Application of Projected Differential Transform Method on Nonlinear PartialDifferential Equations with Proportional Delay in One Variable

0

1 0 1

, , ( , ),..., ( , ), ,

0

, , ,..., ,

, 0 , , ,..., , ,

m m n

n n

n

t x u x t u x t u x t

x x a

f u x t u x t

t a t a

t x a a a m n N

 ∂ ∂  

 ∂ ∂  

   =

 

 

 

∂ ∂

     

 ∂   ∂   

 

≥ ∈

( , ,1 2 , ) u x x xn 1 2 1 1 2

0

( , , , )

( , , , , ) 1

!

n

k n

n k

n

x

u x x x

U x x x k

k x

−

=

 

∂ 

=  

 ∂ 

 

 

ISSN 1818-4952

© IDOSI Publications, 2014

DOI: 10.5829/idosi.wasj.2014.30.03.1841

Application of Projected Differential Transform Method on Nonlinear Partial Differential Equations with Proportional Delay in One Variable

Tarig M. Elzaki

Department of Mathematics, Faculty of Sciences and Arts-Alkamil, King Abdulaziz University, Jeddah-Saudi Arabia

Submitted: Dec 11, 2013; Accepted: Feb 5, 2014; Published:Feb 9, 2014

Abstract:In this paper we solve nonlinear partial differential equations with proportional delay in the variable t using the projected differential transform method. The purpose of the method is obtained analytical or approximate solutions of some nonlinear partial differential equations with proportional delay. This method is more efficient and easy to handle such partial differential equations with proportional delay in comparison to other methods. The result showed the efficiency, accuracy and validation of the projected differential transform method.

Keywords: Projected Differential Transform Method Nonlinear Partial Differential Equations with Proportional Delay Exact Solutions.

INTRODUCTION In this work we will present the projected differential Several numerical and analytical techniques including solve some nonlinear partial differential equations with the Adomian's decomposition method [1], differential proportional delay in the variable t.

transform method [2-9], homotopy perturbation method In this paper we consider the following nonlinear [10-13] and projected differential transform method [14,15], partial differential equations with proportional delay in the have been developed for solving nonlinear delay variablet.

differential equations

Delay partial differential equations arise from various applications, like biology, medicine, control theory,

climate models and many others. Their independent (1)

variables are time t and one or more dimensional variable x , which usually represents position in space but may also represent relative DNA content, size of cells, or their maturation level, or other values. The solutions

(dependent variables) of delay partial differential Projected Differential Transform Methods: In this equations may represent temperature, voltage, or section we introduce the projected differential transform concentrations or densities of various particles, for method [14, 15] which is modified method of the example cells, bacteria, chemicals, animals and so on. differential transform method.

Delay partial differential equations depend on their

solutions at retarded variables or/and they depend on Definition:The basic definition of projected differential smooth averages of the solutions over some retarded transform method of function is defined as intervals. Although delay partial differential equations are

also called partial functional differential equations, the class of functional equations is wider than the class of delay equations.

transform method and some theorems with their proofs to

(2)

( , ,1 2 , ) u x x xn ( , ,1 2 , 1, ) U x x xn− k

( , ,1 2 , 1, ) U x x xn− k

1 2 1 2 1 0

( , , , )_n ( , , ., _n , )( )^k

k o

u x x x ^∞U x x x − k x x

=

=

∑

−

 

( ) ( ) ( ) ( )

(

1 2^{1 2} 1

) (

^{1 2}1 2 1^{1 2}

) (

1 2 1

)

1 , ,..., , ,..., , ,...,

, ,..., , , ,..., , , ,..., ,

n n n

n n n

If z x x x u x x x v x x x

Then z x x x k u x x− x k v x x− x k−

= ±

= ±

( ) ( ) ( )

(

^{1 2}1 2 1

)

^{1 2}

(

1 2 1

)

2 , ,..., , ,...,

, ,..., , , ,..., ,

n n

n n

Ifz x x x cu x x x

Then z x x x − k cu x x x − k

=

=

( ) ( ) ( )

( ) ( ) ( )

1 2 1 2

1 2 1 1 2 1

, ,..., 3 , ,...,

, ,..., , 1 , ,..., , 1

n n

n

n n

du x x x Ifz x x x

dx

Then z x x x − k k u x x x − k

=

= + +

( ) ( ) ( )

( ) ( ) ( )

1 2 1 2

1 2 1 1 2 1

, ,..., 4 , ,...,

, ,..., , ! , ,..., ,

!

n n

n n

n

n n

d u x x x Ifz x x x

dx

Then z x x x k k n u x x x k n

− k −

=

= + +

( ) ( ) ( ) ( )

( )

( ) ( )

1 2 1 2 1 2

1 2 1

1 2 1 1 2 1

0

5 , ,..., , ,..., , ,...,

, ,..., ,

, ,..., , , ,..., ,

n n n

n k

n n

m

If z x x x u x x x v x x x

Then z x x x k

u x x x m v x x x k m

−

− −

=

=

=

∑

−

( ) ( ) ( )

( ) ( )

( )

( )

( )

1 3 2

1 2 2 1

1 2 1 1 2

2 1 2 1 2

1 2 1

1 1 2 1 1

0 0 0 0

2 1 2 1 2 1

1 1

6 , ,..., , ,..., , ,..., ... , ,..., , ,..., ,

... , ,..., , , ,..., ,

... ,

n

n n

n n

n n n

n

k k k

k

n

k k k k

n n

If z x x x u x x x

u x x x u x x x Then

z x x x k

u x x x k

u x x x k k

u x

−

− −

−

−

= = = =

−

−

=

=

−

×

∑ ∑ ∑ ∑

( )

(

1 2 ² 1 ¹ 1

)

^{1 2}

,..., , , ,..., ,

n n n

n n n

x x k k

u x x x k k

− − −

− −

−

−

( , ), ( , ) Z x k U x k

( , ), ( , ) z x t u x t a N∈

( , ) ,t z x t u x

a

 

=  

( , ) ( , ) U x k_k Z x k

= a ( , ) ^m_m ,t ,

z x t u x t a

∂  

=∂  

( )! ( , )

( , )

! ^{k m}

k m U x k m z x k

k a ⁺

+ +

=

( , ) , 1 ,

k k k

kz x t k u x t k k u x t

a a

t t a t

   

∂ = ∂   = ∂   

∂ ∂    ∂   

0 0

( , ) 1 , ! ( , )

k k

k k k k

t t

z x t u x t k U x k

t ₌ a t a ₌ a

∂  = ∂   =

 ∂  ∂  

   

   

0

1 ( , ) 1 ! ( , ) ( , )

( , )

! !

k

k k k

t

z x t k U x k U x k Z x k

k t ₌ k a a

∂   

=  ∂  =  =

0 0

( , ) 1 , ( )! ( , )

k k m

k k m k m k m

t t

z x t u x t k m U x k m

t a t a a

+

+ + +

= =

∂  = ∂   = + +

 ∂  ∂  

   

   

0

1 ( , ) ( , )

!

1 ( )! ( , ) ( )! ( , )

! !

k

k t

k m k m

z x t Z x k

k t

k m U x k m k m U x k m

k a k a

=

+ +

∂ 

=  

 ∂ 

 

+ + + +

 

=  =

Such that is the original function Application of ProjectedDifferential Transform Method and is projected transform function.

And the differential inverse transform of approximating equation (1).

is defined as

(3)

The fundamental theorems of the projected differential transform are

Theorems 1:

Note that c is a constant and n is a nonnegative integer.

on Equation (1): In this section we introduce some theorems in projected differential transform method for

Theorem 2: Let a re the projected differential transform with respect tot of and

, then

(I) If , then

(ii) If then

Proof:

(I) From (2), we have

Therefore

Then by (2), we get

Such that k = 0,1,2,...., (ii) From (i), we have

Then

( , ), ( , ), ( , ) Z x k U x k V x k

( , ), ( , ), ( , ) z x t u x t v x t

( , ) ,t ,t

z x t u x v x

a b

   

=     

0

( , ) ( , )

( , ) ^k _{r k r} , 0,1,2,...,

r

U x r V x k r

Z x k k

a b ⁻

=

=

∑

− = ∞

0

( , ) , ,

1 , 1 ,

k k

k k

k k k r

r k k r k r

r

t t

z x t u x v x

a b

t t

k u x t v x t

r a t a b x a

−

− −

=

 

∂ = ∂      

∂ ∂     

   

  ∂   ∂  

=

∑

   ∂    ∂   

0 0

0

( , ) ! ( , )

( )! ( , ) ! ( , ) ( , )

k k

k r

t r

k

k r r k r

r

z x t k r U x r

t r a

k r V x k r k U x r V x k r

b a b

= =

− −

=

∂  =   

 ∂     

   

 

− − −

 =

 

 

∑

∑

0

( , ) ( , )

( , ) ^k _{r k r} , 0,1,2,...,

r

U x r V x k r

Z x k k

a b ⁻

=

=

∑

− = ∞

( , ), ( , ), ( , ) Z x k U x k V x k

( , ), ( , ), ( , ) z x t u x t v x t , , ,

a b n m N∈

( , ) ⁿ_n ,t ^m_m ,t

z x t u x v x

a a

t x

∂   ∂  

=∂  ∂  

0

( )!( )!

( , ) ( , )

( , ) , 0,1,2,...,

!( )!

k

r n k r m r

r n k r m U x r n V x k r m

Z k x k

r k r a ⁺ b ^{− +}

=

+ − +

+ − +

= = ∞

∑

−

0

( , ) , ,

1 , 1 ,

k k n m

k k n m

k r n k r m

r n r n k r m k r m

r

t t

z x t u x v x

a a

t t t t

k u x t v x t

r a t a b t a

+ − +

+ + − + − +

=

 

∂ = ∂ ∂   ∂  

∂ ∂ ∂  ∂  

   

  ∂   ∂  

=

∑

   ∂    ∂   

0 0

0

( , ) ( )! ( , )

( )! ( , )

!( )!( )! ( , ) ( , )

!( )!

k k

k r n

t r

k r m k

r n k r m r

z x t k r n U x r n

t r a

k r m V x k r m b

k r n k r m U x r n V x k r m r k r a b

= +

=

− +

+ − +

=

∂  =   + + 

 ∂    

   

 

− + − +

 

 

 

+ − + + − +

= −

∑

∑

0

( )!( )!

( , ) ( , )

( , ) , 0,1,2,...,

!( )!

k

r n k r m r

r n k r m U x r n V x k r m

Z k x k

r k r a ⁺ b ^{− +}

=

+ − +

+ − +

= = ∞

∑

−

2 , . , ( , ) , , 0, ( ,0)

2 2

t t

u x u x xu x t x t u x x

t

  ∂   = ≥ =

  ∂  

   

0 1

( 1) ( , ) ( , 1)

2 ( , )

2 .2

k

r k r r

k r U x r U x k r_{− +} xU x k

=

− + − + =

∑

( ,0) U x =x

( ,1) , ( ,2) , ( ,3) ,

2 6

( ,4) ,..., ( , )

24 !

x x

U x x U x U x

x x

U x U x n

n

= = =

= =

0

( , )

! ^{k t}

k

u x t x t xe k

∞

=

=

∑

=

Theorem 3: Let are the projected Therefore differential transform with respect to t of

respectively and a, b, N, then

If , then

Proof: From theorem (2) and Leibnitz, we get

Therefore

And from equation (2), we get

Theorem 4: Let are the projected differential transform with respect to t of

respectively and , then

If , then

Proof:From theorem (7) and Leibnitz, we get

And from equation (2), we get

Applications:In this section we illustrate some examples to explain the presented method. We choose examples have exact solutions.

Example 1:In the first example, consider the following nonlinear first-order partial differential equation with proportional delay in the variable t.

(4)

Using the projected differential transform method with respect to t of Eq. (4), yields

(5)

The projected differential transform method with respect tot of initial condition is , then by this equation and Eq. (5), we have

Substituting in Eq.(3), to get the exact solution of Eq.

(4), in the form

Example 2: Consider the following nonlinear second-order partial differential equation with proportional delay in the variablet.

2

( , ) 42 , . , ( , ) 0, , 0

2 2

u x t u x t u x t u x t x t

x t

t

∂ ∂ − ∂∂   ∂∂  + = ≥

( ,0) ( ,0) , u x

u x x x

t

= ∂ =

∂

0 1

( 2)( 1) ( , 2) ( , )

( , )( 1) ( , 1)

4 0

2 .2

k

r k r r

k k U x k U x k

U x r k r U x k r

x − +

=

+ + + +

∂ − + − +

−

∑

∂ =

( ,0) ( ,0) , u x

u x x x

t

= ∂ =

∂

( ,0) ( ,1) U x =U x =x

( ,2) , ( ,3) , ( ,4) ,..., ( , )

2 6 24 !

x x x x

U x U x U x U x n

= = = =n

0

( , )

! ^{k t}

k

u x t x t xe k

∞

=

=

∑

=

2 2

2 2

( , )

16 , . , cos 0 , , 0

2 2

t t u x t

u x u x x x t

t t x

∂   ∂   + ∂ = ≥

∂   ∂   ∂

( ,0)

( ,0) 0, u x cos

u x x

t

= ∂ =

∂

( 1)( 2)( 1)

k r+ k r− + k r− +

( ,0) ( ,0) 0, u x cos

u x x

t

= ∂ =

∂

( ,0) 0, ( ,1) cos

U x = U x = x

cos cos

( ,2) 0 , ( ,3) , ( ,4) 0, ( ,5) ,...

3! 5!

x x

U x = U x = U x = U x =

( , ) sinh cos u x t = t x

(6) And the projected differential transform with respect

With the initial conditions

(7)

Taking the projected differential transform method with respect to t of Eq. (6), we have

(8)

And the projected differential transform with respect tot of initial conditions will be

(9) Where U(x,k) is the projected differential transform method with respect to t of U(x,t)

Using Eqs. (8) and (9), we have

Substituting in Eq.(3), to get the exact solution of Eq.

(6), in the form

Example 3: Consider the following nonlinear second-order partial differential equation with proportional delay in the variablet.

(10)

With the conditions

(11) Taking the projected differential transform with respect to t of Eq. (10), we have

tot of initial condition will be (13) Using Eqs. (12) and (13), we have

Substituting in Eq.(3), to get the exact solution of Eq.

(10), in the form

CONCLUSION

In this paper we applied the projected differential transformation technique to solve nonlinear partial differential equation with proportional delay in the variablet. The obtained results show that this method is powerful and meaningful for solving partial differential equations with proportional delays. In fact, PDTM is very efficient methods to find the numerical and analytic solutions of partial differential-difference equations, delay partial differential equations as well as integral equations.

REFERENCES

1. Hashim, I.M., SM. Noorani, R. Ahmed, S.A. Bakar, E.S.I. Ismail and A.M. Zakaria, 2006. Accuracy of the Adomian decomposition method applied to the Lorenz system chaos 2005.08.135.

2. Abdel-Hassan, I.H., 2004. Differential transformation technique for solving higher-order initial value problem. Applied math.Comput, 154-299-311.

3. Ayaz, F., 2004. Solution of the system of differential equations by differential transforms method.Applied math.Comput., 147: 54767.

4. Bongsoo Jang, 2009. Solving linear and nonlinear initial value proplems by the projected differential transform method. Ulsan National Institute of Science and Technology (UNIST), Banyeon-ri-100, Ulsan 689-798 korea. Compu. Phys. Communication.

5. Hchen, C. and S.H. Ho, 1999. Solving Partial differential by two dimensiona differential transform method, APPL. Math.Comput., 106: 171-179.

6. Fatma Ayaz, 2004. Solution of the system of differential equations by differential transforms

7. Kanglgil, F. and F. Ayaz, 2008. Solitary wave Solution 12. Jafari, M.A. and A. Aminataei, 2010. Improved for kdv and M kdv equations by differential Homotopy Perturbation Method. International transform method, chaos solutions and fractions Mathematical Forum, 5(32): 1567-1579.

do1:10.1016/j. Chaos. 02. 009. 13. Jagdev Singh, Devendra and Sushila, 2011.

8. Zhou, J.K., 1986. Differential Transformation and its Homotopy Perturbation Sumudu Transform Method Application for Electrical eructs. Hunzhong for Nonlinear Equations. Adv. Theor. Appl. Mech.,

University Press, Wuhan, China. 4(4): 165-175.

9. Montri Thong moon and Sasitornpusjuso, 2009. 14. Salih, M., Elzaki, M.A. Bashir and L. Lajimi, 2011.

The numerical Solutions of differential transform Solution of Linear and Nonlinear Partial Differential method and the Laplace transform method for a Equation by Using Projected Differential Transform system of differential equation. Nonlinear Analysis. Method, Applied Mathematics, Elixir Appl. Math., Hybrid systems d0I: 10. 1016/J. nahs. 10. 006. 40: 5234-5236.

10. Sweilam, N.H. and M.M. Khader, 2009. Exact 15. Tarig, M. Elzaki and Eman M.A. Hilal, 2012. Solution Solutions of some capled nonlinear partial differential of Linear and Nonlinear Partial Differential Equations equations using the homotopy perturbation method. Using Mixture of Elzaki Transform and the Projected Computers and Mathematics with Applications, Differential Transform Method, Mathematical Theory

58: 2134-2141. and Modeling, ISSN 2224-5804 (Paper) ISSN 2225-

11. Sharma, P.R. and Giriraj Methi, 2011. Applications of 0522 (Online), 2(4): 50-59.

Homotopy Perturbation method to Partial differential equations. Asian Journal of Mathematics and Statistics, 4(3): 140-150.