## 1 Introduction

A differential equation in which small parameters multiply the highest order derivative and some or none of the lower order derivatives
is known as a singularly perturbed differential equation. In this paper, a class of linear parabolic singularly perturbed second order differential equation of reaction-diffusion type with
initial and Robin boundary conditions is considered.

For a general introduction to parameter-uniform numerical methods for singular perturbation problems, see 11 , 12 , 18 and 19 . In 13 , a Dirichlet boundary value problem for a linear parabolic singularly perturbed differential equation is studied and a numerical method
comprising of a standard finite difference operator on a fitted piecewise uniform mesh is considered and it is proved to be uniform with respect to the small
parameter in the maximum norm. In 14 , a boundary-value problem for a singularly perturbed parabolic PDE with convection is considered on an interval
in the case of the singularly perturbed Robin boundary condition is considered and using a defect correction technique, an ε-uniformly convergent schemes of
high-order time-accuracy is constructed. The efficiency of the new defect-correction schemes is confirmed by numerical experiments. In 15 , a
one-dimensional steady-state convection dominated convection-diffusion problem with Robin boundary conditions is considered and the numerical solutions
obtained using an upwind finite difference scheme on Shishkin meshes are uniformly convergent with respect to the diffusion cofficient.

Consider the following parabolic initial-boundary value problem for a singularly perturbed linear system of second order differential equations

(1) |

with

(2) |

where with
Here, for all are column vectors, and are matrices,
with for all
The parameters are assumed to be distinct and for convenience, to have the ordering

The problem (1), (2) can also be written in the operator form

where the operators are defined by

where is the identity operator. The reduced problem corresponding to (1), (2) is defined by

(3) |

The problem (1), (2) is said to be singularly perturbed in the following sense.

Each component of the solution of (1), (2) is expected to exhibit twin layers of width at and
while the components have additional twin layers of width the components have
additional twin layers of width and so on.

## 2 Solution of the continuous problem

Standard theoretical results on the existence of the solution of (1), (2) are stated, without proof, in this section. See 16 and 17 for more details. For all it is assumed that the components of satisfy the inequalities

(4) |

and for some

(5) |

It is also assumed, without loss of generality, that

(6) |

## 3 Analytical results

The operator satisfies the following maximum principle:

###### Lemma 3.1

###### Lemma 3.2

A standard estimate of the solution

of the problem (1), (2) and its derivatives is contained in the following lemma.###### Lemma 3.3

The Shishkin decomposition of the solution of the problem (1), (2) is

(11) |

where and are the smooth and singular components of the solution respectively.

Taking into consideration, the sublayers that appear for the components, the smooth component is subjected to further decomposition.

(12) |

as all the components have layers. Since components except have sublayers, the components takes the form,

(13) |

Further, have sublayers and hence that leads to the decomposition,

(14) |

Proceeding like this, it is not hard to see that

i.e.

(15) |

where

(16) |

Then using (11) and (15) in (1), (2), it is found that the smooth component of the solution satisfies

(17) |

with

(18) |

and the singular component of the solution satisfies

(19) |

with

(20) |

Consider the following parabolic initial-boundary value problem for a singularly perturbed linear system of second order differential equations

(21) |

with

(22) |

where is a matrix,
with

The problem (21), (22) can also be written in the operator form

where the operators are defined by

where is the identity operator. The reduced problem corresponding to (21), (22) is defined by

The operator satisfies the following maximum principle:

###### Lemma 3.4

###### Lemma 3.5

A standard estimate of the solution of the problem (21), (22) and its derivatives is contained in the following lemma.

###### Lemma 3.6

Bounds on the smooth component of and its derivatives are contained in

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