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In this paper, Differential Transform Method (DTM) is proposed for the closed form solution of linear and non-linear stiff systems. First, we apply DTM to find the series solution which can be easily converted into exact solution. The method is described and illustrated with different examples and figures are plotted accordingly. The obtained result confirm that DTM is very easy, effective and convenient.

Consider the stiff initial value problem [

on the finite interval, where

and are continuous.

The initial value problem of stiff differential equations occurs in almost every field of science [1-5], particularly, in the fields of:

1) Chemical Reactions: A famous chemical reaction is the Oregenator reaction between HBrO_{2}, Br^{−}, and Ce (IV) described by Field and Noyes in 1984.

2) Reaction-diffusion systems: Problems in which the diffusion is modeled via the Laplace operator may become stiff as they are discretized in space by finite differences or finite elements. Well-known example of such systems which appear so often in mathematical biology.

Several further occurrences of stiffness can be found in electrical circuits, mechanics, meteorology, oceanography and vibrations.

Definition 1: If the solution of the system contains components which change at significantly different rates for given changes in the independent variable, then system is said to be stiff [2,3].

Stiff differential equations are characterized as those whose exact solution has a term of the form, where

is a large positive constant. The key features of stiff equations are that the derivative terms may increase rapidly as t increases [

In the last three decades numerous works have been focusing on the development of more advanced and efficient methods for stiff problems [1,2]. The situation becomes more complicated when stiffness coupled with nonlinearity. Carroll presents an exponential fitted scheme for solving stiff systems of initial value problems [

Differential Transform Method (DTM) is a semi numerical method which gives series solution. But sometimes the series solution can be easily converted into closed form solution. This Method was first introduced by Zhou [

In [

In this paper, we solve the linear and non-linear stiff system via DTM. In Section 2, we give some basic properties of one-dimensional DTM. In Section 3, we have applied the method to linear and non-linear stiff systems.

In this section, we first give some basic properties of one-dimensional differential transform method. Differential transform of a function is defined as follows:

where is the original function and is the transformed function for. The differential inverse transform of is defined as

From Equations (2) and (3) we get

which implies that the concept of DTM is derived from Taylor series expansion, but the method does not evaluate the derivative symbolically. However, relative derivatives are calculated by an iterative procedure which is described by the transformed equations of the original functions.

In this work, we use the lower case letters to represent the original functions and upper case letters to represent the transformed functions. In actual applications, the function is expressed by a finite series Equation (5) can be written as

Here is represented the convergence of natural frequency.

From Equations (2) and (3) we obtain

In this section, we apply DTM to both linear and nonlinear stiff systems.

Problem 1: Consider the linear stiff system:

with initial value,

This system has eigen values of large modulus lying closed to the imaginary axis.

By applying Differential Transformation, we have

The initial conditions of Differential Transformation are given by:

.

For, the series coefficients for and can be obtained as

We used MATHEMATICA to calculate the unknown coefficients and.

Using the inverse Transform, we get

Equations (10) and (11) can be written in the exponential form are given by

and

Thus we get the exact solution by differential transform method.

Problem 2: Consider the non-linear system in the form of initial value problems [

Applying Differential Transform, we have

For the series coefficients for and can be obtained as

We used MATHEMATICA to calculate the unknown coefficients and.

Using the inverse Transform:

This can be written as folows

and

Thus we get the exact solution by differential transform method.

Problem 3: Consider the system of initial value problems [

with initial conditions, i = 1, 2, 3, 4.

Applying Differential Transform, we have

;

;

;

.

With the transformed initial conditions are,.

For the series coefficients for, , and can be obtained as:

Using the inverse Transform:

We obtain

, , ,.

In this section, we have presented three different linear and nonlinear stiff systems via Differential Transform Method and the series solution of Equations (13) and (22) have shown in Figures 1 and 2 respectively.

In this work, DTM has been applied to the exact solution of linear and non-linear stiff system. DTM is a semi numerical method and we obtained a closed form solution such as [

observed that DTM is simpler, effective and reliable.