Differential Transform Method for Solving Engineering and Mathematical Problems Mohammad Mehdi Rashidi, Navid Freidoonimehr, Amir. Basiri Parsa
[email protected] Shanghai Key Lab of Vehicle Aerodynamics and Vehicle Thermal Management Systems, Tongji University, Address: 4800 Cao An Rd., Jiading, Shanghai 201804, China, Email:
[email protected] ENN-Tongji Clean Energy Institute of advanced studies, Shanghai, China
Abstract The Mathematica package of DTM is a free/open source database based on Differential Transform Method (DTM) for solution of initial value problems (IVPs) and boundary value problems (BVPs). This tutorial shows how to use its newest version of DTM to solve and plot the solutions of initial value and boundary value for ordinary differential equations, with using similarity transformation we can solve some partial differential equations also. Keywords: Differential Transform Method; IVPs; BVPs; Partial differential equations
1. Introduction Partial and nonlinear ordinary differential equations occur in the most scientific problems and phenomena. Nonlinear phenomena have important effects on applied mathematics, physics, and issues related to engineering. In most cases, these problems do not have precise analytic solution. Boundarylayer problem is one of the problems that are described through nonlinear differential equations. 1
The basic technique that we used is the DTM, which is based on Taylor series expansion. In 1986, Zhou [1] employed the basic ideas of DTM for solving linear and nonlinear problems in electrical circuit problems. It gives exact values of the n th derivative of an analytical function at a point in terms of known and unknown boundary conditions in a fast manner. The differential transform is an iterative procedure for obtaining analytic Taylor series solutions of differential equations. Chen and Ho [2] developed this method for partial differential equations and Ayaz [3] applied it to the system of differential equations. Jang et al. [4] presented the two-dimensional DTM to the solution of partial differential equations and Hassan [5] adopted the DTM to solve some eigenvalue-problems. This method was successfully applied to various application problems [6-10]. On the other hand, if the DTM is used for solving differential equations with the boundary conditions at infinity, the obtained results were incorrect. The MHD boundary-layer flow is investigated by employing the modified Adomian decomposition method and the Padé approximation by Rashidi [11].
2. The Differential Transform Method Transformation of the kth derivative of function f (t ) in one variable is defined as follows [1] 1 d k f (t ) F (k ) , k ! dt k t t
(1)
0
in Eq. (14), f (t ) is the original function and F (k ) is the transformed function. Differential inverse transform of F (k ) is defined as follows
f (t ) F (k ) (t t 0 ) k ,
(2)
k 0
i
f (t ) F (k ) (t t 0 ) k ,
(3)
k 0
in fact, from (1) and (3), we obtain
2
d k f (t ) (t t 0 )k f (t ) . k k! k 0 dt t t 0 i
(4)
Eq. (4) implies that the concept of differential transformation is derived from the Taylor series expansion. From the definitions (1) and (3), it is easy to obtain the mathematical operations according to Table 1. Table 1: Various differential transform operators.
1. If f (x ) g (x ) h (x ),
then
F (K ) G ( k ) H ( k ).
2. If f (x ) c g (x ),
then
F (K ) c G (k ), c is a constant.
then
F (K )
then
F (K ) G (l )H (k l ).
3. If f (x )
d n g (x ) , dx n
(k n )! G ( k n ). k! k
4. If f (x ) g (x )h (x ),
l 0
5. If f (x ) x n ,
then
x
1, k n , F (K ) ( k n ), ( k n ) 0, k n . G (k 1) , where k 1. k
6. If f (x ) g (t )dt ,
then F (K )
7. If f (x ) g (x )h (x )i (x ),
then F ( K ) G (s ) H ( m ) I ( k s m ).
0
k
k s
s 0 m 0
3. Illustrative examples 3.1. Example 1 : Linear IVP Consider a simple ODE in form of initial value problem (IVP) 1 f (x ) f (x ) 0,
(5)
Subject to
f (0) .
(6)
The exact solution can be obtained easily as follow
f (x ) 1 (1 ) e x
(7)
Taking the differential transform of Eq. (5) by Table 1, we obtain
3
(k ) (k 1)F [k 1] F [k ] 0,
(8)
By applying the DTM into Eq. (6), differential transform of initial value are thus determined into a recurrence equation that finally leads to the solution of a system of algebraic equations. The differential transform of the initial value is as follow F (0) .
(9)
Moreover, substituting Eq. (8) into Eq. (7) and by recursive method we can calculating another values of F [k ] . Finally, by using Eq. (3) the final solution can be obtained. For example for 1 and 50-order approximation the DTM solution of Eq. (5) can be coded by Mathematica as follows
we can use:
to plot the comparison of 50th-order approximate solution of DTM results and exact solution, which is shown in Fig. 1.
4
f x 800
600
400
200
1
2
3
4
5
6
x
Fig. 1. Comparison of 50th-order approximate solution of DTM results and exact solution.
We can conclude that, the results obtained by the DTM have good agreement with the exact solution. Also we can obtain the exact solution with using DTM directly, The corresponding equation by taking the differential transform is
k k 1 F k 1 F k 0. Namely,
F k 1
k F k , k 1
then,
F 1 1 F 0 , and
F k
F k 1 k
F k 2 k k 1
Therefore,
5
F 1 k!
, when k 1.
f x F k x k k 0
F 0 xF 1
x 2 F 1 2!
x 3 F 1
x 4 F 1
3! 4 2 3 4 x x x 1 1 F 0 1 x 2! 3! 4 1 F 0 e x 1= 1 (1 ) e x .
3.2. Example 2: Non-Linear IVP Consider a simple non-linear ODE in form of initial value problem (IVP)
2 f (x ) f 2 (x ) 0,
(10)
Subject to f (0) 0, f (0) 1.
(11)
Taking the differential transform of Eq. (5) by Table 1, we obtain k
2 (k ) (k 2)(k 1) F [k 2] F [i]F [k i] 0,
(12)
i 0
By applying the DTM into Eq. (11), differential transform of initial values are thus determined into a recurrence equation that finally leads to the solution of a system of algebraic equations. The differential transform of the initial values are as follows F (0) 0, F [1] 1.
(13)
Moreover, substituting Eqs. (13) into Eq. (12) and by recursive method we can calculating another values of F [k ] . Finally, by using Eq. (3) the final solution can be obtained. For 50-order approximation the DTM solution of Eq. (10) can be coded by Mathematica as follows
6
we can use:
to plot the comparison of 50th-order approximate solution of DTM results and numerical solution (by using a fourth-order Runge–Kutta and shooting method), which is shown in Fig. 2. f x 80
60
40
20
0.5
1.0
1.5
2.0
2.5
x
Fig. 2. Comparison of 50th-order approximate solution of DTM results and numerical solution.
7
We can conclude that, the results obtained by the DTM have good agreement with the numerical solution. 3.3. Example 3: Finite non-linear BVP Consider a non-linear ODE in form of boundary value problem (BVP) with finite boundary conditions f (x ) f (x )f (x ) 2 f (x ) x 0,
(14)
Subject to f (0) 1, f (1) 2.
(15)
Taking the differential transform of Eq. (14) by Table 1, we obtain k
(k 2)(k 1) F[k 2] F[i](k i 1) F[k i 1] 2 F[k ] (k 1) 0,
(16)
i 0
By applying the DTM into Eq. (15), differential transform of initial values are thus determined into a recurrence equation that finally leads to the solution of a system of algebraic equations. As for a problem with the boundary conditions, differential transform of the upper boundary condition is indeterminate thus we must consider the boundary conditions (Eqs. (15)) as follows F (0) 1, F[1] .
(17)
Moreover, substituting Eqs. (17) into Eq. (16) and by recursive method we can calculating another values of F [k ]. Hence, substituting all F [k ] into Eq. (3), the series solutions are obtained. After finding the series solutions, with using actual upper boundary condition ( f (1) 2 ) we can obtain . For 50-order approximation the DTM solution of Eq. (14) can be coded by Mathematica as follows
8
For this example after running of above code, 3.993361150 is calculated. We can use:
to plot the comparison of 50th-order approximate solution of DTM results and numerical solution (by using a fourth-order Runge–Kutta and shooting method), which is shown in Fig. 3.
9
f x 2.2
2.0
1.8
1.6
1.4
1.2
0.2
0.4
0.6
0.8
1.0
x
Fig. 3. Comparison of 50th-order approximate solution of DTM results and numerical solution.
We can conclude that, the results obtained by the DTM have good agreement with the numerical solution. 3.4. Example 4: Infinite non-linear BVP Consider a non-linear ODE in form of boundary value problem (BVP) with an infinity boundary condition
f (x ) f (x )f (x ) (1 f 2 (x )) 0,
(18)
with constant. The boundary conditions are
10
f (0) 0, f (0) 0, f () 1.
(19)
Taking the differential transform of Eq. (14) by Table 1, we obtain k
(k 3)(k 2)(k 1) F [k 3] F [i](k i 2)(k i 1) F [k i 2] i 0
(k )
k
(i 1) F[i 1](k i 1) F[k i 1] 0,
(20)
i 0
By applying the DTM into Eq. (19), differential transform of initial values are thus determined into a recurrence equation that finally leads to the solution of a system of algebraic equations. As for a problem with the boundary conditions at the infinity, differential transform of infinity boundary conditions are indeterminate thus we must consider the boundary conditions (Eqs. (19)) as follows f (0) 0, f (0) 0, f (0) ,
(21)
Therefore problem change to an initial conditions problem. The differential transform of the boundary conditions are as follows F (0) 0, F [1] 0, F [2] / 2.
(22)
Moreover, substituting Eqs. (22) into Eq. (20) and by recursive method we can calculating another values of F [k ]. Hence, substituting all F [k ] into Eq. (3), the series solutions are obtained. On the other hand, if the DTM is used for solving differential equations with the boundary conditions at infinity, the obtained results were incorrect (when the boundary-layer variable go to infinity, the obtained series solutions are divergent). In addition that, power series aren’t useful for large values of , say (when is independent variable of problem). Boyd [12] and others have formally shown that power series in isolation are not useful for handling the boundary value problems. This can be attributed to the possibility that the radius of convergence may not be sufficiently large to contain the boundaries of the domain. Therefore, the combination of the series solution through the DTM or 11
any other series solution method with the Padé approximation provides an effective tool for handling the boundary value problems on infinite or semi-infinite domains. After finding the series solutions, the Padé approximation [13-15] must be applied, with using asymptotic boundary condition ( f () 1 ) we can obtain . When 1, for 50-order approximation the DTM solution of Eq. (18) with Padé approximation can be coded by Mathematica as follows
For this example, when 1, after running of above code, 1.232590898 is calculated. We can use:
12
to plot the comparison of 50th-order approximate solution of DTM results and numerical solution (by using a fourth-order Runge–Kutta and shooting method), which is shown in Fig. 4. f x 1.0
0.8
0.6
0.4
0.2
1
2
3
4
x
Fig. 4. Comparison of 50th-order approximate solution of DTM results and numerical solution.
We can conclude that, the results obtained by the DTM have good agreement with the numerical solution.
13
3.5. Example 5: Three-Dimensional Squeezing Nanofluid Flow Consider an unsteady 3D rotating Nano-fluid flow of an incompressible electrically conducting viscous fluid between two infinite horizontal plane walls. The nonlinear ordinary differential equations are obtained as follows [16]
f iv 1 s f
1
2.5
2.5 2 f f ff 2Ωg 3 f f 1 M f 0, 2
2.5 2.5 g 1 s f 1 fg f g g g 2Ωf 1 M 2 g 0. 2
(23)
(24)
where is the nanoparticle volume fraction, f and s are the densities of the fluid and of the solid fractions, respectively, a is the characteristic parameter of the flow, a is the rotation parameter, M 2 B02 f a is the magnetic parameter. The boundary conditions are
f (0) w 0 , f (1)
2
,
f (0) 1,
g (0) 0,
(25)
f (1) 0,
g (1) 0,
(26)
Taking the differential transform of Eqs. (23) and (24) by Table 1, we obtain
k 1 k 2 k 3 k 4 f k 4 1 M 2 k 1 k 2 f (k 2) k k r 1 k r 2 r 1 f (r 1) f (k r 2) r 0 k r 1 k r 2 k r 3 f (r ) f (k r 3) 2.5 3 k 1 k 2 f (k 2) 1 s 1 0, f k 2 k r 1 k r 2 2 k 1 g (k 1) r 0 k r 3 ( r ) f ( k r 3) 2.5
k 1 k 2 g (k 2) 1
2.5
(27)
M 2 g (k )
k k r 1 f (r ) g (k r 1) 2 k 1 f (k 1) 2.5 r 0 k r 1 g ( r ) f ( k r 1) 1 s 1 0, k f g (k ) 1 2 k r 1 (r ) g (k r 1) r 0
14
(28)
By applying the DTM into Eqs. (27) and (28), differential transform of initial values are thus determined into a recurrence equation that finally leads to the solution of a system of algebraic equations. As for a problem with the boundary conditions at the infinity, differential transform of infinity boundary conditions are indeterminate thus we must consider the boundary conditions (Eqs. (25) and (26)) as follows
f (0) w 0 , f (0) 1, f (0) q , f (0) r , g (0) 0, g (0) s ,
(29)
Therefore problem change to an initial conditions problem. The differential transform of the boundary conditions are as follows
F (0) w 0 , F [1] 1, F [2] q / 2, F [3] r / 6, G (0) 0,
(30)
G [1] s ,
Moreover, substituting Eq. (30) into Eqs. (27) and (28) and by recursive method we can calculating another values of F [k ] and G [k ] . Hence, substituting all F [k ] and G [k ] into Eq. (3), the series solutions are obtained.
15
In order to plot the comparison of different order approximate solution of DTM results and numerical solution (by using a fourth-order Runge–Kutta and shooting method), which is shown in Fig. 5. 1 DTM, n = 5 DTM, n = 10 DTM, n = 20 DTM, n = 30 DTM, n = 40 Numerical
0.5
0
g
-0.5
-1
-1.5
-2
-2.5 0
0.2
0.4
0.6
0.8
1
Fig. 5. Comparison of different order approximate solutions of DTM results and numerical solution.
3.6. Example 6: Nonlinear Vibration of Euler-Bernoulli Beams In order to obtain the governing equation, we consider three assumptions: the plane deformation is negligible so it is neglected, transverse shear strains are small so they are neglected, and rotation of the cross section is only due to bending. By applying the Galerkin method, the equation of motion is obtained as follows: f t f t f 3 t 0,
(31)
The above equation is the governing equation of nonlinear vibration of Euler-Bernoulli beams. The center of the beam is subjected to the following initial conditions: f 0 A,
f 0 0,
(32)
16
where A denotes the non-dimensional maximum amplitude of oscillation. Taking the differential transform of Eq. (31) by Table 1, we obtain
r k 1 k 2 F k 2 F k F s F r s F k r 0. r 0 s 0 k
(33)
By applying the DTM into Eq. (33), differential transform of initial value are thus determined into a recurrence equation that finally leads to the solution of a system of algebraic equations. The differential transforms of the initial conditions are as follow F 0 A,
F 1 0.
(34)
Moreover, using Eq. (33) and by recursive method we can calculating another values of T [k ] . Finally, by using Eq. (3) the final solution can be obtained.
In order to plot the comparison of different order approximate solution of DTM results and numerical solution (by using a fourth-order Runge–Kutta and shooting method), which is shown in Fig. 6.
17
T (t)
0.0001
0
numerical n=5 n=10 n=20 -0.0001 0
4
8
t Fig. 6. Comparison of different order approximate solutions of DTM results and numerical solution.
18
References [1]
J.K. Zhou, Differential Transformation and Its Applications for Electrical Circuits, Huazhong University Press, Wuhan, China, 1986 (in Chinese).
[2]
C.K. Chen, S.H. Ho, Solving partial differential equations by two dimensional differential transform method, Applied Mathematics and Computation 106 (1999) 171–179.
[3]
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[4]
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[5]
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[6]
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[7]
Z. Odibat, Sh. Momani, A generalized differential transform method for linear partial differential equations of fractional order, Applied Mathematics Letters 21 (2008) 194–199.
[8]
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[9]
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[10]
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[11]
M.M Rashidi, The modified differential transform method for solving MHD boundary-layer equations, Computer Physics Communications 180 (11), 2210-2217.
[12]
J. Boyd, Padé approximant algorithm for solving nonlinear ordinary differential equation boundary value problems on an unbounded domain, Computers in Physics 11 (1997) 299–303.
[13]
M.M Rashidi, T Hayat, A Basiri Parsa, Solving of boundary-layer equations with transpiration effects, governance on a vertical permeable cylinder using modified differential transform method, Heat Transfer-Asian Research 40 (8), 677-692.
[14]
T. Hayat, Q. Hussain, T. Javed, The modified decomposition method and Pad´e approximants for the MHD flow over a nonlinear stretching sheet, Nonlinear Analysis: Real World Applications 10 (2009) 966–973.
[15]
M.M Rashidi, M Keimanesh, Using differential transform method and padé approximant for solving mhd flow in a laminar liquid film from a horizontal stretching surface, Mathematical Problems in Engineering 2010.
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[16]
N. Freidoonimehr, B. Rostami, M.M. Rashidi, E. Momoniat, Analytical Modelling of Three-Dimensional Squeezing Nanofluid Flow in a Rotating Channel on a Lower Stretching Porous Wall, Mathematical Problems in Engineering 2014.
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