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Differential transformation technique for solving higher-order initial value problems
Applied Mathematics and Computation 154 (2004) 299–311 www.elsevier.com/locate/amc
Differential transformation technique for solving higher-order initial value problems I.H. Abdel-Halim Hassan Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt
Abstract In this paper, the differential transformation method is applied to solve the higherorder initial value problems (HOIVPs). The differential transformation method of fixed grid size is used to approximate solutions HOIVPs. The higher order of TaylorÕs series be applied. Thus, there exist a trade-off between the selection of the order of TaylorÕs series and the grid size. The proposed method provides the TaylorÕs series between any adjacent grid points. Numerical examples are used to illustrate the preciseness and effectiveness of the proposed the method. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Higher-order initial value problem; Differential transformation; TaylorÕs series expansion; Runge–Kutta method
1. Introduction A variety of methods, exact, approximate, and purely numerical are available for the solution of initial value problems. Most of these methods are computationally intensive because they are trial-and-error in nature, or need complicated symbolic computations. In [3], the Eular method, the Taylor method, and Runge–Kutta methods serve as an introduction to numerical method for solving initial value problems. However, the Taylor method requires the calculation of high-order derivatives, a difficult symbolic and complex problem. The differential transformation technique is one of the numerical methods for ordinary differential equations. The concept of differential
E-mail address: [email protected] (I.H. Abdel-Halim Hassan). 0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00708-2
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transformation was first proposed by Zhou in 1986 (see [4–7,10]), and it was applied to solve linear and non-linear initial value problems in electric circuit analysis. In a recent work, Jang, Chen and Liy [9] in 2000, introduced the application of the concept of the differential transformation of fixed grid size to approximate solutions of linear and non-linear initial value problems. In this paper, the differential transformation technique of fixed grid size is applied to solve the higher-order initial value problems (HOIVPs). The method can be used to evaluates the approximating solution by the finite Taylor series and by an iteration procedure described by the transformed equations obtained from the original equation using the operations of differential transformation. The mathematical background to convert HOIVPs into a system of first-order initial value problems is described in Section 2. The basic definitions of the differential transformation are introduced in Section 3. The operations properties of the differential transformation are shown in Section 4. The differential transformation technique for solving HOIVPs is illustrated in Section 5. Numerical examples are used to illustrate the effectiveness of the proposed method in Section 6. Through out the calculations we have used a simple computation software.
2. Mathematical background The differential equation for initial value problem can be described as dy ¼ f ðt; yÞ dt
ð2:1Þ
for a 6 t 6 b
subject to the initial condition yðaÞ ¼ a:
ð2:2Þ
In this paper, HOIVPs are considered. To convert a general mth order initial value problems y ðmÞ ðtÞ ¼ f ðt; y; y 0 ; . . . ; y ðm1Þ Þ;
a 6 t 6 b;
ð2:3Þ
with the initial conditions yðaÞ ¼ a1 ;
y 0 ðaÞ ¼ a2 ;
y 00 ðaÞ ¼ a3 ; . . . y ðm1Þ ¼ am ;
ð2:4Þ
into a system of first-order initial value problems. Let u1 ðtÞ ¼ yðtÞ; u2 ðtÞ ¼ y 0 ðtÞ; . . . ; um ðtÞ ¼ y ðm1Þ ðtÞ, using this notation, we obtain the first-order system
I.H. Abdel-Halim Hassan / Appl. Math. Comput. 154 (2004) 299–311
du1 ðtÞ dyðtÞ ¼ ¼ u2 ðtÞ; dt dt 0 du2 ðtÞ dy ðtÞ ¼ ¼ u3 ðtÞ; dt dt .. .
301
ð2:5Þ
ðm2Þ
dum1 ðtÞ dy ðtÞ ¼ ¼ um ðtÞ; dt dt dum ðtÞ dy ðm1Þ ðtÞ ¼ ¼ y ðmÞ ðtÞ ¼ f ðt; y; y 0 ; . . . ; y ðm1Þ Þ; dt dt with initial conditions u1 ðaÞ ¼ yðaÞ ¼ a1 ;
du2 ðaÞ ¼ y 0 ðaÞ ¼ a2 ; . . . dum ðaÞ ¼ y ðm1Þ ¼ am : ð2:6Þ
3. Basic definitions As in Refs. [4,8], the basic definition of the differential transformation are introduced as the follows: Definition 3.1. If xðtÞ is analytic in the domain T , then it will be differentiated continuously with respect to time t, ok xðtÞ ¼ /ðt; kÞ for all t 2 T otk
ð3:1Þ
for t ¼ ti , then /ðt; kÞ ¼ /ðti ; kÞ, where k belongs to the set of non-negative integers, denoted as the K-domain. Therefore, Eq. (3.1) can be rewritten as k o xðtÞ X ðkÞ ¼ /ðti ; kÞ ¼ ; ð3:2Þ otk t¼ti where X ðkÞ is called the spectrum of xðtÞ at t ¼ ti . Definition 3.2. If xðtÞ can be expressed by TaylorÕs series, then xðtÞ can be represented as " # k 1 X ðt ti Þ xðtÞ ¼ X ðkÞ: ð3:3Þ k! k¼0 Eq. (3.3) is called the inverse of xðtÞ, with the symbol D denoting the differential transformation process. Upon combining (3.2) and (3.3), we obtain
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Table 1. The fundamental mathematics operations Time function
Transformed function
zðtÞ ¼ xðtÞ yðtÞ zðtÞ ¼ axðtÞ dxðtÞ zðtÞ ¼ dt d2 xðtÞ zðtÞ ¼ dt2 dm xðtÞ zðtÞ ¼ dtm zðtÞ ¼ xðtÞyðtÞ
ZðkÞ ¼ X ðkÞ Y ðkÞ ZðkÞ ¼ aX ðkÞ
ZðkÞ ¼ ðk þ 1Þðk þ 2Þ ðk þ mÞX ðk þ mÞ P ZðkÞ ¼ k‘¼0 Y ð‘ÞX ðk ‘Þ
zðtÞ ¼ 1 zðtÞ ¼ t
ZðkÞ ¼ dðkÞ ZðkÞ ¼ dðk 1Þ
zðtÞ ¼ tm
ZðkÞ ¼ dðk mÞ; dðk mÞ ¼
zðtÞ ¼ expðktÞ zðtÞ ¼ ð1 þ tÞm zðtÞ ¼ sinðxt þ aÞ zðtÞ ¼ cosðxt þ aÞ
ZðkÞ ¼ ðk þ 1ÞX ðk þ 1Þ ZðkÞ ¼ ðk þ 1Þðk þ 2ÞX ðk þ 2Þ
1 0
kk k! mðm 1Þ ðm k þ 1Þ ZðkÞ ¼ k!
xk pk þa sin ZðkÞ ¼ 2! k!
xk pk þa ZðkÞ ¼ cos 2! k!
if k ¼ m if k ¼ 6 m
ZðkÞ ¼
" # k 1 X ðt ti Þ xðtÞ ¼ X ðkÞ D1 X ðkÞ: k! k¼0
ð3:4Þ
Using the differential transformation, a differential equation in the domain of interest can be transformed to an algebraic equation in the K-domain and the xðtÞ can be obtained by finite-term TaylorÕs series plus a remainder, as " # k n X ðt ti Þ xðtÞ ¼ ð3:5Þ X ðkÞ þ Rnþ1 ðtÞ: k! k¼0 The fundamental mathematics operations performed by differential transformation are listed in Table 1 (see Refs. [1,2,8]). 4. The operation properties of differential transformation As in [8], if xðtÞ, yðtÞ are two uncorrelated functions with time t and X ðkÞ, Y ðkÞ are the transformed functions corresponding to xðtÞ, yðtÞ and the basic properties are shown as follows: 1. If X ðkÞ ¼ D½xðtÞ , Y ðkÞ ¼ D½yðtÞ , and c1 and c2 are independent of t and k, then
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D½c1 xðtÞ þ c2 yðtÞ ¼ c1 X ðkÞ þ c2 Y ðkÞ:
303
ð4:1Þ
2. If zðtÞ ¼ xðtÞyðtÞ, xðtÞ ¼ D1 ½X ðkÞ , yðtÞ ¼ D1 ½Y ðkÞ and denote the convolution, then D½zðtÞ ¼ D½xðtÞyðtÞ ¼ X ðkÞ Y ðkÞ ¼
k X
Y ð‘ÞX ðk ‘Þ:
ð4:2Þ
‘¼0
Therefore, the transform of xm ðtÞ, m is a positive integer, can be obtained as D½xm ðtÞ ¼ X m ðkÞ ¼ X m1 ðkÞ X ðkÞ ¼
k X
X m1 ð‘ÞX ðk ‘Þ:
ð4:3Þ
‘¼0
5. Differential transformation method The objective of this section is to find the solution of the HOIVPs (2.3) and (2.4) at the equally spaced grid points ti , i ¼ 0; 1; 2; . . . ; N where ti ¼ a þ ih
for all i ¼ 0; 1; 2; . . . ; N and h ¼
ba : N
ð5:1Þ
The domain ½a; b is divided into N sub-domains and the approximation function in each sub-domain are ui ; i ¼ 0; 1; 2; . . . ; N 1 respectively (see Ref. [9]). Taking the differential transformation of the system (2.5), we get 1 U2 ðkÞ; ðk þ 1Þ 1 U3 ðkÞ; U2 ðk þ 1Þ ¼ ðk þ 1Þ 1 U4 ðkÞ; U3 ðk þ 1Þ ¼ ðk þ 1Þ .. . 1 Um1 ðk þ 1Þ ¼ Um ðkÞ; ðk þ 1Þ Um ðk þ 1Þ ¼ F ðY ðkÞÞ; U1 ðk þ 1Þ ¼
ð5:2Þ
where F ð Þ denotes the transformed function of f ðt; y; y 0 ; . . . ; y ðm1Þ Þ. From the initial conditions (2.6) we can obtained that U1 ð0Þ ¼ yðaÞ ¼ a1 ;
U2 ð0Þ ¼ y 0 ðaÞ ¼ a2 ; . . . Um ð0Þ ¼ y ðm1Þ ¼ am : ð5:3Þ
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In the first sub-domain yðtÞ can be described by ys ðtÞ. From (3.3), (5.2) and (5.3), ys ðtÞ can be represented in terms of its nth order TaylorÕs polynomial about a, that is ys ðtÞ ¼ U1i ð0Þ þ U1i ð1Þðt aÞ þ U1i ð2Þðt aÞ2 þ þ U1i ðnÞðt aÞn ; ð5:4Þ where the subscript ‘‘s’’ denotes that the TaylorÕs polynomial is expand about a. Once TaylorÕs polynomial is obtained, yðt1 Þ can be evaluated as yðt1 Þ ’ y0 ðt1 Þ ¼
n X
U10 ðjÞhj ;
h ¼ ðt1 t0 Þ:
ð5:5Þ
j¼0
The final value y0 ðt1 Þ of the first sub-domain is the initial value of the second sub-domain i.e. y1 ðt1 Þ ¼ y0 ðt1 Þ ¼ U11 ð0Þ (see [9]) yðt2 Þ ’ y1 ðt2 Þ ¼
n X
U11 ðjÞhj ;
h ¼ ðt2 t1 Þ:
ð5:6Þ
j¼0
In the same manner, yðt3 Þ can be represented as: yðt3 Þ ’ y2 ðt3 Þ ¼
n X
U12 ðjÞhj ;
h ¼ ðt3 t2 Þ:
ð5:7Þ
j¼0
Hence, the solution on the grid points tiþ1 can be obtained yðtiþ1 Þ ’ yi ðtiþ1 Þ ¼
n X
U1i ðjÞhj ;
h ¼ ðtiþ1 ti Þ; i ¼ 0; 1; 2; . . . ; N 1:
j¼0
ð5:8Þ
6. Solution of higher-order initial value problems In this section, the differential transformation technique is applied to solve HOIVPs. To illustrate the method proposed in this paper, two HOIVPs are considered as follows: Problem 1. Consider the second-order initial value problem y 00 ðtÞ 2y 0 ðtÞ þ 2yðtÞ ¼ expð2tÞ sinðtÞ;
0 6 t 6 1;
ð6:1Þ
with initial conditions yð0Þ ¼ 0:4; 0
y ð0Þ ¼ 0:6:
ð6:2Þ ð6:3Þ
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305
As above, in Section 2, with u1 ðtÞ ¼ yðtÞ and u2 ðtÞ ¼ y 0 ðtÞ. Eq. (6.1) transformed into the system of the first-order differential equation u01 ðtÞ ¼ u2 ðtÞ;
ð6:4Þ
u02 ðtÞ
ð6:5Þ
¼ expð2tÞ sinðtÞ 2u1 ðtÞ þ 2u2 ðtÞ
and the initial conditions (6.2) and (6.3) becomes u1 ð0Þ ¼ 0:4;
ð6:6Þ
u2 ð0Þ ¼ 0:6:
ð6:7Þ
Let h ¼ 0:1 and N ¼ 10. The differential equation of the system (6.4) and (6.5) between ti and tiþ1 can be represented as u01 ðtH Þ ¼ u2 ðtH Þ;
ð6:8Þ
u02 ðtH Þ ¼ expð2tH þ 2ti Þ sinðtH þ ti Þ 2u1 ðtH Þ þ 2u2 ðtH Þ;
ð6:9Þ
where tH ¼ t ti . Taking the differential transformation of (6.8) and (6.9), respectively, we get U1i ðk þ 1Þ ¼
1 U2i ðkÞ ðk þ 1Þ
ð6:10Þ
and 1 U2i ðk þ 1Þ ¼ ðk þ 1Þ
( expð2ti Þ cosðti Þ
þ expð2ti Þ sinðti Þ
k X ‘¼0
p 2‘ sin ðk ‘Þ 2 ð‘!Þðk ‘Þ!
p 2‘ cos ðk ‘Þ 2 ð‘!Þðk ‘Þ! ‘¼0 )
k X
2U1i ðkÞ þ 2U2i ðkÞ ;
ð6:11Þ
with U10 ð0Þ ¼ 0:4;
U20 ð0Þ ¼ 0:6:
ð6:12Þ
The set of U1i ðk þ 1Þ of (6.10) are presented in Table 2. The approximation of u1 ðtÞ ¼ yðtÞ on the grid points can be obtained from (6.10) and (5.8). The actual solution of (6.1)–(6.3) is yðtÞ ¼ u1 ðtÞ ¼ 0:2 expð2tÞðsinðtÞ 2 cosðtÞÞ:
ð6:13Þ
Fig. 1, shows errors involved with different order of differential transformation method, along with the result obtained by the Runge–Kutta fourth order method (see Table 3).
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Table 2 Different order of the differential transformation method (DTM) of Problem 1 ti
DTM of order 4
DTM of order 5
DTM of order 10
DTM of order 15
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
)0.400000000 )0.461733059 )0.525559267 )0.588600399 )0.646610831 )0.693564763 )0.721149613 )0.718150376 )0.669708650 )0.556440473 )0.353397162
)0.400000000 )0.461732981 )0.525559066 )0.588600073 )0.646610347 )0.693563995 )0.721148557 )0.718148972 )0.669706840 )0.556438203 )0.353394446
)0.400000000 )0.461732979 )0.525559063 )0.588600070 )0.646610344 )0.693563992 )0.721148554 )0.718148969 )0.669706837 )0.556438201 )0.353394444
)0.400000000 )0.461732979 )0.525559063 )0.588600070 )0.646610344 )0.693563992 )0.721148554 )0.718148969 )0.669706837 )0.556438201 )0.353394444
Fig. 1. Computational errors corresponding to different order of differential transformation method and Runge–Kutta of fourth order method to the Problem 1 with actual solution yðtÞ ¼ 0:2 expð2tÞðsin t 2 cos tÞ.
Problem 2. Consider the third-order initial value problem y 000 ðtÞ þ 2y 00 ðtÞ y 0 ðtÞ 2yðtÞ ¼ expðtÞ;
0 6 t 6 3;
ð6:14Þ
with initial conditions yð0Þ ¼ 1;
ð6:15Þ
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307
Table 3 Errors involved with different order of the DTM along with the result obtained by the RKM of Problem 1 Errors
ti 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
jyðti Þ RKMj
jyðti Þ DT4j
jyðti Þ DT5j
jyðti Þ DT10j
jyðti Þ DT15j
0.0000000000 3.7332754e)5 8.3926639e)5 1.3955412e)4 2.0308873e)4 2.7212788e)4 3.4251283e)4 4.0772521e)4 4.5813778e)4 4.8009448e)4 4.5479347e)4
0.000000 5.900e)6 1.670e)5 3.990e)5 5.310e)5 8.630e)5 1.013e)4 1.476e)4 1.850e)4 2.473e)4 3.062e)4
0.00000 1.90e)6 3.40e)6 7.40e)6 4.70e)6 9.50e)6 4.30e)6 7.30e)6 4.00e)6 2.03e)5 3.46e)5
0.00000 2.10e)6 3.70e)6 7.00e)6 4.40e)6 9.20e)6 4.60e)6 6.90e)6 3.70e)6 2.01e)6 3.44e)6
0.00000 2.10e)6 3.70e)6 7.00e)6 4.40e)6 9.20e)6 4.60e)6 6.90e)6 3.70e)6 2.01e)6 3.44e)6
RKM: The Runge–Kutta fourth-order method and yðti Þ the actual solution on the grid points.
y 0 ð0Þ ¼ 2: 00
y ð0Þ ¼ 0:
ð6:16Þ ð6:17Þ
With u1 ðtÞ ¼ yðtÞ, u2 ðtÞ ¼ y 0 ðtÞ and u3 ðtÞ ¼ y 00 ðtÞ. Eq. (6.14) transformed into the system of the first-order differential equation u01 ðtÞ ¼ u2 ðtÞ;
ð6:18Þ
u02 ðtÞ
ð6:19Þ
¼ u3 ðtÞ;
u03 ðtÞ ¼ 2u3 ðtÞ þ u2 ðtÞ þ 2u1 ðtÞ þ expðtÞ
ð6:20Þ
and the initial conditions (6.15)–(6.17) becomes u1 ð0Þ ¼ 1;
ð6:21Þ
u2 ð0Þ ¼ 2;
ð6:22Þ
u3 ð0Þ ¼ 0:
ð6:23Þ
Let h ¼ 0:2 and N ¼ 15. The differential equation of the system (6.18), (6.19) and (6.20) between ti and tiþ1 can be represented as u01 ðtH Þ ¼ u2 ðtH Þ;
ð6:24Þ
u02 ðtH Þ ¼ u3 ðtH Þ;
ð6:25Þ
u03 ðtH Þ ¼ 2u3 ðtH Þ þ u2 ðtH Þ þ 2u1 ðtH Þ þ expðtH þ ti Þ;
ð6:26Þ
where tH ¼ t ti . Taking the differential transformation of (6.24)–(6.26), respectively, we get
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U1i ðk þ 1Þ ¼
1 U2i ðkÞ; ðk þ 1Þ
ð6:27Þ
U2i ðk þ 1Þ ¼
1 U3i ðkÞ ðk þ 1Þ
ð6:28Þ
and 1 U3i ðk þ 1Þ ¼ ðk þ 1Þ
1 2U3i ðkÞ þ U2i ðkÞ þ 2U1i ðkÞ þ expðti Þ ; k! ð6:29Þ
with U10 ð0Þ ¼ 1; U20 ð0Þ ¼ 2;
ð6:30Þ
U30 ð0Þ ¼ 0: The set of U1i ðk þ 1Þ of (6.27) are presented in Table 4. The approximation of u1 ðtÞ ¼ yðtÞ on the grid points can be obtained from (6.27) and (5.8). The actual solution of (6.1)–(6.3) is yðtÞ ¼ u1 ðtÞ ¼
43 1 4 1 expðtÞ þ expðtÞ expð2tÞ þ ½t expðtÞ : 36 4 9 6
ð6:31Þ
Table 4 Different order of the differential transformation method (DTM) of Problem 2 ti
DTM of order 4
DTM of order 5
DTM of order 10
DTM of order 15
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
1.00000000 1.40633333 1.84917442 2.36189778 2.97753987 3.73160354 4.66457225 5.82438402 7.26907260 9.06975381 11.3141407 14.1107716 17.5941670 21.9311637 27.3287178 34.0435550
1.00000000 1.40637600 1.84923760 2.36197255 2.97762545 3.73170425 4.66469625 5.82454315 7.26928196 9.07003341 11.3145155 14.1112739 17.5948384 21.9320567 27.3298999 34.0451137
1.00000000 1.40637383 1.84923496 2.36197035 2.97762430 3.73170461 4.66469831 5.82454695 7.26928808 9.07004262 11.3145290 14.1112931 17.5948638 21.9320896 27.3299428 34.0451693
1.00000000 1.40637383 1.84923496 2.36197035 2.97762430 3.73170461 4.66469831 5.82454695 7.26928808 9.07004262 11.3145290 14.1112931 17.5948638 21.9320896 27.3299428 34.0451693
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309
Fig. 2. Computational errors corresponding to different order of differential transformation method and Runge–Kutta of fourth order method to the Problem 2 with actual solution yðtÞ ¼ 43=36 expðtÞ þ 1=4 expðtÞ 4=9 expð2tÞ þ 1=6t expðtÞ.
Table 5 Errors involved with different order of the DTM along with the result obtained by the RKM of Problem 2 ti 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
Errors jyðti Þ RKMj
jyðti Þ DT4j
jyðti Þ DT5j
jyðti Þ DT10j
jyðti Þ DT15j
0.0000000000 3.7051385e)5 5.3496545e)5 6.1340427e)5 6.7810840e)5 7.7492288e)5 9.3657449e)5 1.1911737e)4 1.5679347e)4 2.1014117e)4 2.8350990e)4 3.8249793e)4 5.1434509e)4 6.8840014e)4 9.1669703e)4 1.2146776e)3
0.0000000 4.0670e)5 6.0580e)5 7.2220e)5 8.4130e)5 1.0046e)4 1.2575e)4 1.6398e)4 2.1640e)4 2.9119e)4 3.8930e)4 5.2840e)4 7.0300e)4 9.3630e)4 1.2422e)3 1.6350e)3
0.000000 2.000e)6 2.600e)6 2.550e)6 1.450e)6 2.500e)7 1.750e)6 4.850e)6 7.040e)6 1.159e)5 1.450e)5 2.610e)5 3.160e)5 4.330e)5 6.010e)5 7.630e)5
0.000000 1.700e)7 4.000e)8 3.500e)7 3.000e)7 6.100e)7 3.100e)7 1.050e)6 9.200e)7 2.380e)6 1.000e)7 6.900e)6 6.200e)6 1.040e)5 1.720e)5 2.070e)5
0.000000 1.700e)7 4.000e)8 3.500e)7 3.000e)7 6.100e)7 3.100e)7 1.050e)6 9.200e)7 2.380e)6 1.000e)7 6.900e)6 6.200e)6 1.040e)5 1.720e)5 2.070e)5
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Fig. 2, shows errors involved with different order of differential transformation method, along with the result obtained by the Runge–Kutta fourth order method (see Table 5). As indicated in Figs. 1 and 2, the computational error decreases as the order of Taylor series increases. The order of computational error corresponding to the Runge–Kutta method and the differential transformation method of the same order is the same.
7. Conclusion This paper shows that the differential transformation technique can be applied to solve the HOIVPs. The proposed approach can be applied to both second- and third-order initial value problems. Convert the general mth order differential equation with initial conditions into a system of 1st order differential equations. Apply the differential transformation to the system of 1st order differential equations, respectively, and their corresponding initial conditions. Repeat the iterative procedure until the desired nth order Taylor series expansion is obtained. An nth order approximation obtained using the Taylor transformation method of the form yðtiþ1 Þ ¼ yðti Þ þ hUðti ; yðti Þ; hÞ þ Oðhnþ1 Þ where h is the grid size, U the incremental function of yðtÞ and Oðhnþ1 Þ denotes the error with an order of ðn þ 1Þ. The order of computational errors corresponding to different orders of differential transformation method and the Runge–Kutta fourth-order method with the exact solution of the main problem are compared graphically.
References [1] I.H. Abdel-Halim Hassan, On solving some eigenvalue problems by using a differential transformation, Applied Mathematics and Computation 127 (2002) 1–22. [2] I.H. Abdel-Halim Hassan, Different applications for the differential transformation in the differential equations, Applied Mathematics and Computation 129 (2002) 183–201. [3] R.L. Burrden, J.D. Faires, Numerical Analysis, PWS-KENT Publishing Company, Boston, 1993. [4] C.L. Chen, Y.C. Liu, Differential transformation technique for steady nonlinear heat conduction problems, Applied Mathematics and Computation 95 (1998) 155–164. [5] C.L. Chen, Y.C. Liu, Solution of two point boundary value problems using the differential transformation method, Journal of Optimization Theory and Applications 99 (1998) 23–35. [6] C.L. Chen, S.H. Lin, C.K. Chen, Application of Taylor transformation to nonlinear predictive control problem, Applied Mathematical Modeling 20 (1996) 699–710. [7] C.K. Chen, S.H. Ho, Application of differential transformation to eigenvalue problems, Applied Mathematics and Computation 79 (1996) 173–188. [8] M.J. Jang, C.L. Chen, Analysis of the response of a strongly nonlinear damped system using a differential transformation technique, Applied Mathematics and Computation 88 (1997) 137– 151.
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[9] M.J. Jang, C.L. Chen, Y.C. Liy, On solving the initial value problems using the differential transformation method, Applied Mathematics and Computation 115 (2000) 145–160. [10] M. Malik, H.H. Dang, Vibration analysis of continuous system by differential transformation, Applied Mathematics and Computation 96 (1998) 17–26.
Differential transformation technique for solving higher-order initial value problems I.H. Abdel-Halim Hassan Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt
Abstract In this paper, the differential transformation method is applied to solve the higherorder initial value problems (HOIVPs). The differential transformation method of fixed grid size is used to approximate solutions HOIVPs. The higher order of TaylorÕs series be applied. Thus, there exist a trade-off between the selection of the order of TaylorÕs series and the grid size. The proposed method provides the TaylorÕs series between any adjacent grid points. Numerical examples are used to illustrate the preciseness and effectiveness of the proposed the method. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Higher-order initial value problem; Differential transformation; TaylorÕs series expansion; Runge–Kutta method
1. Introduction A variety of methods, exact, approximate, and purely numerical are available for the solution of initial value problems. Most of these methods are computationally intensive because they are trial-and-error in nature, or need complicated symbolic computations. In [3], the Eular method, the Taylor method, and Runge–Kutta methods serve as an introduction to numerical method for solving initial value problems. However, the Taylor method requires the calculation of high-order derivatives, a difficult symbolic and complex problem. The differential transformation technique is one of the numerical methods for ordinary differential equations. The concept of differential
E-mail address: [email protected] (I.H. Abdel-Halim Hassan). 0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00708-2
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transformation was first proposed by Zhou in 1986 (see [4–7,10]), and it was applied to solve linear and non-linear initial value problems in electric circuit analysis. In a recent work, Jang, Chen and Liy [9] in 2000, introduced the application of the concept of the differential transformation of fixed grid size to approximate solutions of linear and non-linear initial value problems. In this paper, the differential transformation technique of fixed grid size is applied to solve the higher-order initial value problems (HOIVPs). The method can be used to evaluates the approximating solution by the finite Taylor series and by an iteration procedure described by the transformed equations obtained from the original equation using the operations of differential transformation. The mathematical background to convert HOIVPs into a system of first-order initial value problems is described in Section 2. The basic definitions of the differential transformation are introduced in Section 3. The operations properties of the differential transformation are shown in Section 4. The differential transformation technique for solving HOIVPs is illustrated in Section 5. Numerical examples are used to illustrate the effectiveness of the proposed method in Section 6. Through out the calculations we have used a simple computation software.
2. Mathematical background The differential equation for initial value problem can be described as dy ¼ f ðt; yÞ dt
ð2:1Þ
for a 6 t 6 b
subject to the initial condition yðaÞ ¼ a:
ð2:2Þ
In this paper, HOIVPs are considered. To convert a general mth order initial value problems y ðmÞ ðtÞ ¼ f ðt; y; y 0 ; . . . ; y ðm1Þ Þ;
a 6 t 6 b;
ð2:3Þ
with the initial conditions yðaÞ ¼ a1 ;
y 0 ðaÞ ¼ a2 ;
y 00 ðaÞ ¼ a3 ; . . . y ðm1Þ ¼ am ;
ð2:4Þ
into a system of first-order initial value problems. Let u1 ðtÞ ¼ yðtÞ; u2 ðtÞ ¼ y 0 ðtÞ; . . . ; um ðtÞ ¼ y ðm1Þ ðtÞ, using this notation, we obtain the first-order system
I.H. Abdel-Halim Hassan / Appl. Math. Comput. 154 (2004) 299–311
du1 ðtÞ dyðtÞ ¼ ¼ u2 ðtÞ; dt dt 0 du2 ðtÞ dy ðtÞ ¼ ¼ u3 ðtÞ; dt dt .. .
301
ð2:5Þ
ðm2Þ
dum1 ðtÞ dy ðtÞ ¼ ¼ um ðtÞ; dt dt dum ðtÞ dy ðm1Þ ðtÞ ¼ ¼ y ðmÞ ðtÞ ¼ f ðt; y; y 0 ; . . . ; y ðm1Þ Þ; dt dt with initial conditions u1 ðaÞ ¼ yðaÞ ¼ a1 ;
du2 ðaÞ ¼ y 0 ðaÞ ¼ a2 ; . . . dum ðaÞ ¼ y ðm1Þ ¼ am : ð2:6Þ
3. Basic definitions As in Refs. [4,8], the basic definition of the differential transformation are introduced as the follows: Definition 3.1. If xðtÞ is analytic in the domain T , then it will be differentiated continuously with respect to time t, ok xðtÞ ¼ /ðt; kÞ for all t 2 T otk
ð3:1Þ
for t ¼ ti , then /ðt; kÞ ¼ /ðti ; kÞ, where k belongs to the set of non-negative integers, denoted as the K-domain. Therefore, Eq. (3.1) can be rewritten as k o xðtÞ X ðkÞ ¼ /ðti ; kÞ ¼ ; ð3:2Þ otk t¼ti where X ðkÞ is called the spectrum of xðtÞ at t ¼ ti . Definition 3.2. If xðtÞ can be expressed by TaylorÕs series, then xðtÞ can be represented as " # k 1 X ðt ti Þ xðtÞ ¼ X ðkÞ: ð3:3Þ k! k¼0 Eq. (3.3) is called the inverse of xðtÞ, with the symbol D denoting the differential transformation process. Upon combining (3.2) and (3.3), we obtain
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Table 1. The fundamental mathematics operations Time function
Transformed function
zðtÞ ¼ xðtÞ yðtÞ zðtÞ ¼ axðtÞ dxðtÞ zðtÞ ¼ dt d2 xðtÞ zðtÞ ¼ dt2 dm xðtÞ zðtÞ ¼ dtm zðtÞ ¼ xðtÞyðtÞ
ZðkÞ ¼ X ðkÞ Y ðkÞ ZðkÞ ¼ aX ðkÞ
ZðkÞ ¼ ðk þ 1Þðk þ 2Þ ðk þ mÞX ðk þ mÞ P ZðkÞ ¼ k‘¼0 Y ð‘ÞX ðk ‘Þ
zðtÞ ¼ 1 zðtÞ ¼ t
ZðkÞ ¼ dðkÞ ZðkÞ ¼ dðk 1Þ
zðtÞ ¼ tm
ZðkÞ ¼ dðk mÞ; dðk mÞ ¼
zðtÞ ¼ expðktÞ zðtÞ ¼ ð1 þ tÞm zðtÞ ¼ sinðxt þ aÞ zðtÞ ¼ cosðxt þ aÞ
ZðkÞ ¼ ðk þ 1ÞX ðk þ 1Þ ZðkÞ ¼ ðk þ 1Þðk þ 2ÞX ðk þ 2Þ
1 0
kk k! mðm 1Þ ðm k þ 1Þ ZðkÞ ¼ k!
xk pk þa sin ZðkÞ ¼ 2! k!
xk pk þa ZðkÞ ¼ cos 2! k!
if k ¼ m if k ¼ 6 m
ZðkÞ ¼
" # k 1 X ðt ti Þ xðtÞ ¼ X ðkÞ D1 X ðkÞ: k! k¼0
ð3:4Þ
Using the differential transformation, a differential equation in the domain of interest can be transformed to an algebraic equation in the K-domain and the xðtÞ can be obtained by finite-term TaylorÕs series plus a remainder, as " # k n X ðt ti Þ xðtÞ ¼ ð3:5Þ X ðkÞ þ Rnþ1 ðtÞ: k! k¼0 The fundamental mathematics operations performed by differential transformation are listed in Table 1 (see Refs. [1,2,8]). 4. The operation properties of differential transformation As in [8], if xðtÞ, yðtÞ are two uncorrelated functions with time t and X ðkÞ, Y ðkÞ are the transformed functions corresponding to xðtÞ, yðtÞ and the basic properties are shown as follows: 1. If X ðkÞ ¼ D½xðtÞ , Y ðkÞ ¼ D½yðtÞ , and c1 and c2 are independent of t and k, then
I.H. Abdel-Halim Hassan / Appl. Math. Comput. 154 (2004) 299–311
D½c1 xðtÞ þ c2 yðtÞ ¼ c1 X ðkÞ þ c2 Y ðkÞ:
303
ð4:1Þ
2. If zðtÞ ¼ xðtÞyðtÞ, xðtÞ ¼ D1 ½X ðkÞ , yðtÞ ¼ D1 ½Y ðkÞ and denote the convolution, then D½zðtÞ ¼ D½xðtÞyðtÞ ¼ X ðkÞ Y ðkÞ ¼
k X
Y ð‘ÞX ðk ‘Þ:
ð4:2Þ
‘¼0
Therefore, the transform of xm ðtÞ, m is a positive integer, can be obtained as D½xm ðtÞ ¼ X m ðkÞ ¼ X m1 ðkÞ X ðkÞ ¼
k X
X m1 ð‘ÞX ðk ‘Þ:
ð4:3Þ
‘¼0
5. Differential transformation method The objective of this section is to find the solution of the HOIVPs (2.3) and (2.4) at the equally spaced grid points ti , i ¼ 0; 1; 2; . . . ; N where ti ¼ a þ ih
for all i ¼ 0; 1; 2; . . . ; N and h ¼
ba : N
ð5:1Þ
The domain ½a; b is divided into N sub-domains and the approximation function in each sub-domain are ui ; i ¼ 0; 1; 2; . . . ; N 1 respectively (see Ref. [9]). Taking the differential transformation of the system (2.5), we get 1 U2 ðkÞ; ðk þ 1Þ 1 U3 ðkÞ; U2 ðk þ 1Þ ¼ ðk þ 1Þ 1 U4 ðkÞ; U3 ðk þ 1Þ ¼ ðk þ 1Þ .. . 1 Um1 ðk þ 1Þ ¼ Um ðkÞ; ðk þ 1Þ Um ðk þ 1Þ ¼ F ðY ðkÞÞ; U1 ðk þ 1Þ ¼
ð5:2Þ
where F ð Þ denotes the transformed function of f ðt; y; y 0 ; . . . ; y ðm1Þ Þ. From the initial conditions (2.6) we can obtained that U1 ð0Þ ¼ yðaÞ ¼ a1 ;
U2 ð0Þ ¼ y 0 ðaÞ ¼ a2 ; . . . Um ð0Þ ¼ y ðm1Þ ¼ am : ð5:3Þ
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In the first sub-domain yðtÞ can be described by ys ðtÞ. From (3.3), (5.2) and (5.3), ys ðtÞ can be represented in terms of its nth order TaylorÕs polynomial about a, that is ys ðtÞ ¼ U1i ð0Þ þ U1i ð1Þðt aÞ þ U1i ð2Þðt aÞ2 þ þ U1i ðnÞðt aÞn ; ð5:4Þ where the subscript ‘‘s’’ denotes that the TaylorÕs polynomial is expand about a. Once TaylorÕs polynomial is obtained, yðt1 Þ can be evaluated as yðt1 Þ ’ y0 ðt1 Þ ¼
n X
U10 ðjÞhj ;
h ¼ ðt1 t0 Þ:
ð5:5Þ
j¼0
The final value y0 ðt1 Þ of the first sub-domain is the initial value of the second sub-domain i.e. y1 ðt1 Þ ¼ y0 ðt1 Þ ¼ U11 ð0Þ (see [9]) yðt2 Þ ’ y1 ðt2 Þ ¼
n X
U11 ðjÞhj ;
h ¼ ðt2 t1 Þ:
ð5:6Þ
j¼0
In the same manner, yðt3 Þ can be represented as: yðt3 Þ ’ y2 ðt3 Þ ¼
n X
U12 ðjÞhj ;
h ¼ ðt3 t2 Þ:
ð5:7Þ
j¼0
Hence, the solution on the grid points tiþ1 can be obtained yðtiþ1 Þ ’ yi ðtiþ1 Þ ¼
n X
U1i ðjÞhj ;
h ¼ ðtiþ1 ti Þ; i ¼ 0; 1; 2; . . . ; N 1:
j¼0
ð5:8Þ
6. Solution of higher-order initial value problems In this section, the differential transformation technique is applied to solve HOIVPs. To illustrate the method proposed in this paper, two HOIVPs are considered as follows: Problem 1. Consider the second-order initial value problem y 00 ðtÞ 2y 0 ðtÞ þ 2yðtÞ ¼ expð2tÞ sinðtÞ;
0 6 t 6 1;
ð6:1Þ
with initial conditions yð0Þ ¼ 0:4; 0
y ð0Þ ¼ 0:6:
ð6:2Þ ð6:3Þ
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305
As above, in Section 2, with u1 ðtÞ ¼ yðtÞ and u2 ðtÞ ¼ y 0 ðtÞ. Eq. (6.1) transformed into the system of the first-order differential equation u01 ðtÞ ¼ u2 ðtÞ;
ð6:4Þ
u02 ðtÞ
ð6:5Þ
¼ expð2tÞ sinðtÞ 2u1 ðtÞ þ 2u2 ðtÞ
and the initial conditions (6.2) and (6.3) becomes u1 ð0Þ ¼ 0:4;
ð6:6Þ
u2 ð0Þ ¼ 0:6:
ð6:7Þ
Let h ¼ 0:1 and N ¼ 10. The differential equation of the system (6.4) and (6.5) between ti and tiþ1 can be represented as u01 ðtH Þ ¼ u2 ðtH Þ;
ð6:8Þ
u02 ðtH Þ ¼ expð2tH þ 2ti Þ sinðtH þ ti Þ 2u1 ðtH Þ þ 2u2 ðtH Þ;
ð6:9Þ
where tH ¼ t ti . Taking the differential transformation of (6.8) and (6.9), respectively, we get U1i ðk þ 1Þ ¼
1 U2i ðkÞ ðk þ 1Þ
ð6:10Þ
and 1 U2i ðk þ 1Þ ¼ ðk þ 1Þ
( expð2ti Þ cosðti Þ
þ expð2ti Þ sinðti Þ
k X ‘¼0
p 2‘ sin ðk ‘Þ 2 ð‘!Þðk ‘Þ!
p 2‘ cos ðk ‘Þ 2 ð‘!Þðk ‘Þ! ‘¼0 )
k X
2U1i ðkÞ þ 2U2i ðkÞ ;
ð6:11Þ
with U10 ð0Þ ¼ 0:4;
U20 ð0Þ ¼ 0:6:
ð6:12Þ
The set of U1i ðk þ 1Þ of (6.10) are presented in Table 2. The approximation of u1 ðtÞ ¼ yðtÞ on the grid points can be obtained from (6.10) and (5.8). The actual solution of (6.1)–(6.3) is yðtÞ ¼ u1 ðtÞ ¼ 0:2 expð2tÞðsinðtÞ 2 cosðtÞÞ:
ð6:13Þ
Fig. 1, shows errors involved with different order of differential transformation method, along with the result obtained by the Runge–Kutta fourth order method (see Table 3).
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Table 2 Different order of the differential transformation method (DTM) of Problem 1 ti
DTM of order 4
DTM of order 5
DTM of order 10
DTM of order 15
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
)0.400000000 )0.461733059 )0.525559267 )0.588600399 )0.646610831 )0.693564763 )0.721149613 )0.718150376 )0.669708650 )0.556440473 )0.353397162
)0.400000000 )0.461732981 )0.525559066 )0.588600073 )0.646610347 )0.693563995 )0.721148557 )0.718148972 )0.669706840 )0.556438203 )0.353394446
)0.400000000 )0.461732979 )0.525559063 )0.588600070 )0.646610344 )0.693563992 )0.721148554 )0.718148969 )0.669706837 )0.556438201 )0.353394444
)0.400000000 )0.461732979 )0.525559063 )0.588600070 )0.646610344 )0.693563992 )0.721148554 )0.718148969 )0.669706837 )0.556438201 )0.353394444
Fig. 1. Computational errors corresponding to different order of differential transformation method and Runge–Kutta of fourth order method to the Problem 1 with actual solution yðtÞ ¼ 0:2 expð2tÞðsin t 2 cos tÞ.
Problem 2. Consider the third-order initial value problem y 000 ðtÞ þ 2y 00 ðtÞ y 0 ðtÞ 2yðtÞ ¼ expðtÞ;
0 6 t 6 3;
ð6:14Þ
with initial conditions yð0Þ ¼ 1;
ð6:15Þ
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307
Table 3 Errors involved with different order of the DTM along with the result obtained by the RKM of Problem 1 Errors
ti 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
jyðti Þ RKMj
jyðti Þ DT4j
jyðti Þ DT5j
jyðti Þ DT10j
jyðti Þ DT15j
0.0000000000 3.7332754e)5 8.3926639e)5 1.3955412e)4 2.0308873e)4 2.7212788e)4 3.4251283e)4 4.0772521e)4 4.5813778e)4 4.8009448e)4 4.5479347e)4
0.000000 5.900e)6 1.670e)5 3.990e)5 5.310e)5 8.630e)5 1.013e)4 1.476e)4 1.850e)4 2.473e)4 3.062e)4
0.00000 1.90e)6 3.40e)6 7.40e)6 4.70e)6 9.50e)6 4.30e)6 7.30e)6 4.00e)6 2.03e)5 3.46e)5
0.00000 2.10e)6 3.70e)6 7.00e)6 4.40e)6 9.20e)6 4.60e)6 6.90e)6 3.70e)6 2.01e)6 3.44e)6
0.00000 2.10e)6 3.70e)6 7.00e)6 4.40e)6 9.20e)6 4.60e)6 6.90e)6 3.70e)6 2.01e)6 3.44e)6
RKM: The Runge–Kutta fourth-order method and yðti Þ the actual solution on the grid points.
y 0 ð0Þ ¼ 2: 00
y ð0Þ ¼ 0:
ð6:16Þ ð6:17Þ
With u1 ðtÞ ¼ yðtÞ, u2 ðtÞ ¼ y 0 ðtÞ and u3 ðtÞ ¼ y 00 ðtÞ. Eq. (6.14) transformed into the system of the first-order differential equation u01 ðtÞ ¼ u2 ðtÞ;
ð6:18Þ
u02 ðtÞ
ð6:19Þ
¼ u3 ðtÞ;
u03 ðtÞ ¼ 2u3 ðtÞ þ u2 ðtÞ þ 2u1 ðtÞ þ expðtÞ
ð6:20Þ
and the initial conditions (6.15)–(6.17) becomes u1 ð0Þ ¼ 1;
ð6:21Þ
u2 ð0Þ ¼ 2;
ð6:22Þ
u3 ð0Þ ¼ 0:
ð6:23Þ
Let h ¼ 0:2 and N ¼ 15. The differential equation of the system (6.18), (6.19) and (6.20) between ti and tiþ1 can be represented as u01 ðtH Þ ¼ u2 ðtH Þ;
ð6:24Þ
u02 ðtH Þ ¼ u3 ðtH Þ;
ð6:25Þ
u03 ðtH Þ ¼ 2u3 ðtH Þ þ u2 ðtH Þ þ 2u1 ðtH Þ þ expðtH þ ti Þ;
ð6:26Þ
where tH ¼ t ti . Taking the differential transformation of (6.24)–(6.26), respectively, we get
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I.H. Abdel-Halim Hassan / Appl. Math. Comput. 154 (2004) 299–311
U1i ðk þ 1Þ ¼
1 U2i ðkÞ; ðk þ 1Þ
ð6:27Þ
U2i ðk þ 1Þ ¼
1 U3i ðkÞ ðk þ 1Þ
ð6:28Þ
and 1 U3i ðk þ 1Þ ¼ ðk þ 1Þ
1 2U3i ðkÞ þ U2i ðkÞ þ 2U1i ðkÞ þ expðti Þ ; k! ð6:29Þ
with U10 ð0Þ ¼ 1; U20 ð0Þ ¼ 2;
ð6:30Þ
U30 ð0Þ ¼ 0: The set of U1i ðk þ 1Þ of (6.27) are presented in Table 4. The approximation of u1 ðtÞ ¼ yðtÞ on the grid points can be obtained from (6.27) and (5.8). The actual solution of (6.1)–(6.3) is yðtÞ ¼ u1 ðtÞ ¼
43 1 4 1 expðtÞ þ expðtÞ expð2tÞ þ ½t expðtÞ : 36 4 9 6
ð6:31Þ
Table 4 Different order of the differential transformation method (DTM) of Problem 2 ti
DTM of order 4
DTM of order 5
DTM of order 10
DTM of order 15
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
1.00000000 1.40633333 1.84917442 2.36189778 2.97753987 3.73160354 4.66457225 5.82438402 7.26907260 9.06975381 11.3141407 14.1107716 17.5941670 21.9311637 27.3287178 34.0435550
1.00000000 1.40637600 1.84923760 2.36197255 2.97762545 3.73170425 4.66469625 5.82454315 7.26928196 9.07003341 11.3145155 14.1112739 17.5948384 21.9320567 27.3298999 34.0451137
1.00000000 1.40637383 1.84923496 2.36197035 2.97762430 3.73170461 4.66469831 5.82454695 7.26928808 9.07004262 11.3145290 14.1112931 17.5948638 21.9320896 27.3299428 34.0451693
1.00000000 1.40637383 1.84923496 2.36197035 2.97762430 3.73170461 4.66469831 5.82454695 7.26928808 9.07004262 11.3145290 14.1112931 17.5948638 21.9320896 27.3299428 34.0451693
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309
Fig. 2. Computational errors corresponding to different order of differential transformation method and Runge–Kutta of fourth order method to the Problem 2 with actual solution yðtÞ ¼ 43=36 expðtÞ þ 1=4 expðtÞ 4=9 expð2tÞ þ 1=6t expðtÞ.
Table 5 Errors involved with different order of the DTM along with the result obtained by the RKM of Problem 2 ti 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
Errors jyðti Þ RKMj
jyðti Þ DT4j
jyðti Þ DT5j
jyðti Þ DT10j
jyðti Þ DT15j
0.0000000000 3.7051385e)5 5.3496545e)5 6.1340427e)5 6.7810840e)5 7.7492288e)5 9.3657449e)5 1.1911737e)4 1.5679347e)4 2.1014117e)4 2.8350990e)4 3.8249793e)4 5.1434509e)4 6.8840014e)4 9.1669703e)4 1.2146776e)3
0.0000000 4.0670e)5 6.0580e)5 7.2220e)5 8.4130e)5 1.0046e)4 1.2575e)4 1.6398e)4 2.1640e)4 2.9119e)4 3.8930e)4 5.2840e)4 7.0300e)4 9.3630e)4 1.2422e)3 1.6350e)3
0.000000 2.000e)6 2.600e)6 2.550e)6 1.450e)6 2.500e)7 1.750e)6 4.850e)6 7.040e)6 1.159e)5 1.450e)5 2.610e)5 3.160e)5 4.330e)5 6.010e)5 7.630e)5
0.000000 1.700e)7 4.000e)8 3.500e)7 3.000e)7 6.100e)7 3.100e)7 1.050e)6 9.200e)7 2.380e)6 1.000e)7 6.900e)6 6.200e)6 1.040e)5 1.720e)5 2.070e)5
0.000000 1.700e)7 4.000e)8 3.500e)7 3.000e)7 6.100e)7 3.100e)7 1.050e)6 9.200e)7 2.380e)6 1.000e)7 6.900e)6 6.200e)6 1.040e)5 1.720e)5 2.070e)5
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Fig. 2, shows errors involved with different order of differential transformation method, along with the result obtained by the Runge–Kutta fourth order method (see Table 5). As indicated in Figs. 1 and 2, the computational error decreases as the order of Taylor series increases. The order of computational error corresponding to the Runge–Kutta method and the differential transformation method of the same order is the same.
7. Conclusion This paper shows that the differential transformation technique can be applied to solve the HOIVPs. The proposed approach can be applied to both second- and third-order initial value problems. Convert the general mth order differential equation with initial conditions into a system of 1st order differential equations. Apply the differential transformation to the system of 1st order differential equations, respectively, and their corresponding initial conditions. Repeat the iterative procedure until the desired nth order Taylor series expansion is obtained. An nth order approximation obtained using the Taylor transformation method of the form yðtiþ1 Þ ¼ yðti Þ þ hUðti ; yðti Þ; hÞ þ Oðhnþ1 Þ where h is the grid size, U the incremental function of yðtÞ and Oðhnþ1 Þ denotes the error with an order of ðn þ 1Þ. The order of computational errors corresponding to different orders of differential transformation method and the Runge–Kutta fourth-order method with the exact solution of the main problem are compared graphically.
References [1] I.H. Abdel-Halim Hassan, On solving some eigenvalue problems by using a differential transformation, Applied Mathematics and Computation 127 (2002) 1–22. [2] I.H. Abdel-Halim Hassan, Different applications for the differential transformation in the differential equations, Applied Mathematics and Computation 129 (2002) 183–201. [3] R.L. Burrden, J.D. Faires, Numerical Analysis, PWS-KENT Publishing Company, Boston, 1993. [4] C.L. Chen, Y.C. Liu, Differential transformation technique for steady nonlinear heat conduction problems, Applied Mathematics and Computation 95 (1998) 155–164. [5] C.L. Chen, Y.C. Liu, Solution of two point boundary value problems using the differential transformation method, Journal of Optimization Theory and Applications 99 (1998) 23–35. [6] C.L. Chen, S.H. Lin, C.K. Chen, Application of Taylor transformation to nonlinear predictive control problem, Applied Mathematical Modeling 20 (1996) 699–710. [7] C.K. Chen, S.H. Ho, Application of differential transformation to eigenvalue problems, Applied Mathematics and Computation 79 (1996) 173–188. [8] M.J. Jang, C.L. Chen, Analysis of the response of a strongly nonlinear damped system using a differential transformation technique, Applied Mathematics and Computation 88 (1997) 137– 151.
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[9] M.J. Jang, C.L. Chen, Y.C. Liy, On solving the initial value problems using the differential transformation method, Applied Mathematics and Computation 115 (2000) 145–160. [10] M. Malik, H.H. Dang, Vibration analysis of continuous system by differential transformation, Applied Mathematics and Computation 96 (1998) 17–26.