International Journal of Conceptions on Computing and Information Technology
Vol.2, Issue 1, Jan’ 2014; ISSN: 2345 - 9808
The solution of radial diffusivity equation using
reduced differential transform method
Benedict Iserom Ita
Physical and Theoretical Chemistry Group,
Dept. of Pure and Applied Chemistry,
University of Calabar, Calabar, Nigeria.
iserom2001@yahoo.com
Abstract— In this paper, a seemingly new mathematical
technique, the reduced differential transform method
(RDTM) has been applied to solve the radial diffusivity
equation of the form:
𝝏𝟐 𝒇 𝟏 𝝏𝒇 𝝏𝒇
+
=
𝝏𝒓𝟐 𝒓 𝝏𝒓 𝝏𝒕
Subject to various initial conditions. The result indicates
this method (RDTM) to be very effective and simple. It is
also consistent with the He’s homotopy perturbation
method.
Keywords- Adomian decomposition method, He’s
perturbation method, Radial diffusivity, Reduced differential
transform method, Variational iteration method.
I.
INTRODUCTION
Many physical problems are clearly understood using
mathematical models involving partial differential equations
(PDEs). Most of these equations are very difficult to solve for
their exact analytical solutions except in few cases. Therefore,
we often times attempt to develop new techniques to enable us
obtain an approximate solution as close as possible to the
exact ones. The contemporary mathematical researches try to
investigate such phenomenon using homotopy analysis
method (HAM) [1], Adomian decomposition method (ADM)
[2], the variational iteration method (VIM) [3], He’s
homotopy perturbation method [4], differential transformation
method [5], just to cite a few methods.
The main aim of this paper is to apply the reduced
differential transform method (RDTM) [6 – 16] to solve the
radial diffusivity equation given as:
𝜕2 𝑓
𝜕𝑟 2
+
1 𝜕𝑓
𝑟 𝜕𝑟
=
𝜕𝑓
𝜕𝑡
(1)
Subject to the initial conditions given as:
a) 𝑓(𝑟, 0) = 𝑟
(2)
𝑏) 𝑓(𝑥, 𝑜) = 𝑠𝑖𝑛𝑟, 𝑐)𝑟(𝑥, 0) = 1 + 𝑟 2
Similar equation to equation (1) has been solved by
Neyrameh et al [4] using He’s homotopy perturbation method.
We also compare our results with that obtained by Neyrameh
et al [4] to show the simplicity and efficacy of the RDTM to
solve the radial diffusivity equation.
The reduced differential transform method (RDTM)
The basic definitions of the RDTM are well elaborated to
include [7 – 10]:
Definition 1: If the function 𝑓(𝑥, 𝑡) is analytic and
continuously differentiable with respect to both space and
time, then we write that:
𝐹𝑘 (𝑥) =
1
[
𝜕𝑘
𝑘! 𝜕𝑡 𝑘
𝑓(𝑥, 𝑡)]
(3)
𝑡=0
In equation (3), the t-dimensional spectrum function 𝐹𝑘 (𝑥)
is the transformed function whereas 𝑓(𝑥, 𝑡) is the original
function.
Definition 2 [7 – 10]: The differential inverse of 𝐹𝑘 (𝑥)is
defined as:
𝑓(𝑥, 𝑡) = ∑∞𝑘=0 𝐹𝑘 (𝑥)𝑡 𝑘
(4)
A combination of equations (3) and (4) then gives:
1
𝑓(𝑥, 𝑡) = ∑∞𝑘=0 [
𝜕𝑘
𝑘! 𝜕𝑡 𝑘
𝑓(𝑥, 𝑡)]
𝑡=0
(5)
As stated by Taha [16], the concept of the reduced
differential transform method is derived from the power series
expansion. Moreover, the fundamental theorems of reduced
differential transform method (RDTM) stated without proofs
are given below [16]:
Theorem 1: If 𝑚(𝑥, 𝑡) = 𝑓(𝑥, 𝑡) ± 𝑔(𝑥, 𝑡) is the original
function, then the transformed function will be given as
𝑀𝑘 (𝑥) = 𝐹𝑘 (𝑥) ± 𝐺𝑘 (𝑥).
Theorem 2: If the original function is given as 𝑚(𝑥, 𝑡) =
𝛽𝑓(𝑥, 𝑡) then the transformed function will be gives as
𝑀𝑘 (𝑥) = 𝛽𝐹𝑘 (𝑥); 𝛽 is a constant.
Theorem 3: If the original function is given as 𝑚(𝑥, 𝑡) =
𝑥 𝑎 𝑡 𝑏 then the transformed function will be given as 𝑀𝑘 (𝑥) =
1, 𝑘 = 0
𝑥 𝑎 𝛿(𝑘 − 𝑏), 𝛿(𝑘) = {
.
0, 𝑘 ≠ 0
104 | P a g e
International Journal of Conceptions on Computing and Information Technology
Vol.2, Issue 1, Jan’ 2014; ISSN: 2345 - 9808
Theorem 4: If the original function is given as 𝑚(𝑥, 𝑡) =
𝑥 𝑎 𝑡 𝑏 𝑓(𝑥, 𝑡) then the transformed function will be given as
𝑀𝑘 (𝑥) = 𝑥 𝑎 𝐹𝑘−𝑏 (𝑥).
Equation (12) in closed form becomes [4]:
Theorem 5: If the original function is given as 𝑚(𝑥, 𝑡) =
𝑓(𝑥, 𝑡)𝑔(𝑥, 𝑡) then the transformed function will be given
as𝑀𝑘 (𝑥) = ∑𝑘𝑟=0 𝐺𝑟 𝐹𝑘−𝑟 (𝑥) = ∑𝑘𝑟=0 𝐹𝑟 𝐺𝑘−𝑟 (𝑥).
Equation (13) gives the exact solution to the problem.
𝑓(𝑥, 𝑡) = 𝑟 + ∑∞
𝑥=1
1
1
𝑠𝑖𝑛𝑟 + 𝑐𝑜𝑠𝑟⁄𝑟 3 − 𝑐𝑜𝑠𝑟⁄𝑟
2
2
1
𝑐𝑜𝑠𝑟
+ (−𝑐𝑜𝑠𝑟 −
⁄𝑟 2 − 𝑠𝑖𝑛𝑟⁄𝑟 ) 𝑟
2
=
Theorem 7: If the original function is given as 𝑚(𝑥, 𝑡) =
𝑓(𝑥, 𝑡) then the transformed function will be given as
𝑀𝑘 (𝑥) =
And for condition (c) we get
𝜕
𝐹 (𝑥).
𝜕𝑥 𝑘
c)r(x, 0) = 1 + r 2 , f1 = 4, f2 = 0, f3 = 0, f4 = 0, fn = 0, n
>1
Theorem 8: If the original function is given as 𝑚(𝑥, 𝑡) =
𝜕2
𝜕𝑥 2
𝑓(𝑥, 𝑡) then the transformed function will be given
as𝑀𝑘 (𝑥) =
𝜕2
𝜕𝑥
2 𝐹𝑘 (𝑥).
Solution of the radial diffusivity equation using RDTM
The radial diffusivity equation solved is of the form [4]:
𝜕2 𝑓
𝜕𝑟 2
+
1 𝜕𝑓
𝑟 𝜕𝑟
=
𝜕𝑓
(6)
𝜕𝑡
II.
REFERENCES
a)𝑓(𝑟, 0) = 𝑟
(7)
𝐹 (𝑟)
𝜕𝑟 2 𝑘
1 𝜕
+
𝐹 (𝑟)
𝑟 𝜕𝑟 𝑘
(8)
Where 𝐹𝑘 (𝑟) is the transformed function. The initial
condition (7) reveals that
𝐹0 (𝑟) = 𝑟
(9)
Making use of equation (9), equation (8) yields successive
values of 𝐹𝑘 (𝑟) as:
𝐹1 (𝑟) =
1
(10a)
𝑟
1
𝐹2 (𝑟) =
2𝑟 3
𝐹3 (𝑟) =
2𝑟 5
𝐹4 (𝑟) =
8𝑟 7
𝐹5 (𝑟) =
(10b)
3
(10c)
75
(10d)
735
(10e)
8𝑟 9
The results of equations 10 (a – e) reveal that
𝑘
𝑓(𝑥, 𝑡) = ∑∞
𝑘=0 𝐹𝑘 (𝑥)𝑡
(11)
yields:
𝑡
𝑡2
𝑟
2𝑟 3
𝑓(𝑥, 𝑡) = 𝑟 + +
+
3𝑡 3
2𝑟 5
+
75𝑡 4
8𝑟 7
+
735𝑡 5
8𝑟 9
+⋯
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The recursion formula for the reduced differential
transform of equation (6) is:
𝜕2
CONCLUSION
In this paper, we have successfully applied a seemingly
novel technique, the reduced differential transform method to
solve the radial diffusivity equation. The result obtained is
exactly the same as that obtained by Neyrameh [4] using He’s
homotopy perturbation method. We also state therefore that the
reduced differential transform method is effective, reliable and
easy to use.
Subject to the initial condition given as:
(𝑘 + 1)𝐹𝑘+1 (𝑟) =
(13)
𝑏) 𝑓(𝑥, 𝑜) = 𝑠𝑖𝑛𝑟, 𝑓1
Theorem 6: If the original function is given as 𝑚(𝑥, 𝑡) =
𝑓(𝑥, 𝑡) then the transformed function will be given
𝜕𝑡 𝑟
as𝑀𝑘 (𝑥) = (𝑘 + 1) … (𝑘 + 𝑟)𝐹𝑘+𝑟 (𝑥).
𝜕𝑥
𝑥!×𝑟 2𝑥−1
Similarly for condition (b) we get
𝜕𝑟
𝜕
12 ×32 ×⋯×(2𝑥−3)2 𝑡 𝑥
(12)
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International Journal of Conceptions on Computing and Information Technology
Vol.2, Issue 1, Jan’ 2014; ISSN: 2345 - 9808
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