A Uniformly Convergent Scheme for A System of Two Coupled Singularly Perturbed Reaction-Diffusion Robin Type Boundary Value Problems with Discontinuous Source Term
Pathan Mahabub Basha1,, Vembu Shanthi1
1Department of Mathematics, National Institute of Technology, Tiruchirappalli, India
Abstract | |
1. | Introduction |
2. | Continuous Problem |
3. | Discrete Problem |
4. | Error Analysis |
5. | Numerical Results |
6. | Conclusions |
Acknowledgement | |
References |
Abstract
In this paper, a uniformly convergent scheme for a system of two coupled singularly perturbed reaction-diffusion Robin type mixed boundary value problems (MBVPs) with discontinuous source term is presented. A fitted mesh method has been used to obtain the difference scheme for the system of MBVPs on a piecewise uniform Shishkin mesh. A cubic spline scheme is used for Robin boundary conditions and the classical central difference scheme is used for the differential equations at the interior points. An error analysis is carried out and numerical results are provided to show that the method is uniformly convergent with respect to the singular perturbation parameter which supports the theoretical results.
Keywords: singular perturbation problem, weakly coupled system, discontinuous source term, Robin boundary conditions, Shishkin mesh, fitted mesh method, uniform convergence
Received January 1, 2015, Revised February 25, 2015; Accepted September 08, 2015
Copyright © 2015 Science and Education Publishing. All Rights Reserved.Cite this article:
- Pathan Mahabub Basha, Vembu Shanthi. A Uniformly Convergent Scheme for A System of Two Coupled Singularly Perturbed Reaction-Diffusion Robin Type Boundary Value Problems with Discontinuous Source Term. American Journal of Numerical Analysis. Vol. 3, No. 2, 2015, pp 39-48. https://pubs.sciepub.com/ajna/3/2/2
- Basha, Pathan Mahabub, and Vembu Shanthi. "A Uniformly Convergent Scheme for A System of Two Coupled Singularly Perturbed Reaction-Diffusion Robin Type Boundary Value Problems with Discontinuous Source Term." American Journal of Numerical Analysis 3.2 (2015): 39-48.
- Basha, P. M. , & Shanthi, V. (2015). A Uniformly Convergent Scheme for A System of Two Coupled Singularly Perturbed Reaction-Diffusion Robin Type Boundary Value Problems with Discontinuous Source Term. American Journal of Numerical Analysis, 3(2), 39-48.
- Basha, Pathan Mahabub, and Vembu Shanthi. "A Uniformly Convergent Scheme for A System of Two Coupled Singularly Perturbed Reaction-Diffusion Robin Type Boundary Value Problems with Discontinuous Source Term." American Journal of Numerical Analysis 3, no. 2 (2015): 39-48.
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At a glance: Figures
1. Introduction
Singular perturbation problems (SPPs) arise in various fields of science and engineering which include fluid mechanics, fluid dynamics, quantum mechanics, control theory, semiconductor device modeling, chemical reactor theory, elasticity, hydrodynamics, gas porous electrodes theory, etc. SPPs are characterized by the presence of a small parameter that multiplies the highest derivative term. This leads to boundary and/or interior layers in the solution of such problems. A much attention has been drawn on these problems to obtain good approximate solutions for the past few decades. Since classical numerical methods fail to produce good approximations for these equations, it is inevitable to go for non-classical methods. There are several articles available at the literature but they are mainly based on singularly perturbed problems containing one equation. Some authors have developed robust numerical methods for a system of singularly perturbed convection-reaction-diffusion problems on smooth data. Very few researchers can be seen for problems with non-smooth data which frequently arises in electro analytic chemistry, predator-prey population dynamics, etc. as a perfect application. Oseen equations form a convection-diffusion system where as linearized Navier-Stokes equations yield a reaction-diffusion system at large Reynolds number.
For a parameter-uniform methods pertaining to singular perturbation problems, one can refer the books [1, 2, 3]. A standard finite difference method is proved uniformly convergent on a fitted piece wise uniform Shishkin mesh for a single equation reaction-diffusion problem [2]. The same approach for coupled system of two singularly perturbed reaction-diffusion problems, with diffusion coefficients was originally proposed by Shishkin [4] and identified three different cases
For case-
, Matthews et al. [5] proved almost first order convergence using classical finite difference scheme on Shishkin mesh for a system of singularly perturbed reaction-diffusion equations subject to Dirichlet boundary conditions. Tamilselvan et al. [6] developed a numerical method using fitted piecewise uniform Shishkin mesh for the coupled system of singularly perturbed reaction-diffusion equations for case-
with discontinuous source term subject to Dirichlet boundary conditions and obtained almost first order uniform convergence. Singularly perturbed linear second order ordinary differential equations of reaction-diffusion type with discontinuous source term subject to Dirichlet boundary conditions having diffusion parameters with different magnitudes was studied by Paramasivam et al. [7]. In that paper, the authors constructed a numerical method using classical finite difference scheme on Shishkin mesh with first order parameter-uniform accuracy. Using mesh equidistribution technique, Das and Natesan [8] studied the singularly perturbed system of reaction-diffusion problems subject to Dirichlet boundary conditions on smooth data having diffusion parameters with different magnitudes. In that article, the central difference scheme is used to discretize the problem on adaptively generated mesh and obtained an optimal second order parameter uniform convergence.
In recent years, system of singularly perturbed Robin type reaction-diffusion problems has attracted a lot of attention for many researchers. Das and Natesan [9] achieved perfect second order accuracy for a single second order Robin type reaction-diffusion problems using adaptively generated grid for smooth case. In that article, the authors proposed the cubic spline difference scheme for mixed boundary conditions and the classical central difference scheme for the differential equation at the interior points to get second order parameter uniform convergence. Das and Natesan [10] also proposed an efficient hybrid numerical scheme, which uses cubic spline difference scheme in the inner region and central difference scheme in the outer region, for singularly perturbed system of Robin type reaction-diffusion problems on Shishkin meh for smooth case. It has been shown that the scheme is -uniform convergent with almost second order accuracy. Two hybrid difference schemes on the Shishkin mesh were constructed by Mythili Priyadharshini and Ramanujam [11] for solving the singularly perturbed coupled system of convection-diffusion equations with mixed type boundary conditions on smooth data which generate
-uniform convergent numerical approximations to the solution. Recently, Mahabub Basha and Shanthi [12] have considered a numerical method for singularly perturbed coupled system of convection-diffusion Robin type boundary value problems with discontinuous source term. Motivated by the above works, in this article, we have developed a uniformly convergent numerical method on the Shishkin mesh for a system of two coupled singularly perturbed reaction-diffusion Robin type boundary value problems with discontinuous source term.
This paper is organized as follows: In Section-2, some analytical results of the solution of singularly perturbed MBVP with discontinuous source term are presented. The numerical method is described in Section-3. Error analysis is carried out in Section-4. Numerical examples are provided in Section-5 and conclusions are given in Section-6.
Throughout this paper, denotes a generic positive constant independent of the singular perturbation parameter
, the nodal points
and the number of mesh intervals N which may not be same at each occurrence. Let
The norm which is suitable for studying the convergence of numerical solution to the exact solution of the singular perturbation problem is the maximum norm
. Further,
and
2. Continuous Problem
2.1. Statement of the ProblemFind such that
![]() | (1) |
![]() | (2) |
with the boundary conditions
![]() | (3) |
![]() | (4) |
![]() | (5) |
![]() | (6) |
where is a small parameter
,
and
are such that
![]() | (7) |
![]() | (8) |
and also
![]() | (9) |
Here and
.
It is also assumed that the source terms are sufficiently smooth on
At the point
, the functions
have jump discontinuity. In general, this discontinuity gives rise to interior layers in the solution of the problem. Since
are discontinuous at d, the solution
of (1)-(6) does not necessary to have a continuous second order derivative at the point d. i.e.,
But the first derivative of the solution exists and is continuous.
The above system (1)-(6) can be written in matrix form as
![]() |
with the boundary conditions
![]() |
where and
The jump at d is denoted in any function with
.
Remark-1: The presence of multiplying the derivative terms in the mixed boundary conditions amplifies the significance of the boundary layers at both ends. In the absence of
, the layers are sufficiently weak [9, 13].
In this section, the existence of a solution, the maximum principle and stability result are established for the MBVP (1)-(6).
Theorem-1: The MBVP (1)-(6) has a solution with
.
Proof: The proof is by construction. Let and
be the particular solutions of the following system of equations
![]() |
and
![]() |
respectively.
Also let and
be the solutions of the following MBVPs:
![]() |
and
![]() |
respectively.
Here and
Then can be written as
![]() |
where
![]() |
and
are matrices with constant entries.
On and
cannot have internal maximum or minimum [14].
Hence
Choose the matrices and
so that
i.e., we impose the conditions
and
.
For the matrices and
to exist it requires
![]() |
This implies
Theorem-2: (Maximum principle) Suppose Further suppose that
satisfies
and
. Also let
and
on
Then if there exists a function
such that
and
then
Proof: Define
Assume that the theorem is not true.
Then and there exists a point
such that
or
or both.
Further, or
Also
Case-(i): for
It implies that
attains a minimum at
Therefore,
which is a contradiction.
Case-(ii): i.e.,
Therefore,
![]() |
since ( attains a minimum at
which is a contradiction.
Case-(iii): i.e.,
Since
attains a minimum at
then
which is a contradiction.
Case-(iv): for
It implies that
attains a minimum at
Therefore,
which is a contradiction.
Case-(v): Similar to Case-(ii), it leads to a contradiction.
Case-(vi): Similar to Case-(iii), it leads to a contradiction.
Case-(vii): Similar to Case-(i), it leads to a contradiction.
Case-(viii): Similar to Case-(iv), it leads to a contradiction.
Hence,
Corollary-1: Consider the differential equations (1)-(2) subject to the conditions (7)-(8). Let where
![]() |
and
![]() |
Then the above maximum principle is true for the MBVP (1)-(6).
Remark-2: The MBVP (1)-(6) has a solution and it is unique.
Theorem-3: (Stability result) Consider the differential equations (1)-(2) subject to the quasi-monotonicity and diagonally dominant conditions (7)-(8). If then
![]() |
Proof: Let
Define the functions , where
. It is easy to prove that
and
by a proper choice of
Therefore, by the maximum principle the required result follows.
Remark-3: The MBVP (1)-(6) is well-posed. i.e., the problem has a unique stable solution.
2.3. Derivative EstimatesIn this section, the derivative estimates for the MBVP (1)-(6) are provided.
Theorem-3: Let be the solution of the MBVP (1)-(6). Then for k=1, 2 and
and
Proof: This theorem can be proved by using the results of [10] and [15].
Remark-4: The sharper bounds on the derivatives of the solution are obtained by decomposing the solution into smooth and singular components as
where the smooth component
is given by
![]() |
and the singular component is given by
![]() |
where
![]() |
The solution can be constructed by the procedure given in [16]. Therefore, the singular component is well defined.
Theorem-4: The smooth and singular components and
of
satisfy the bounds
![]() |
and
![]() |
where
![]() |
and
Proof: This theorem can be proved by using the results of [15, 16, 17] and by following the technique of [16, 17, 18].
3. Discrete Problem
A fitted mesh method for problem (1)-(6) is now described. On a piecewise uniform mesh of N mesh intervals is constructed as follows:
The interval is subdivided into three subintervals
and
for some
that satisfies
On
and
a uniform mesh with
mesh intervals is placed while
has a uniform mesh with
mesh intervals. The subintervals
and
of
are treated analogously for some
satisfying
The interior points of the mesh are denoted by
Clearly
and
Note that this mesh is a uniform mesh when
and
The transition parameters
and
are functions of N and
and are chosen as
and
The six mesh widths are given by
On the piecewise uniform mesh , a cubic spline scheme is used for Robin boundary conditions and the classical central difference scheme is used for the differential equations at the interior points. Then the fitted mesh method for MBVP (1)-(6) is:
![]() | (10) |
![]() | (11) |
i.e.,
and at the scheme is given by [19, 20]
![]() |
![]() | (12) |
![]() | (13) |
![]() | (14) |
![]() | (15) |
![]() | (16) |
and can be obtained from the one sided limits
![]() | (17) |
![]() | (18) |
for j=1,2, respectively of the first order derivatives of cubic spline function, given in [9, 10]. Substituting from
and
to (17) and (18), we get the approximation of the one sided first order derivatives at both boundary points. Hence, the discretization of the Robin boundary conditions of (13)-(16) reduce to
![]() | (19) |
![]() | (20) |
![]() | (21) |
![]() | (22) |
The following discrete maximum principle and discrete stability result can be proved analogous to the continuous results stated in Theorem-2 and Theorem-3.
Theorem-5: (Discrete maximum principle) For any mesh function assume that
and
Also let
Then if there exists a mesh function
such that
and
then
Corollary-2: Consider the discrete problem (10)-(15) subject to the conditions (7)-(8). Let where
![]() |
and
![]() |
Then the above discrete maximum principle is true for (10)-(15).
Theorem-6: (Discrete stability result)If is any mesh function, then
![]() |
4. Error Analysis
Using the results of Theorem-4, the procedure adopted in [10, 17] and the basic ideas of the proofs of some theorems presented in [16] for the derivation of estimates for the truncation error, the following inequalities can be derived for the MBVP (1)-(6):
![]() | (23) |
At the point using the procedure adopted in [19, 20] with appropriate barrier functions, it is easy to see that
![]() | (24) |
The truncation errors of the solution y at boundary points x=0,1, for the discrete problem (10)-(16), where Robin boundary conditions are discretized by using spline approximation from (17)-(22), lead to the following estimates [9]:
![]() | (25) |
Theorem-7: The error of the numerical scheme (10)-(16) at inner grid points satisfies
![]() | (26) |
for N sufficiently large.
Proof: Using (23), (24) and (25), the desired result follows.
Remark-5: (Adjoint system) Consider the MBVP (1)-(6).
Suppose that the quasi-monotonicity condition (7) is not satisfied by the system. Then the following system is adjoined to (1)-(2):
![]() | (27) |
![]() | (28) |
![]() | (29) |
![]() | (30) |
![]() | (31) |
![]() | (32) |
![]() | (33) |
![]() | (34) |
and
![]() |
and
![]() |
Since is a solution of (1)-(6),
is a solution of the above adjoint system (27)-(34). The results derived for (1)-(6) still hold good even if the quasi-monotonicity condition is not met.
5. Numerical Results
In this section, two examples are given to illustrate the computational methods discussed in this paper.
Consider the following singularly perturbed Robin type boundary value problems with discontinuous source term:
Example-1:
![]() |
where and
Example-2:
![]() |
where and
The maximum errors and the orders of convergence for the solution of the above two examples are presented for various values of ε and N in the Table 1- Table 2 and Table 3- Table 4 respectively. For a finite set of values maximum point-wise errors
are computed as
for
where
is the piecewise linear interpolant of the mesh function
onto
From these values, the
uniform maximum error is calculated by
Further, the order of convergence is computed by
.
Table 1. Maximum point-wise errors Eε,1N. ε- uniform error E1N and ε- uniform order of convergence P1N for different values of the mesh points N for the solution y1 of Example-1
Table 4. Maximum point-wise errors Eε,2N. ε-uniform error E2N and ε- uniform order of convergence P2N for different values of the mesh points N for the solution y2 of Example-2






6. Conclusions
A system of two coupled singularly perturbed reaction-diffusion Robin type boundary value problem with discontinuous source term was examined. A difference scheme using fitted mesh method on piecewise uniform Shishkin mesh was constructed for solving the problem which gives uniform convergence. A cubic spline scheme is used for Robin boundary conditions and the classical central difference scheme is used for the differential equations at the interior points. From the obtained numerical results, it is noted that the rate of convergence is approaching to almost the second order as N increases and are in agreement with the theoretical results.
Remark-6: The authors are in the process of extending the same analysis for convection-diffusion problems considered in [21].
Acknowledgement
The authors are thankful to the anonymous referee for his constructive comments and valuable suggestions in improving the quality of the paper.
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