clearly, it can be seen that For sufficiently small h , the matrix A is irreducible and monotone which implies 1 A exists.Hence from Eq.( 24), we have 1 ( ) i E A T h .From the theory of matrices we have, Where K is constant independent of h .Therefore

Numerical Results
To demonstrate the applicability of the method, we consider two convective diffusion with dominated convection term exhibiting boundary layer at left end of the interval and one right end of the interval.The exact solution of the boundary value problem (1) with constant coefficients is given by In Table 1, represents the comparison between the present method and the method in [7] with fixed H and G .It is observed that the present method gives more accurate results than the method in [7].Table 2 indicates the maximum absolute error for different N and H , it is observed from the results, the method is H uniform convergent.Figure 1 displays the numerical solution for different H values, the solution of the problem exhibits the boundary layer behavior to the left side while the perturbation parameter tends to zero.and G values, it is observed that the boundary layer behavior is not only on perturbation parameter and also depends on the delay parameter.

Determination of fitting factor for right end problem
We assume that ( ) then, the Eq.( 3) exhibits boundary layer at 1 3) and by applying the same procedure as in Section(2) and simplifying, we get is a constant fitting factor for right end boundary value problem.subject to the interval conditions 0 y x on 0 1 2 x ,y Gd d Table 3 gives the comparison between present method and the method in [7] for fixed H and G .It is observed that the method yields results better than the existing method in [7]. Figure 2 the interval [0,1] , where M is positive constant, then the problem (3) exhibits boundary layer at 0 x .From the theory of singular perturbations[4],Now introducing an exponentially fitting factor V U to the Eq.(3), we get 0 y x a x b x y x b x y x cc c

Figure 2 .
Figure 2. indicates the numerical solution for different H and G values, it is observed that the boundary layer indicates the numerical solution for different H and G values.The perturbation parameter tends to zero the solution of the problem exhibits boundary layer behavior to the right side of the interval.Authors would like to thank National Board for Higher Mathematics (NBHM), Government of India for providing financial support under the grant number 2/48(12)/2013/NBHM(R.P.)/R&D II/1084. with

Table 1 .
Numerical solution for Example 1. with

Table 2 .
Maximum absolute errors for the Example 1.

Table 3 .
Numerical solution for Example 2.