YILDIZ TECHNICAL UNIVERSITYCIVIL ENGINEERING DEPARTMENT

MECHANICS DIVISIONAPPLIED ENGINEERING

MATHEMATICS2nd midterm homework on

DIFFERENTIAL TRANSFORM METHODBY

SAIF ALDIEN SAMI SHAKIRStudent ID No: 15596006

LIST OF CONTENT

1. INTRODUTION

2. DEFINITIONS AND THEOREMS OF DTM

2.1. THE ONE-DIMENSIONAL DIFFERENTIAL TRANSFORM METHOD

2.2. THE TWO-DIMENSIONAL DIFFERENTIAL TRANSFORM METHOD

2.3. THE THREE-DIMENSIONAL DIFFERENTIAL TRANSFORM METHOD

3. EXAMPLES

4. REFERENCES

1. INTRODUTION

• The differential transform method (DTM) is a semi-analytical-numerical method that uses Taylor series for the solution of differential equations. The DTM is an alternative procedure for obtaining the Taylor series solution of various types of differential equations. Moreover, in this method, it is possible to obtain highly accurate results or exact solutions for differential equations.The concept of the differential transform method was first proposed by [Zhou (1986)], who solved linear and nonlinear initial value problems in electric circuit analysis.

• By using this method, it is possible to solve different types of problems in different fields. For example, Ayaz, F. [2] studied the solutions of the systems of differential equations by differential transform method. Hassan, I. [3] studied different applications for the differential transformation in the differential equations. Hassan, I. and Vedat S. E. [4] used DTM to solutions of different types of the linear and non-linear higher-order boundary value problems. Jafari, H., Sadeghi, S. and Biswas, A. [5] used DTM for solving multidimensional partial differential equations. Arikoglu, A. and Ozkol, I. [6,7] studied the solution of fractional differential equations and solutions of integral and integro-differential equation systems by using differential transform method. Jang, M., and others [8] used two-dimensional differential transform method for partial differential equations. Finally, Bakhshi, M. and Asghari-Larimi, M. [9] and Garg, M. and Manohar, P. [10] used Three-Dimensional DTM for solving nonlinear three-dimensional volterra integral equation and for space-time fractional diffusion equation in two space variables with variable coefficients, respectively.

• In this presentation we will explain the different dimensional DTM and takes some examples in different kind of problems.

2. DEFINITIONS AND THEOREMS OF DTM

• 2.1. The One-Dimensional Differential Transform Method

2.1.1. Definitions: The one–dimensional differential transform method of a function y(x) at the point (x=x0) is defined as follows

(1)

Where: y(x) is the original function and Y(K) is the transformed function

The differential inverse transform of Y(K) is defined as follows:

(2)

From equations (1) and (2) we obtain equation (3):

(3)

2.1.2.Theorems: The following theorems that can be deduced from Eqs. (2) and (3) are given below:

Table (1): Operations of differential transformation.

2.2. The Two-Dimensional Differential Transform Method

• Definitions: If w(x, t) is analytic and continuously differentiable with respect to time (t) in the domain of interest, then:

(4)

Where the spectrum function W(k, h) is the transformed function, which is also called the T -function. Now we define the differential inverse transform of W(k, h) as following

(5)

From equations (4) and (5) we obtain equation (6)

(6)

Where x0 = 0 and t0 = 0

2.2.2. Theorems: Now from the above definitions and Eqs. (5) and (6), we can obtain some of the fundamental mathematical operations performed by two-dimensional differential transform in Table (2)

Table (2): Operations of two-dimensional differential transformation

2.3. The Three-Dimensional Differential Transform Method

• 2.3.1. Definitions: Consider the analytical function of three variables u(x, y, z). Three dimensional Differential Transform method of u(x, y, z)is denoted by U(k, h, l) is defined as following:

(7)

Where u(x, y, z) is original function and U(k, h, l) is called transformed function and the inverse differential transform for U(k, h, l) in Equation (7) is defined as following:

(8)

By combining Equations (7) and (8) with (x0, y0, z0) = (0,0,0), the function u(x, y, z) can be written as:

(9)

Theorems: If U(k, h, l), F(k, h, l) and G(k, h, l) are three-dimensional differential transform of the functions u(x, y, z), f(x, y, z), g(x, y, z) respectively, and from Eqs. (8) and (9), we can obtain some of the fundamental mathematical operations as following:

3. EXAMPLES• Example (1): Consider the following system of simultaneous

linear differential equations.

(1)

(2)

With the condition x(0) = 8, y(0) = 3.

Taking the differential transform method to equation (1) and (2), using above mentioned theorem we obtain:

(3)

(4)2X(k)

With initial conditions X(0) = 8 and Y(0) = 3

The approximation solution when n = 3 (number of terms) using the following equations to get the solution of x(t) and y(t):

By using the Laplace transform method, the exact solution of example (1) is

x(t) = 5e-t + 3e4t

y(t) = 5e-t – 2e4t

Example (2): Consider the following non-homogenous differential equations:

(1)

(2)

(3)

With the condition x(0) = 1, y(0) = 0, z(0) = 2.

Taking the differential transform method to equation (1), (2) and (3), using above mentioned theorem we obtain:

(4)

(5)

(6)

With initial conditions X(0) = 1, Y(0) = 0, and Z(0) = 2

0 12

The approximation solution when n = 3 (number of terms) using the following equations to get the solution of x(t), y(t) and z(t):

x(t) = 1 + t + t2 + t3 + t412

16

124

y(t) = t - t316

z(t) = 2 + t + t3 + t4 16

112

By using the Laplace transform method, the exact solution of example (2) is:

• x(t) = et • y(t) = sin (t)• z(t) = et + cos (t)

Example (3): Consider second order homogenous differential equation:

(1)

With the initial condition y(0) = -1and y 7( = 0)׳ .

We apply DTM on Eq.(1), with initial conditions Y(0) = -1 and Y(1) = 7.

(2)

19

2Put k = 0, Y(2) =

Put k = 1, Y(3) =1

2

Put k = 2, Y(4) = -89

24

Now we applied the formula of y(t) as following

y(t) = -1 +7 t + t2 + t3 - t419

2

1

2

89

24

Using the Laplace transform method, the exact solution of this example is:

• y(t) = - et cos (2t) + 4e2t sin (2t)

Example (4): Consider second order non-homogenous differential equation:

(1)

With the initial condition y(0) = -2 and y 8( = 0)׳ .

We apply DTM on Eq.(1), with initial conditions U(0) = -2 and U(1) = 8, we get:

(2)

Put k = 0, Y(2) = - 15

Put k = 1, Y(3) = 413

Put k = 2, Y(4) = - 10712

Now we applied the formula of y(t) as following:

y(t) = -2 +8 t -15 t2 + t3 - t4413

10712

Using the Laplace transform method, the exact solution of this example is:

y(t) = 6e-t - 8e-2t

4. REFERENCES1. Zhou, J. K. (1986), Differential transformation and its applications for

electrical circuits. Huazhong University Press, Wuhan, China, (in Chinese).2. Ayaz, F. (2004), Solutions of the systems of differential equations by

differential transform method, Applied Mathematics and Computation, 147, 547-567.

3. Hassan, I. (2002), Different applications for the differential transformation in the differential equations, Applied Mathematics and Computation, 129, 183-201.

4. Hassan, I. and Vedat S. E. (2009), Solutions of Different Types of the linear and Non-linear Higher-Order Boundary Value Problems by Differential Transformation Method, European Journal of Pure and Applied Mathematics, Vol. 2, No. 3, (426-447).

5. Jafari, H., Sadeghi, S. and Biswas, A. (2009), The differential transform method for solving multidimensional partial differential equations, Indian Journal of Science and Technology, Vol. 5, No. 2.

6. Arikoglu, A. and Ozkol, I. (2007), Solution of fractional differential equations by using differential transform method. Chaos, Soliton and Fractals, 34, 1473-1481.

7. Arikoglu, A. & Ozkol, I. (2008). Solutions of integral and integro-differential equation systems by using differential transform method. Comput. Math. Appl., 56, 2411-2417.

8. Jang, M., Chen, C. and Liy, Y. (2001), Two-dimensional differential transform for partial differential equations. Appl. Math. Comput., 121, 261-270.

9. Bakhshi, M. and Asghari-Larimi, M. (2012), Three-Dimensional differential transform method for solving nonlinear three-dimensional volterra integral equations, The Journal of Mathematics and Computer Science Vol. 4, No.2, 246 – 256.

10. Garg, M. and Manohar, P. (2015), Three-Dimensional generalized differential transform method for space-time fractional diffusion equation in two space variables with variable coefficients, Palestine Journal of Mathematics, Vol. 4(1) , 127–135.

11. Vedat, S. E. , Zaid, M. O. and Shaher, M. (2012), The Multi-Step Differential Transform Method and Its Application to Determine the Solutions of Non-Linear Oscillators, Adv. Appl. Math. Mech., Vol. 4, No. 4, pp. 422-438.

12. Soltanalizadeh, B. (2011), Differential transformation method for solving one-space-dimensional telegraph equation, Computational and Applied Mathematics Vol. 30, No. 3, pp. 639–653.