THE EXACT SOLUTION OF DELAY DIFFERENTIAL EQUATIONS USING COUPLING VARIATIONAL ITERATION WITH TAYLOR SERIES AND SMALL TERM

  Bulletin of Mathematics Vol. 04, No. 01 (2012), pp. 1–15.

  

THE EXACT SOLUTION OF DELAY

DIFFERENTIAL EQUATIONS USING

COUPLING VARIATIONAL ITERATION WITH

TAYLOR SERIES AND SMALL TERM

Y.M. Rangkuti and M.S.M. Noorani

_Abstract.

  This study investigated the applicability of coupling the variational it-

eration method (VIM) with Taylor series and small term for exact of delay differ-

ential equations (DDEs). VIM uses general Lagrange multipliers for constructing

the correction functional for the problems. For this work, it is assumed that terms

with delay are considered as restricted variations and also gives Taylor approach

and ignoring a small term on VIM. The analytical solutions for various examples

are obtained by this method. The results obtained show that these algorithms are

accurate and efficient for the solution of DDEs.

  1. INTRODUCTION In mathematics, delay differential equations (DDEs) is type of differ- ential equations in which the derivative of the unknown function at a certain time t is given in terms of the values at an earlier time ξ(t). In this paper, delay differential equations are considered, DDEs in the form: u

  (n)

  (t) = f (t, u

  

(i)

  (t), u(ξ(t))), (1) Received 28-11-2011, Accepted 05-12-2011.

  2010 Mathematics Subject Classification: 65D25; 40A25; 92B05.

Key words and Phrases: variational iteration method; delay differential equations; Taylor series;

Small term; Lagrange multiplier.

  

Y.M. Rangkuti & M.S.N. Noorani – The exact solution of delay differential equation

where n and i (n > i) denote the orders of the derivative.

  DDEs has been used for many years in biochemical, medical, control system, and biological models. Bunsenberg and Tang [1] modelled an em- bryonic cell cycle by delay equations. Marchuk [4] developed a hierarchy of immune response models of increasing complexity to account for various details of within–host defense responses. Patel et al. [5] proposed an iter- ative scheme for the optimal control systems that are described by DDEs with a quadratic cost functional. Glass and Mackey [6] made a systematic study of several important and interesting physiological models with time delays. Recently, DDEs have been used to model, such phenomena as HIV-1 therapy for fighting a virus with another virus [7] and the input–to–state stability of a time–invariant system with multiple noncommensurate and distributed time delays [8].

  Delay problems always lead to an infinite spectrum of frequencies. Hence, they are solved by numerical methods, asymptotic solution, approx- imations and graphical approaches. Asl and Ulsoy [10] used Lambert func- tions to show the analytical solution of homogeneous DDEs. Evans and Raslan [11] and Adomian and Rach [12, 13] employed the Adomian De- composition Method (ADM) to compute an approximate solution of DDEs.

  Shakeri and Dehghan [14] using the Homotopy Perturbation Method (HPM) to gain an approximate solution of DDEs and Alomari et al. [15] using the Homotopy Analysis Method (HAM) obtained approximate solutions of DDEs. Since purely numerical methods involve discretization, the ADM can involve complicated evaluation of the Adomian polynomials, the HPM could yield complicated set of equations to solve and the HAM requires the calculation of an auxiliary parameter, they could possibly be inefficient.

  Currently, great attention is being given to exhibit the power of the variation iteration method (VIM). He [16, 17, 18], who proposed VIM, solved linear and nonlinear models using VIM successfully without any discretiza- tion. In particular, He [18] demonstrated an application of VIM to a first- order DDE modelling a population growth model. Other researchers have also demonstrated the power of this method. Shakeri and Dehghan [19] solved by VIM a system of two nonlinear integro-differential equations which arises in biology. Yulita et al. have solved fractional heat– and wave–like equations using VIM, obtaining good approximate solutions comparable to those obtained with ADM [20] and obtained the approximate solution of fractional Zakhanov–Kuznetsov equations using VIM, showing that a few iterations of the VIM recursive formula can yield good solutions [21]. Re- centy, VIM has been applied for solving family of Kuramoto-Sivashinsky

  

Y.M. Rangkuti & M.S.N. Noorani – The exact solution of delay differential equation

  equations [22] and nonlinear singular boundary value problems arising in various physical equations [23].

  In this paper, we use VIM to find the exact solutions of DDEs. To do so, we assume the delay term as restricted variations in VIM. This assump- tion is made in an effort to make it easier to find the Lagrange multipliers necessary for identifying the correction functional; imposing this assumption has noticeable effects on the ease with which VIM can be used for solving such equations. The exact solution will be reached in next section.

  2. VARITIONAL ITERATION METHOD To illustrate the basic concepts of the VIM we consider the following general equation with a delay term:

  Lu(t) + N [u(t), u(ξ(t))] = g(t), (2) where L is linear operator and N is non-linear operator, ξ(t) is the delay term, and g(t) is a nonhomogeneous term.

  The general Lagrange multiplier method was proposed by Inokuti et al. [24]. He [16, 17, 18] modified the general Lagrange multiplier method to an iteration method using a correction functional which can be written as: _{t k k k} Z u (t) = u (t) + λ(s) [Lu (3)

• 1

  (s) + N [e u(t), e u(ξ(t))] − g(s)] ds, where λ is a general Lagrange multiplier [24], which can be identified opti- mally via variational theory, the subscript k indexes the order of approxima- _{k k} u is considered as restricted variations [16, 17, 18], i.e. δu = 0. tion and e

  The Lagrange multipliers can be easily and precisely obtained for linear problems. However, for nonlinear problems, they are not easy to obtain. In _k u are considered as restricted variations, a notion

  VIM, nonlinear terms e drawn from variational theory that allows the Lagrange multiplier can to be more readily determined. _k u (ξ(t)) should also be con-

  In this paper, we assume that the function f u k (ξ(t)) = 0. With this assumption, sidered as restricted variations, so that δf the Lagrange multiplier can be more easily determined. Taylor series and then ignoring the small term [25] in each iteration is applied. Therefore, we can successively approximate or even reach the exact solution by using u(t) = lim u k (t). (4)

  t→∞

  

Y.M. Rangkuti & M.S.N. Noorani – The exact solution of delay differential equation

  3. ILLUSTRATIVE EXAMPLES In this section, various examples are presented with known exact solutions in order to show the efficiency and high accuracy of this algorithm.

3.1 Example 1

  Consider the first order linear DDEs of the form: _{t t} 1 t ₂

  1 D u(t) = e + u u(t), (5) 0 ≤ t ≤ 1,

  2

  2

  2 with initial condition u(0) = 1. (6) _t According Evans and Raslan [11] the exact solution is given by u(t) = e . To solve Eq. (5), we first consider the correction functional which can be expressed as follow: _t

  Z _{s s} 1 s ₂

  1 u k (t) = u k (t) + λ(s) D u k e u k u k (s) ds, (7)

• 1

  (s) − f −

  2

  2

  2 where λ is a general Lagrange multiplier, which can be identified optimally _{k s} u is considered as restricted variations, i.e. via variational theory and f _s

  2 u k = 0.

  δf

2 We have

  _t Z _{k k s k k}

  1 δu (t) = δu (t) + δ λ(s) D u u (s) ds, (8)

• 1

  (s) −

  2 From Eq. 8, we obtain the following stationary conditions: _{k s} δu = 0

  =t

  (t) : 1 + λ(t)|

  1

  ′ δu k (s) : λ (s) + λ(s) = 0.

  2 The general Lagrange multipliers, therefore, can be identified as:

  (s−t) ₂

  . (9) λ(s) = −e

• 1
• 1
• 3
• 1
• 7

  4950 e

  2

  8 t

  1

  (t) = 1 + t −

  1

  Afterward, we expand each iteration of classical VIM results using Taylor series. This in turn yields gives the components: u

  1265 e ^{15t 16} , (14) and so on.

  − ^t ₂

  230053511 14610750 e

  3025 e ^{7t 8} −

  7425 e ^{t 4}

  t − 53897

  − ^t ₂

  − 38387

  2

  251 125 e ^{3t 4}

  32 189 e ^{13t 16} +

  45 e ^{5t 8} −

  76

  30 e ^{t 4} t −

  47

  −

  2

  t

  − ^t ₂

  20 e

  37

  2 e ^{t 2} −

  (15) u

  (t) = 1 + t +

  8 e ^{t 4} t

  (17) u

  , (18) and so on. _{t 2}

  5

  1966080 t

  − 8861

  4

  24 t

  3

  6 t

  2

  2 t

  1

  (t) = 1 + t +

  4

  4

  1

  12288 t

  − 253

  3

  6 t

  2

  2 t

  1

  (t) = 1 + t +

  3

  (16) u

  3

  384 t

  2

  2 t

  2

  1

  13t ³

  − ^t ₂

  3

  5 e ^{3t 4} , (12) u

  t − e ^{t 4}

  

−

^t ₂

  2 e

  3t

  2 e ^{t 2} −

  − ^t ₂

  10 e

  37

  (t) = 4 −

  2

  2 e ^{t 2} , (11) u

  2 e

  4950 e

  3

  (t) = 2 −

  1

  2 u k (s) ds. (10) We start by taking the linearized solution u(t) = Ce ^{t 2} as the initial approx- imation u ; the initial condition Eq. (5) gives us C = 1. Furthermore, using the iteration formula Eq. (10), we can directly obtain the other components using a Maple package as follows: u

  1

  2 −

  2 e ^{s 2} u k s

  1

  (s) −

  (s−t) ₂ D s u k

  Z ^t e

  (t) = u k (t) −

  By substituting Eq. (9) into Eq. (7), we obtain the following iteration formula: u k

  

Y.M. Rangkuti & M.S.N. Noorani – The exact solution of delay differential equation

  (t) = 8 − 38387

  − ¹ ₂

  9 e ^{5t 8} t −

  2 e ^{t 4} t −

  1

  −

  3

  t

  − ^t ₂

  4 e

  1

  (t) = 16 −

  4

  55 e ^{7t 8} , (13) u

  4

  9 e ^{5t 8} +

  4

  1

  2 e ^{t 2} −

  −

  2

  t

  − ^t ₂

  4 e

  3

  25 e ^{3t 4} −

  19

  15 e ^{t 4} +

  47

  t −

  − ^t ₂

  10 e

  37

• 15
• 996
• 32
• 13
• 1
• 1
• 1

  

Y.M. Rangkuti & M.S.N. Noorani – The exact solution of delay differential equation

  in u

  1 and u 2 . The small terms are ignoring in the components u 1 and u 2 ,

  1

  2

  then the remain terms u and u , that are u = 1 + t and u = 1 + t + t

  1

  2

  1

  2

  2

  to satisfy the convergent series. it is repeat until n tend to infinity. This in turn yields gives the components: u (t) = 1 + t + small term, (19)

  1

  1

  2

  u (t) = 1 + t + t + small term, (20)

  2

  2

  1

  1

  2

  3

  u (t) = 1 + t + + t t + small term, (21)

  3

  2

  6

  1

  1

  1

  2

  3

  4

  u +

  4 (t) = 1 + t + t + t t + small term, (22)

  2

  6

  24 and so on. Therefore, the series solution is express as:

  2

  3

  4

  5

  t t t t

• u(t) = 1 + t + +
• · · · , 2! 3! 4! 5! which converges to u(t) = sin t. This result is the same as that obtained by Alomari et al. [15] used Homotopy Analysis Method.

3.2 Example 2

  In this example a second order linear DDEs is considered as follows: 3 t

2 D t u(t) = u(t) + u + f (t), (23)

  0 ≤ t ≤ 1,

  4

  2

  2

• 2 with initial conditions as follows: where f (t) = −t _t u(0) = 0, D u(0) = 0. (24)

2 The exact solution u(t) = t was found by [15].

  The correction functional for Eq. (23) can be expressed as follows: _t Z 3 s _{k k k}

  2

  u (t) = u (t) + λ(s) D s u u ds, (25)

• 1

  (s) − u(s) − e − f(s)

  4

  2 where λ is a general Lagrange multiplier, which can be identified optimally _{k s} u is considered as restricted variations, i.e. via variational theory; here, e _{k s}

  2 u = 0.

  δf

  2

• 1
• 1

  √ 3t

  −

  2

  9 t

  27

  40

  (t) = −

  2

  ), (29) u

  − ¹ ₂ √ 3t

  81 e

  9 (e

  4

  −

  2

  3 t

  9

  8

  1 (t) =

  we directly obtain the other components as follows: u

  4

  − ¹ ₂ √ 3t

  e ^{√ −3 t 2} as the initial approximation u . The initial conditions given by Eq.(23) gives us C

  −

  − ¹ ₈ √ 3t

  √ 3t

  − 4096 3645 e ^{1 8}

  − ¹ ₄ √ 3t

  √ 3t

  729 e ^{1 4}

  √ 3t

  − ¹ ₂ √

3t

  4 3645 e

  2

  √ 3t

  27 t

  27

  56

  (t) =

  3

  , (30) u

  − ¹ ₄ √ 3t

  √ 3t

  81 e ^{1 4}

  1 = C 2 = 0. Furthermore, using the above iteration formula Eq. (28),

  2

  , (31) and so on. dan u

  (t)| ^s

  4 λ(s) = 0. The general Lagrange multipliers, therefore, can be identified as:

  3

  (s) −

  ′′

  = 0, δu k (s) : λ

  =t

  δu ^k (t) : λ(t)| ^s

  =t = 0,

  ′

  1

  δu ^k (t) : 1 − λ

  4 u(s) ds. (26) From this equation, we obtain the following stationary conditions:

  3

  (s) −

  2 _{s u k}

  λ(s) D

  (t) = δu k (t) + δ Z ^t

  We have δu k

  

Y.M. Rangkuti & M.S.N. Noorani – The exact solution of delay differential equation

  λ(s) =

  3 √

  e ^{√ 3t 2} + C

  − e

  1

  We start by taking the linearized solution u(t) = C

  2 − f(s) ds. (28)

  4 u(s) −u s

  3

  u k (s) −

  2 _s

  i D

  − ¹ ₂ √ 3(s−t)

  √

3(s−t)

  3 h e ^{1 2}

  3 Z ^t h e ^{1 2}

  3 √

  1

  (t) = u k (t) +

  By substituting Eq. (27) into Eq. (25), we obtain the following iteration formula: u k

  i . (27)

  − ¹ ₂ √ 3(s−t)

  − e

  √

3(s−t)

• 4
• e
¹ 2<
• 8
• e
¹ 2<
• 64
• e
• 28
• e
¹ 2<
• 64
• e
• e

  2

  2

  (t) = t

  2

  −

  1 82575360 t

  8

  , (34) u

  4

  (t) = t

  , (35) u

  , (33) u

  5 (t) = t

  2

  , (36) Based on [25], − ^{t 4}

  48

  dan − ^{t 6}

  23.040

  are considered as a small term appearing between the components u

  1

  2

  3

  8

  (t) = t

  −

  

Y.M. Rangkuti &amp; M.S.N. Noorani – The exact solution of delay differential equation

  Afterward, we expand each iteration of classical VIM results using Taylor series. This in turn yields gives the components: u

  1

  (t) = t

  2

  −

  1

  48 t

  

4

  1 1920 t

  1 1376256 t

  6

  , (32) u

  2

  (t) = t

  2

  −

  1 23040 t

  6

  −

  . This in turn yields gives the compo- nents: u

• small term, (37) u

  1

• small term, (38) u
• small term, (39) u

  1

  = t

  2

  and u

  2

  = t

  2

  for obtaining the convergent series [25]. It repeats until n tends to infinity. Consequently, the exact solution converges to t

  2

  , which is the same as that arrived at by Alomari et al. [15].

  Now, we consider first order nonlinear DDEs in the form: D t u k

  (t) = 1 − 2u

  2

  t

  2 , 0 ≤ t ≤ 1,

  (42) with the initial condition: u(0) = 0. (43)

  The exact solution is given by Evans and Raslan [11]:

  , that is u

  dan u

  2

  2

  2

  (t) = t

  2

  3

  (t) = t

  2

  4 (t) = t

  , (40) u

  1

  5

  (t) = t

  2

  , (41) and so on. Then these small terms can be canceled between the components u

  1

  and u

  2

  , and justify the remaining terms of u

3.3 Example 3

  

Y.M. Rangkuti &amp; M.S.N. Noorani – The exact solution of delay differential equation

  The correction functional for Eq. (42) can be expressed as follows: _t Z h i s _{k k s k}

  2

  u (t) = u (t) + λ(s) D u u k ds, (45)

• 1

  (s) + 2e

  2 where λ is a general Lagrange multiplier, which can be identified optimally _s

  2 u is considered as restricted variations, i.e. _k via variational theory, and e _s

  2

  2 u = 0. _k

  δe

2 We have

  _t Z _s

  δu k (t) = δu k (t) + δ λ(s) D u k (s)ds. (46)

• 1

  From this equation we obtain the following stationary conditions: δu k s = 0,

  =t

  (t) : 1 + λ(t)| _k ′ δu (s) : λ (s) = 0.

  The general Lagrange multipliers, therefore, can be identified as: (47) λ(s) = −1. By substituting Eq. (47) into Eq. (45), we obtain the following iteration formula: _t

  Z h i s _{k k s k}

  2

  u (t) = u D u (s) + 2u k ds. (48)

• 1

  (t) −

  2 As before, we can take the linearized solution u(t) = t + C as the initial approximation u (t). The condition y(0) = 0 gives us C = 0. Furthermore, by using the iteration formula in Eq. (48), we can directly obtain the other components as follows:

  3

  t u , (49)

  1

  (t) = t − 3!

  3

  5

  t t (50) + u

  2

  (t) = t − − small term, 3! 5!

  3

  5

  7

  t t t

• u + small term, (51)

  3

  (t) = t − − 3! 5! 7!

  3

  5

  7

  9

  t t t t

• 3
• u

  (52)

  (t) = t − − − small term, 3! 5! 7! 9! Y.M. Rangkuti &amp; M.S.N. Noorani – The exact solution of delay differential equation

  This in turn yields gives the components:

  

3

  t u

  1

  (t) = t − 3!

  

3

  5

  7

  t t t u + ,

  2

  (t) = t − − 3! 5! 8064

  7

  /8064 is considered as a small term appearing between the compo- where −t nents u and u . We then can cancel these small terms between the compo-

  1

  2 _{t t 3 5}

  and u

• nents u and justify the remaining terms of u , i.e. u , to

  1

  2

  2

  2

  = t−

  3! 5!

  get a convergent series [25]. Then the series solution using VIM is expressed as:

  3

  

5

  7

  9

  t t t t u(t) = t − − − · · · ,

  3! 5! 7! 9! which converges to u(t) = sin t. This result is the same as that obtained by Evans and Raslan [11] using ADM.

3.4 Example 4

  Consider this third order nonlinear DDEs as follows: t

  3

2 D u k (t) = 2 u (53)

  _t − 1, , 0 ≤ t ≤ 0,

  2 with the initial conditions:

_{t k k}

  2

  u(0) = 0, D u (0) = 0, D t u (0) = 0. (54) The correction functional for Eq. (53) can be expressed as follows: _t

  Z s _{k k k}

  

3

  2

  u +1 (t) = u (t) + λ(s) D s u u k + 1 ds, (55) (s) − 2e

  2 where λ is a general Lagrange multiplier, which can be identified optimally _{k k} u u = via variational theory and e is considered as restricted variations, i.e. δf

  0. We have _t Z _{k k k}

  3

  δu (t) = δu (t) + δ λ(s) D s u (s)ds. (56)

• 1

Y.M. Rangkuti &amp; M.S.N. Noorani – The exact solution of delay differential equation

  From this equation, we obtain the following stationary conditions: δu k s = 0

  =t

  (t) : 1 + λ(t)| _k

′

δu (s) : λ (s) = 0.

  The general Lagrange multipliers, therefore, can be identified as:

  1

  2

  . (57) λ(s) = − (t − s)

  2 By substituting Eq. (57) into Eq. (55), we obtain the following iteration formula: _t Z 1 s _{k k k}

  2

  3

  2

  u (t) = u D s u k + 1 ds. (58)

• 1

  (t) − (t − s) (s) − 2u

  2 _t

  2 ₃ We start with an initial approximation u , which satisfies the (t) = t −

  6

  initial conditions. Furthermore, using the iteration formula in Eq. (58), we can directly obtain the other components as follows:

  3

  5

  7

  t t t u + small term, + (59)

  1

  (t) = t − − 3! 5! 7!

  3

  5

  7

  9

  11

  t t t t t

• u + small term, (60) +

  2

  (t) = t − − − 3! 5! 7! 9! 11!

  3

  5

  7

  9

  11

  13

  15

  t t t t t t t

• u +
• small term, (61)

  3

  (t) = t − − − − 3! 5! 7! 9! 11! 13! 15! and so on.

  As in example 3, this in turn yields the components:

  3

  t u (t) = t −

  3!

  3

  5

  7

  5

  t t t t

• u , +

  1

  (t) = t − − 3! 5! 7! 580608

  5

  where t /580608 is considered as a small term appearing between the com- ponents u and u . We then can cancel this small term between the compo-

  1 _{t t t 3 5 7}

  nents u and u

  1 , i.e. u

  1

• 1 and justify the remaining terms of u

  = t− −

  3! 5! 7!

  to satisfy a convergent series [25]. Consequently, the exact solution converges to u(t) = sin t, which is the same as that arrived at by Evans and Raslan [11].

  Y.M. Rangkuti &amp; M.S.N. Noorani – The exact solution of delay differential equation

  4. ACKNOWLEDGMENT The financial support received from the Universiti Kebangsaan Malaysia research fund UKM-DLP-2011-01 is gratefully acknowledged.

  5. CONCLUSIONS In this paper, VIM has been successfully employed by assuming that the delay terms are also considered as restricted variations. This assumption indeed made it easier to find the Lagrange multipliers necessary for iden- tifying the correction functional; imposing this assumption had noticeable effects on the ease with which VIM can be used for solving such equations. For computations and plots, the Maple Package has been used. Further- more, exact solution of delay differential equations was obtained in good agreement with those obtained by previous researchers. The method also gives more realistic series solutions that converge very rapidly in these prob- lems. As an advantage of VIM over the other methods, we easily integrated the equation without calculating the Adomian polynomials, solving set of higher-order equations and finding an auxiliary parameter. Thus showing that VIM is a simple, straightforward and yet accurate method for solving delay differential equations and delay partial differential equations. Further- more, it can be considered an alternative method for solving a wide class of linear and nonlinear problems which arise in various fields of pure and applied sciences.

  

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  Y.M. Rangkuti : Department Mathematics, Faculty Mathematics and Natural Sci-

ence, Universitas Negeri Medan, UNIMED 20221 Medan Sumatera Utara, Indone-

sia.

  E-mail: yulitamolliq@yahoo.com

  Y.M. Rangkuti &amp; M.S.N. Noorani – The exact solution of delay differential equation M.S.M. Noorani : School of Mathematical Sciences, Universiti Kebangsaan Malaysia, UKM 43600 Bangi Selangor, Malaysia.

  E-mail: msn@ukm.my