INTRODUCTION
Aim of the study is to solve the differential equations using differential transform method which are often encounter in applied sciences and engineering. There are different methods available exact, approximate and numerical for the solution of differential equations. Most of these methods are computationally exhaustive because they require lot of time and space.
The concept of differential transform was first proposed by Zhou1 and it was applied to solve linear and non-linear initial value problems in electric circuit analysis. Jang et al.2 stated that the differential transform is an iterative method for obtaining Taylor series solutions of differential equations. Although, the Taylor series method requires more computational work for large orders, the present method reduces the size of computational domain and is applicable to many problems easily3-6. A method of differential transform was used to obtain approximate solutions of the linear and non-linear equations related to engineering problems and observed that the numerical results are in good agreement with the analytical solutions. In this study, the differential transformation technique is applied to solve differential equations. The differential transformation technique can be used to obtain both numerical and analytical solutions of both linear and non-linear differential equations.
The differential transformation technique uses the polynomials as the approximation to the exact solution. The high-order Taylor series method can also be applied to differential equations. However, the Taylor method requires the calculation of high-order derivatives, a difficult symbolic and complex problem7-10.
MATERIALS AND METHODS
Basic definitions: The basic idea of differential transform method is illustrated.
Definition 1: The one dimensional differential transform of function v(t) is defined as follows:
Definition 2: If v(t) can be expressed by Taylor series, then v(t) is given by:
Equation 2 is called the inverse of V(k). Now combining Eq. 1 and 2:
Properties of DTM: If U(k) and V(k) are corresponding transformed functions of u(t) and v(t). Then we have following DTM properties:
| • |
If z(t) = u(t)±v(t) then: |
| • |
If z(t) = u(t)v(t) then: |
| • |
If z(t) = sin(wt+α) then: |
RESULTS ANS DISCUSSION
Numerical example 1: Consider the differential equation:
with the initial condition:
Now, taking the DTM to Eq. 4 and 5:
and
The numerical results of example 1 by DTM are presented in Table 1 and the approximate solution is:
Using the Laplace transform method, the exact solution is given by:
Example 2: Consider the differential equation:
with initial conditions:
Taking the DTM of Eq. 12 and 13:
The numerical results for example 2 by DTM are presented in Table 2 and the approximate solution is given by:
Using the Laplace transform, the exact solution is given by:
Example 3: Consider the simultaneous differential equations:
with initial conditions:
Taking the DTM of Eq. 18 and 19:
The numerical results for example 3 by DTM are presented in Table 3 and the approximate solution for x(t) is given by:
| Table 1: | Numerical results of example 1 by DTM |
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| Table 2: | Numerical results for example 2 by DTM |
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| Table 3: | Numerical results for example 3 by DTM |
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and approximate solution for y(t) is given by:
Using Laplace transform the exact solution is given by:
Example 4: Consider the following simultaneous differential equation:
with initial conditions:
Taking the differential transform of both the given equations:
The numerical results for example 4 by DTM are presented in Table 4. Therefore the approximate solution for x(t) is:
= X(0)+X(1)t+X(2)t2+X(3)t3+... |
(35) |
| Table 4: | Numerical results for example 4 by DTM |
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and the approximate solution for y(t) is:
= Y(0)+Y(1)t+Y(2)t2+Y(3)t3+... |
(38) |
Using Laplace transform the exact solution is given by:
and
CONCLUSION
The solution obtained by DTM and Laplace transform are compared. It is observed that DTM save time and space and easy to implement. Differential transform method gives solution very fast and does not require linearization, perturbation or any other assumption.
ACKNOWLEDGMENT
Author is very thankful to the reviewer and editor of the Journal for their valuable suggestions to improve the research paper.